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Dynamic Programming and the Bellman EquationLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGSy8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GRzYmRklGTEZPL0ZSUSVhdXRvRidGK0Yv The following was implemented in Maple by Marcus Davidsson (2009) davidsson_marcus@hotmail.com and is partly based upon the work by Zietz (2004) Dynamic Programming: An Introduction by Example JSFH JSFH JSFH JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnL0YzUSdub3JtYWxGJw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== We assume that utility from consumption at time t is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JInVHNiI2IyZJImNHRiU2I0kidEdGJS1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlRiY=

We now note that we discount utility from consumption over time. The discounting is done through a discount factor B^t where B=1/(1+r) and r is the interest rate. Note that since B^t < 1 it means that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUklbXN1YkdGJDYlLUYsNiVRImNGJ0YvRjItRiM2Ji1GLDYlUSJ0RidGL0YyLyUrZXhlY3V0YWJsZUdGMS8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RicvRjNRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGRUZHRkpGSi1GLDYlUSJERicvRjBRJmZhbHNlRidGSkZFRkdGSg== will be decreasing over time. 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyomLUkidUc2IjYjJkkiY0dGJjYjSSJ0R0YmIiIiSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmRiwqJilJIkJHRiZGK0YsLUkjbG5HRi5GJ0Ys

Our objective is to maximize the sum of the discounted utility form consumption LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUklbXN1YkdGJDYlLUYsNiVRImNGJ0YvRjItRiM2JC1GLDYlUSJ0RidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRkVGRS1GLDYlUSJERicvRjBRJmZhbHNlRidGRUZF over time as seen below 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyomSSRtYXhHJSpwcm90ZWN0ZWRHIiIiLUkiVUc2IjYjJkkiY0dGKTYjSSJ0R0YpRiYtSSRTdW1HNiRGJUkoX3N5c2xpYkdGKTYjKiYtSSJ1R0YpRipGJkkiREdGMUYm

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== If we plug in the previous expression in this equation we get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyomSSRtYXhHJSpwcm90ZWN0ZWRHIiIiLUkiVUc2IjYjJkkiY0dGKTYjSSJ0R0YpRiYtSSRTdW1HNiRGJUkoX3N5c2xpYkdGKTYjKiYpSSJCR0YpRi5GJi1JI2xuR0YxRipGJg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== We now assume that the equation of motion of our assets is given by (Note that we now use the notation a (asset) for capital) 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImFHNiI2IywmSSJ0R0YlIiIiRilGKSwmKiYsJkYpRilJInJHRiVGKUYpJkYkNiNGKEYpRikmSSJjR0YlRi8hIiI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8= where a_(t+1) is the amount of assets in the next period, r is the interest rate, a_(t) is the amount of asset today and c_(t) is todays consumption. The equation says that the amount of assets in the next period a_(t+1) is given by the interest we earn form our assets that we have today a_(t) minus todays consumption We now assume that we have a three time period model. We note that we solve the problem by using backward induction which means that we start from the last period, period three, and work our way forward to the beginning to period 0 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== The equation of motion for assets in period 3 is given by: 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImFHNiI2IyIiJCwmKiYsJiIiIkYrSSJyR0YlRitGKyZGJDYjIiIjRitGKyZJImNHRiVGLiEiIg==

We assume that that all assets in the last period (period 3) must be used up which means that a(3)=0. If we plug that into our previous equation we get 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyIiISwmKiYsJiIiIkYnSSJyRzYiRidGJyZJImFHRik2IyIiI0YnRicmSSJjR0YpRiwhIiI=

We now solve for c(2) which gives us 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIyomLCYiIiJGKkkickdGJUYqRiomSSJhR0YlRiZGKg==

This expression says that the amount of consumption in period two c_2 must be equal to the amount of interest we earn form our assets in period two a_2. We now note that the equation of motion for assets in period 2 is given by: 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImFHNiI2IyIiIywmKiYsJiIiIkYrSSJyR0YlRitGKyZGJDYjRitGK0YrJkkiY0dGJUYuISIi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== We now plug in this equation in the previous equation which gives us 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIyomLCYiIiJGKkkickdGJUYqRiosJiomRilGKiZJImFHRiU2I0YqRipGKiZGJEYwISIiRio=

This is the expression for the optimal amount of consumption in period two We now note that our general Bellman equation or value function is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJIlZHNiI2I0kidEdGJSomJkkkbWF4RyUqcHJvdGVjdGVkRzYjJkkiY0dGJUYmIiIiNyMsJi1JI2xuRzYkRitJKF9zeXNsaWJHRiVGLEYvKiZJIkJHRiVGLyZGJDYjLCZGJ0YvRi9GL0YvRi9GLw==

This means that the value function for period two is given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJIlZHNiI2IyIiIyomJkkkbWF4RyUqcHJvdGVjdGVkRzYjJkkiY0dGJUYmIiIiNyMsJi1JI2xuRzYkRitJKF9zeXNsaWJHRiVGLEYvKiZJIkJHRiVGLyZGJDYjIiIkRi9GL0Yv

Since previously we know that all assets are gone in period three a(3)=0 which means that the value function for period three must also be equal to zero V_3=0. This means that our previous equation can be written as 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJIlZHNiI2IyIiIyomJkkkbWF4RyUqcHJvdGVjdGVkRzYjJkkiY0dGJUYmIiIiNyMtSSNsbkc2JEYrSShfc3lzbGliR0YlRixGLw==

Note that in this initial example when we are only one period from the last period (period 3) we already have the expression for the optimal value of consumption in period two which means that the max expression disappears. This gives us: 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJIlZHNiI2IyIiIy1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMmSSJjR0YlRiY=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== If we plug in the expression for the optimal amount of consumption in period two given by 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 we get: 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJIlZHNiI2IyIiIy1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMqJiwmIiIiRjBJInJHRiVGMEYwJkkiYUdGJUYmRjA=

If we plug in the expression for the equation of motion for assets in period 2 is given by 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we get: 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJIlZHNiI2IyIiIy1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMqJiwmIiIiRjBJInJHRiVGMEYwLCYqJkYvRjAmSSJhR0YlNiNGMEYwRjAmSSJjR0YlRjYhIiJGMA==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== We now note that the value function for period one is given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJIlZHNiI2IyIiIiomJkkkbWF4RyUqcHJvdGVjdGVkRzYjJkkiY0dGJUYmRic3IywmLUkjbG5HNiRGK0koX3N5c2xpYkdGJUYsRicqJkkiQkdGJUYnJkYkNiMiIiNGJ0YnRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== We now plug in the value function equation for period two in the above expression. This gives us: 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJIlZHNiI2IyIiIiomJkkkbWF4RyUqcHJvdGVjdGVkRzYjJkkiY0dGJUYmRic3IywmLUkjbG5HNiRGK0koX3N5c2xpYkdGJUYsRicqJkkiQkdGJUYnLUYyNiMqJiwmRidGJ0kickdGJUYnRicsJiomRjpGJyZJImFHRiVGJkYnRidGLSEiIkYnRidGJ0Yn

We now maximize the above equation by taking the partial derivative with respect to c_1 and set it equal to zero 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LywmKiQmSSJjRzYiNiMiIiIhIiJGKSoqSSJCR0YnRiksJkYpRipJInJHRidGKkYpLCZGKUYpRi5GKUYqLCYqJkYvRikmSSJhR0YnRihGKUYpRiVGKkYqRikiIiE=

We now solve for c_1 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIiooJkkiYUdGJUYmRicsJkYnRidJInJHRiVGJ0YnLCZGJ0YnSSJCR0YlRichIiI=

This is the expression for the optimal amount of consumption in period one The value function for period one is now given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJIlZHNiI2IyIiIiwmLUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyZJImNHRiVGJkYnKiZJIkJHRiVGJy1GKjYjKiYsJkYnRidJInJHRiVGJ0YnLCYqJkY2RicmSSJhR0YlRiZGJ0YnRi8hIiJGJ0YnRic=

If we substitute in the expression for the optimal amount of consumption in period one we get: 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJIlZHNiI2IyIiIiwmLUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyooJkkiYUdGJUYmRicsJkYnRidJInJHRiVGJ0YnLCZGJ0YnSSJCR0YlRichIiJGJyomRjVGJy1GKjYjKiZGMkYnLCYqJkYyRidGMEYnRidGL0Y2RidGJ0Yn

which can be written as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEpc2ltcGxpZnlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSIlRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSFGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVkY9RkA=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJIlZHNiI2IyIiIiwmLUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyooJkkiYUdGJUYmRicsJkYnRidJInJHRiVGJ0YnLCZGJ0YnSSJCR0YlRichIiJGJyomRjVGJy1GKjYjKiopRjIiIiNGJ0YwRidGNUYnRjRGNkYnRic=

We now note that the equation of motion for assets in period 1 is given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImFHNiI2IyIiIiwmKiYsJkYnRidJInJHRiVGJ0YnJkYkNiMiIiFGJ0YnJkkiY0dGJUYtISIi

if we substitute in that into our previous equation we get 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJIlZHNiI2IyIiIiwmLUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyooLCYqJiwmRidGJ0kickdGJUYnRicmSSJhR0YlNiMiIiFGJ0YnJkkiY0dGJUY2ISIiRidGMkYnLCZGJ0YnSSJCR0YlRidGOkYnKiZGPEYnLUYqNiMqKilGMiIiI0YnRjBGJ0Y8RidGO0Y6RidGJw==

We now note that the value function for period zero is given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJIlZHNiI2IyIiISomJkkkbWF4RyUqcHJvdGVjdGVkRzYjJkkiY0dGJUYmIiIiNyMsJi1JI2xuRzYkRitJKF9zeXNsaWJHRiVGLEYvKiZJIkJHRiVGLyZGJDYjRi9GL0YvRi8=

We now plug in the value function equation for period one in the above expression. This gives us: 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJIlZHNiI2IyIiISomJkkkbWF4RyUqcHJvdGVjdGVkRzYjJkkiY0dGJUYmIiIiNyMsJi1JI2xuRzYkRitJKF9zeXNsaWJHRiVGLEYvKiZJIkJHRiVGLywmLUYzNiMqKCwmKiYsJkYvRi9JInJHRiVGL0YvJkkiYUdGJUYmRi9GL0YtISIiRi9GPkYvLCZGL0YvRjdGL0ZCRi8qJkY3Ri8tRjM2IyoqKUY+IiIjRi9GPEYvRjdGL0ZDRkJGL0YvRi9GL0Yv

We now maximize the above equation by taking the partial derivative with respect to c_0 and set it equal to zero 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LywmKiQmSSJjRzYiNiMiIiEhIiIiIiIqJkkiQkdGJ0YrLCYqJCwmKiYsJkYrRitJInJHRidGK0YrJkkiYUdGJ0YoRitGK0YlRipGKkYqKiZGLUYrRjBGKkYqRitGK0Yp

We now solve for c_0 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiISooJkkiYUdGJUYmIiIiLCZGK0YrSSJyR0YlRitGKywoRitGK0kiQkdGJUYrKiQpRi8iIiNGK0YrISIi

This is the expression for the optimal amount of consumption in period zero We can now summaries what we have so far LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8= Consumption 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiISooJkkiYUdGJUYmIiIiLCZGK0YrSSJyR0YlRitGKywoRitGK0kiQkdGJUYrKiQpRi8iIiNGK0YrISIi

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIiooJkkiYUdGJUYmRicsJkYnRidJInJHRiVGJ0YnLCZGJ0YnSSJCR0YlRichIiI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYoLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUpcmVhZG9ubHlHRjQvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR0ZFLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlxvLUkobWZlbmNlZEdGJDYkLUYjNiotRjs2JFEiMUYnRj4tRk82LVEiK0YnRj5GUkZURlZGWEZaRmZuRmhuL0Zbb1EsMC4yMjIyMjIyZW1GJy9GXm9GW3AtRi82JVEickYnRjJGNUZARkNGRkZIRj5GPi1GTzYtUTEmSW52aXNpYmxlVGltZXM7RidGPkZSRlRGVkZYRlpGZm5GaG4vRltvUSYwLjBlbUYnL0Zeb0ZkcC1GLDYlLUYvNiVRImFGJ0YyRjVGOEZLLUZPNi1RIUYnRj5GUkZURlZGWEZaRmZuRmhuRmpuRl1vRkBGQ0ZGRkhGPg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIyomLCYiIiJGKkkickdGJUYqRiomSSJhR0YlRiZGKg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8= Assets 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImFHNiI2IyIiIiwmKiYsJkYnRidJInJHRiVGJ0YnJkYkNiMiIiFGJ0YnJkkiY0dGJUYtISIi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSS8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZYLUkobWZlbmNlZEdGJDYkLUYjNiYtRjs2JFEiMUYnRj4tRkQ2LVEiK0YnRj5GR0ZKRkxGTkZQRlJGVC9GV1EsMC4yMjIyMjIyZW1GJy9GWkZhby1GLzYlUSJyRidGMkY1Rj5GPi1GRDYtUScmc2RvdDtGJ0Y+RkdGSkZMRk5GUEZSRlQvRldRJjAuMGVtRicvRlpGam8tRiw2JUYuLUYjNiRGam5GPkZALUZENi1RKCZtaW51cztGJ0Y+RkdGSkZMRk5GUEZSRlRGYG9GYm8tRiw2JS1GLzYlUSJjRidGMkY1Rl5wRkAtRkQ2LVEhRidGPkZHRkpGTEZORlBGUkZURlZGWS8lK2V4ZWN1dGFibGVHRklGPg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImFHNiI2IyIiIywmKiYsJiIiIkYrSSJyR0YlRitGKyZGJDYjRitGK0YrJkkiY0dGJUYuISIi

If we substitute in the equations of motion for the assets into the consumption equations we get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHRj0tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXLUkmbWZyYWNHRiQ2KC1GIzYmLUYsNiUtRi82JVEiYUYnRjJGNUY4RkAtRkM2LVExJkludmlzaWJsZVRpbWVzO0YnRj5GRkZJRktGTUZPRlFGUy9GVlEmMC4wZW1GJy9GWUZiby1JKG1mZW5jZWRHRiQ2JC1GIzYmLUY7NiRRIjFGJ0Y+LUZDNi1RIitGJ0Y+RkZGSUZLRk1GT0ZRRlMvRlZRLDAuMjIyMjIyMmVtRicvRllGYHAtRi82JVEickYnRjJGNUY+Rj5GPi1GIzYmLUZDNi1RIUYnRj5GRkZJRktGTUZPRlFGU0ZVRlgtRiM2KUZpb0ZccC1GLzYlUSJCRidGMkY1RlxwLUYjNiQtSSVtc3VwR0YkNiVGXHEtRjs2JFEiMkYnRj4vJTFzdXBlcnNjcmlwdHNoaWZ0R0Y9Rj5GZ3BGPkZncEY+LyUubGluZXRoaWNrbmVzc0dGW3AvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGXXIvJSliZXZlbGxlZEdGSEY+

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiISooJkkiYUdGJUYmIiIiLCZGK0YrSSJyR0YlRitGKywoRitGK0kiQkdGJUYrKiQpRi8iIiNGK0YrISIi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVElc3Vic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYvLUklbXN1YkdGJDYlLUYsNiVRImFGJ0YvRjItRiM2JC1JI21uR0YkNiRRIjFGJy9GM1Enbm9ybWFsRidGRi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiPUYnRkYvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlEvJSlzdHJldGNoeUdGUS8lKnN5bW1ldHJpY0dGUS8lKGxhcmdlb3BHRlEvJS5tb3ZhYmxlbGltaXRzR0ZRLyUnYWNjZW50R0ZRLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGam4tRjY2JC1GIzYmRkItRkw2LVEiK0YnRkZGT0ZSRlRGVkZYRlpGZm4vRmluUSwwLjIyMjIyMjJlbUYnL0Zcb0Zlby1GLDYlUSJyRidGL0YyRkZGRi1GTDYtUScmc2RvdDtGJ0ZGRk9GUkZURlZGWEZaRmZuL0ZpblEmMC4wZW1GJy9GXG9GXnAtRjs2JUY9LUYjNiQtRkM2JEZKRkZGRkZILUZMNi1RKCZtaW51cztGJ0ZGRk9GUkZURlZGWEZaRmZuRmRvRmZvLUY7NiUtRiw2JVEiY0YnRi9GMkZicEZILUZMNi1RIixGJ0ZGRk8vRlNGMUZURlZGWEZaRmZuRl1wL0Zcb1EsMC4zMzMzMzMzZW1GJy1GOzYlRltxRkBGSEZLLUkmbWZyYWNHRiQ2KC1GIzYmRjotRkw2LVExJkludmlzaWJsZVRpbWVzO0YnRkZGT0ZSRlRGVkZYRlpGZm5GXXBGX3BGXW9GRi1GIzYmRkJGYW8tRiw2JVEiQkYnRi9GMkZGLyUubGluZXRoaWNrbmVzc0dGRS8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Znci8lKWJldmVsbGVkR0ZRLyUrZXhlY3V0YWJsZUdGUUZGRkZGXHNGRg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIiooLCYqJiwmRidGJ0kickdGJUYnRicmSSJhR0YlNiMiIiFGJ0YnJkYkRi8hIiJGJ0YrRicsJkYnRidJIkJHRiVGJ0Yy

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIyomLCYiIiJGKkkickdGJUYqRiosJiomRilGKiZJImFHRiU2I0YqRipGKiZGJEYwISIiRio=

finally we substitute in the equation for a_1 in the above equation we get 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIyomLCYiIiJGKkkickdGJUYqRiosJiomRilGKiwmKiZGKUYqJkkiYUdGJTYjIiIhRipGKiZGJEYyISIiRipGKiZGJDYjRipGNUYq

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== 1.1) The Optimal Level of Consumption LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== We now make some assumptions about a_0, r and B The amount of assets in the first period a_0 is given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IiUrNQ==

The interest rate is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEkMC4xRidGOS8lK2V4ZWN1dGFibGVHRj1GOQ==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIiIiEiIg==

Which means that the discount factor is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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We can now solve for the optimal level of consumption with the equations from the previous section 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIrTyFbNi0lISIo

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYoLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUpcmVhZG9ubHlHRjQvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSomY29sb25lcTtGJ0Y+LyUmZmVuY2VHRkUvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGXG8tSSZtZnJhY0dGJDYoLUYjNiotSShtZmVuY2VkR0YkNiQtRiM2Ki1GIzYqLUZlbzYkLUYjNipGOi1GTzYtUSIrRidGPkZSRlRGVkZYRlpGZm5GaG4vRltvUSwwLjIyMjIyMjJlbUYnL0Zeb0ZjcC1GLzYlUSJyRidGMkY1RkBGQ0ZGRkhGPkY+LUZPNi1RMSZJbnZpc2libGVUaW1lcztGJ0Y+RlJGVEZWRlhGWkZmbkZobi9GW29RJjAuMGVtRicvRl5vRlxxLUYsNiUtRi82JVEiYUYnRjJGNS1GIzYoLUY7NiRGTUY+RkBGQ0ZGRkhGPkZLRkBGQ0ZGRkhGPi1GTzYtUSgmbWludXM7RidGPkZSRlRGVkZYRlpGZm5GaG5GYnBGZHAtRiw2JUYuRmNxRktGQEZDRkZGSEY+Rj5GaHBGW3BGQEZDRkZGSEY+LUYjNipGOkZfcC1GLzYlUSJCRidGMkY1RkBGQ0ZGRkhGPi8lLmxpbmV0aGlja25lc3NHRj0vJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZXIvJSliZXZlbGxlZEdGRS1GTzYtUSFGJ0Y+RlJGVEZWRlhGWkZmbkZobkZqbkZdb0Y+

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIrTyFbNi0lISIo

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIrTyFbNi0lISIo

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== The value of c_0, c_1 and c_2 above are the optimal level of consumption in each time period. Note that these values are constant over time and equal for each time period. Note also that these optimal values are the values of c_0,c_1 and c_2 that maximize the sum of the discounted utility from consumption LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEidUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtSSVtc3ViR0YkNiUtRiw2JlEiY0YnRi9GMkY1LUYjNiUtRiw2JlEidEYnRi9GMkY1Ri8vRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGL0ZIRi9GSC1GLDYmUSJERidGLy9GM1EmZmFsc2VGJ0ZIRi9GSA== over time. If you dont feel comfortable to take this at face value we will later see in the section "Equivalence of Dynamic Programming and Lagrangian Optimization" that this indeed is true. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== 1.2) Discounted Utility We are now going to solve for the discounted utility and plot such utility 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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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%;

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

The sum of the discounted utility form consumption over time is given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIrISkpSC9rIiEiKQ==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== 1.3) Equations of Motion for Assets The amount of assets in period zero a_0 is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHRj0tSSNtb0dGJDYtUSomY29sb25lcTtGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlctRjs2JFElMTAwMEYnRj5GPg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IiUrNQ==

The amount of assets in the first period a_1 is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIraz4mKXlwISIo

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== The amount of assets in the second period a_2 is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIrQyIqZWJPISIo

The amount of assets in the third period a_3 is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IiIh

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCYqJiwmIiIiRiVJInJHNiJGJUYlLCYqJkYkRiUmSSJhR0YnNiMiIiFGJUYlJkkiY0dGJ0YsISIiRiVGJSZGLzYjRiVGMA==

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LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCYqJiwmIiIiRiVJInJHNiJGJUYlLCYqJkYkRiUsJiomRiRGJSZJImFHRic2IyIiIUYlRiUmSSJjR0YnRi4hIiJGJUYlJkYxNiNGJUYyRiVGJSZGMTYjIiIjRjI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== We know since previously that all assets in the last period (period 3) must be used up which means that 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyIiISwmKiYsJiIiIkYnSSJyRzYiRidGJywmKiZGJkYnLCYqJkYmRicmSSJhR0YpNiNGI0YnRicmSSJjR0YpRjAhIiJGJ0YnJkYyNiNGJ0YzRidGJyZGMjYjIiIjRjM=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== This means that the above equation fulfills the criteria for a constraint in the Lagrange optimization which is that the constraint must be expressed in an equal to zero form. Note also that all the other constraint a_1 and a_2 are already included into this aggregated constraint. We therefore define our constraint as: 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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Our Lagrangian is therefore given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCotSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjJkkiY0dGKDYjIiIhIiIiKiZJIkJHRihGLi1GJDYjJkYrNiNGLkYuRi4qJilGMCIiI0YuLUYkNiMmRis2I0Y3Ri5GLiomSSdsYW1iZGFHRihGLiwmKiYsJkYuRi5JInJHRihGLkYuLCYqJkZARi4sJiomRkBGLiZJImFHRihGLEYuRi5GKiEiIkYuRi5GM0ZIRi5GLkY6RkhGLkYu

The first order conditions are given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== 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print(); # input placeholder

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiISomSSdsYW1iZGFHRiUhIiIsKCIiIkYsKiYiIiNGLEkickdGJUYsRiwqJClGL0YuRixGLEYq

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVElZGlmZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRIkxGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUklbXN1YkdGJDYlLUYsNiVRImNGJ0YvRjItRiM2JC1JI21uR0YkNiRRIjFGJ0ZBRkEvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZBRkEtRj42LVEhRidGQUZDL0ZHRkVGSEZKRkxGTkZQL0ZTUSwwLjI3Nzc3NzhlbUYnL0ZWRmZvLyUrZXhlY3V0YWJsZUdGRUZB

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCYqJkkiQkc2IiIiIiZJImNHRiU2I0YmISIiRiYqJkknbGFtYmRhR0YlRiYsJkYmRipJInJHRiVGKkYmRiY=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEjQkJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUShpc29sYXRlRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNictRiw2JVEiJUYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkhGaG4vRk5GaW4tSSVtc3ViR0YkNiUtRiw2JVEiY0YnRi9GMi1GIzYkLUkjbW5HRiQ2JFEiMUYnRjlGOS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjlGOS1GNjYtUSFGJ0Y5RjtGPkZARkJGREZGRkhGSkZNLyUrZXhlY3V0YWJsZUdGPUY5

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIiooSSJCR0YlRidJJ2xhbWJkYUdGJSEiIiwmRidGJ0kickdGJUYnRis=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCYqJilJIkJHNiIiIiMiIiImSSJjR0YmNiNGJyEiIkYoSSdsYW1iZGFHRiZGLA==

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0knbGFtYmRhRzYiKiYpSSJCR0YkIiIjIiIiJkkiY0dGJDYjRighIiI=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCYqJiwmIiIiRiVJInJHNiJGJUYlLCYqJkYkRiUsJiomRiRGJSZJImFHRic2IyIiIUYlRiUmSSJjR0YnRi4hIiJGJUYlJkYxNiNGJUYyRiVGJSZGMTYjIiIjRjI=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIyomLCYiIiJGKkkickdGJUYqRiosJiomRilGKiwmKiZGKUYqJkkiYUdGJTYjIiIhRipGKiZGJEYyISIiRipGKiZGJDYjRipGNUYq

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== We now substitute in CC in AA and BB. This gives us: 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiISooKUkiQkdGJSIiIyEiIiZGJDYjRisiIiIsKEYvRi8qJkYrRi9JInJHRiVGL0YvKiQpRjJGK0YvRi9GLA==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEkQkJCRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVElc3Vic0YnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYpLUYsNiVRKSZsYW1iZGE7RicvRjBGPUY5LUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUYsNiVRJHJoc0YnRi9GMi1GUzYkLUYjNiQtRiw2JVEjQ0NGJ0YvRjJGOUY5LUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRicvRk5RLDAuMzMzMzMzM2VtRictRiw2JVEjQkJGJ0YvRjJGOUY5LUY2Ni1RIUYnRjlGO0Y+RkBGQkZERkZGSEZKRk0vJStleGVjdXRhYmxlR0Y9Rjk=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIiooSSJCR0YlISIiJkYkNiMiIiNGJywmRidGJ0kickdGJUYnRio=

We know that c_2 is given by DD so we get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjRERGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRic=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIyomLCYiIiJGKkkickdGJUYqRiosJiomRilGKiwmKiZGKUYqJkkiYUdGJTYjIiIhRipGKiZGJEYyISIiRipGKiZGJDYjRipGNUYq

If we substitute in DD in to BBB and AAA we get: 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIiomSSJCR0YlISIiLCYqJiwmRidGJ0kickdGJUYnRicsJiomRi1GJyZJImFHRiU2IyIiIUYnRicmRiRGM0YqRidGJ0YjRipGJw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVElc3Vic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYqLUklbXN1YkdGJDYlLUYsNiVRImNGJ0YvRjItRiM2JS1JI21uR0YkNiRRIjJGJy9GM1Enbm9ybWFsRidGL0YyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RidGRi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGUS8lKXN0cmV0Y2h5R0ZRLyUqc3ltbWV0cmljR0ZRLyUobGFyZ2VvcEdGUS8lLm1vdmFibGVsaW1pdHNHRlEvJSdhY2NlbnRHRlEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Zqbi1GLDYlUSRyaHNGJ0YvRjItRjY2JC1GIzYlLUYsNiVRI0RERidGL0YyLyUrZXhlY3V0YWJsZUdGUUZGRkYtRkw2LVEiLEYnRkZGTy9GU0YxRlRGVkZYRlpGZm4vRmluUSYwLjBlbUYnL0Zcb1EsMC4zMzMzMzMzZW1GJy1GLDYlUSRBQUFGJ0YvRjJGZ29GRkZGRmdvRkY=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiISoqKUkiQkdGJSIiIyEiIiwmIiIiRi5JInJHRiVGLkYuLCYqJkYtRi4sJiomRi1GLiZJImFHRiVGJkYuRi5GI0YsRi5GLiZGJDYjRi5GLEYuLChGLkYuKiZGK0YuRi9GLkYuKiQpRi9GK0YuRi5GLA==

We need to solve for c_1 and c_0 in both these equations since c_1 and c_0 are located on both sides on the equality sign 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIiomLCwmSSJhR0YlNiMiIiFGJyooIiIjRidGKkYnSSJyR0YlRidGJyZGJEYsISIiKiZGKkYnKUYwRi9GJ0YnKiZGMEYnRjFGJ0YyRicsJkkiQkdGJUYnRidGJ0Yy

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiISomLCoqJiZJImFHRiVGJiIiIilJInJHRiUiIiNGLUYtKihGMEYtRitGLUYvRi1GLUYrRi0mRiQ2I0YtISIiRi0sKiomKUkiQkdGJUYwRi1GL0YtRi1GL0YtKiRGN0YtRi1GLUYtRjQ=

We can see that we still have recursive elements left in X so we need to substitute in Y in X 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIiomLCwmSSJhR0YlNiMiIiFGJyooIiIjRidGKkYnSSJyR0YlRidGJyomLCoqJkYqRicpRjBGL0YnRidGLkYnRipGJ0YjISIiRicsKiomKUkiQkdGJUYvRidGMEYnRidGMEYnKiRGOEYnRidGJ0YnRjVGNUYzRicqKEYwRidGMkYnRjZGNUY1RicsJkY5RidGJ0YnRjU=

We need to solve for c_1 since c_1 are located on both sides on the equality sign 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIiomLCgqKCZJImFHRiU2IyIiIUYnKUkickdGJSIiI0YnSSJCR0YlRidGJyoqRjFGJ0YrRidGMkYnRjBGJ0YnKiZGK0YnRjJGJ0YnRicsKEYyRidGJ0YnKiQpRjJGMUYnRichIiI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== We can see that we still have recursive elements left in Y so if we substitute in X in Y 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiISomLCoqJiZJImFHRiVGJiIiIilJInJHRiUiIiNGLUYtKihGMEYtRitGLUYvRi1GLUYrRi0qJiwsRitGLUYxRi1GIyEiIkYqRi0qJkYvRi1GI0YtRjRGLSwmSSJCR0YlRi1GLUYtRjRGNEYtLCoqJilGN0YwRi1GL0YtRi1GL0YtKiRGOkYtRi1GLUYtRjQ=

We need to solve for c_0 since c_0 are located on both sides on the equality sign 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiISomLCYmSSJhR0YlRiYiIiIqJkYqRixJInJHRiVGLEYsRiwsKEkiQkdGJUYsRixGLCokKUYwIiIjRixGLCEiIg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== Finally we note that we have some recursive elements left DD so we substitute in the expressions for XX and YY 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIyomLCYiIiJGKkkickdGJUYqRiosJiomRilGKiwmKiZGKUYqJkkiYUdGJTYjIiIhRipGKiomLCZGMEYqKiZGMEYqRitGKkYqRiosKEkiQkdGJUYqRipGKiokKUY4RidGKkYqISIiRjtGKkYqKiYsKCooRjBGKilGK0YnRipGOEYqRioqKkYnRipGMEYqRjhGKkYrRipGKiomRjBGKkY4RipGKkYqRjdGO0Y7Rio=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== We can now summaries what we have so far LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjWVlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRic=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiISomLCYmSSJhR0YlRiYiIiIqJkYqRixJInJHRiVGLEYsRiwsKEkiQkdGJUYsRixGLCokKUYwIiIjRixGLCEiIg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjWFhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRic=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIiomLCgqKCZJImFHRiU2IyIiIUYnKUkickdGJSIiI0YnSSJCR0YlRidGJyoqRjFGJ0YrRidGMkYnRjBGJ0YnKiZGK0YnRjJGJ0YnRicsKEYyRidGJ0YnKiQpRjJGMUYnRichIiI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkRERERicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYn

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIyomLCYiIiJGKkkickdGJUYqRiosJiomRilGKiwmKiZGKUYqJkkiYUdGJTYjIiIhRipGKiomLCZGMEYqKiZGMEYqRitGKkYqRiosKEkiQkdGJUYqRipGKiokKUY4RidGKkYqISIiRjtGKkYqKiYsKCooRjBGKilGK0YnRipGOEYqRioqKkYnRipGMEYqRjhGKkYrRipGKiomRjBGKkY4RipGKkYqRjdGO0Y7Rio=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== We can now make the same assumptions that we did in the previous example The amount of assets in the first period a_0 is given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IiUrNQ==

The interest rate is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEkMC4xRidGOS8lK2V4ZWN1dGFibGVHRj1GOQ==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIiIiEiIg==

Which means that the discount factor is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIrIjQ0NDQqISM1

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== This means that the amount of consumption in each period are given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjWVlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRic=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiISQiK08hWzYtJSEiKA==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjWFhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRic=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIiQiK08hWzYtJSEiKA==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkRERERicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYn

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIyQiK08hWzYtJSEiKA==

If we plug this into our objective function we get 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 %; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjT2JGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRic=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIrISkpSC9rIiEiKQ==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== Which is the same results as we have had previously The sum of the discounted utility form consumption over time is 16,4 and consumption in all periods are given by 402.11 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== 2.1 ) The Most Straight Forward and Simplest Way of Doing Things LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

PCYvSSdsYW1iZGFHNiIqKCksJiQiIiIiIiFGKkkickdGJUYqIiIkISIiJkkiYUdGJTYjRitGLiwoKiQpSSJCR0YlIiIjRipGKkY1RipGKUYqRiovJkkiY0dGJUYxKihGKEYqRi9GKkYyRi4vJkY5NiNGKioqRjVGKilGKEY2RipGL0YqRjJGLi8mRjk2I0Y2KipGL0YqRjRGKiwqKiYkRi1GK0YqRixGKkYqKiQpRixGLUYqRioqJkZGRiopRixGNkYqRipGKUYqRipGMkYu

PCYvSSdsYW1iZGFHNiIkIit6JlxfMCMhIzcvJkkiY0dGJTYjIiIhJCIrTyFbNi0lISIoLyZGKzYjIiIiRi4vJkYrNiMiIiNGLg==

NyQkIiRrIiEiIjclLyZJImNHNiI2IyIiISQiJC0lRiwvJkYpNiMiIiJGLS8mRik2IyIiI0Yt

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEocmVzdGFydEYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEifkYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk8tRjk2LlEiO0YnRi9GPEY+L0ZCRjRGQ0ZFRkdGSUZLRk0vRlFRLDAuMjc3Nzc3OGVtRidGLy8lK2V4ZWN1dGFibGVHRkBGPA== The equations for consumption and assets from the previous example are given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiISooJkkiYUdGJUYmIiIiLCZGK0YrSSJyR0YlRitGKywoRitGK0kiQkdGJUYrKiQpRi8iIiNGK0YrISIi

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIiooJkkiYUdGJUYmRicsJkYnRidJInJHRiVGJ0YnLCZGJ0YnSSJCR0YlRichIiI=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImNHNiI2IyIiIyomLCYiIiJGKkkickdGJUYqRiomSSJhR0YlRiZGKg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSS8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZYLUkobWZlbmNlZEdGJDYkLUYjNiZGOi1GRDYtUSIrRidGPkZHRkpGTEZORlBGUkZUL0ZXUSwwLjIyMjIyMjJlbUYnL0ZaRl5vLUYvNiVRInJGJ0YyRjVGPkY+LUZENi1RJyZzZG90O0YnRj5GR0ZKRkxGTkZQRlJGVC9GV1EmMC4wZW1GJy9GWkZnby1GLDYlRi4tRiM2JC1GOzYkRkJGPkY+RkAtRkQ2LVEoJm1pbnVzO0YnRj5GR0ZKRkxGTkZQRlJGVEZdb0Zfby1GLDYlLUYvNiVRImNGJ0YyRjVGW3BGQC1GRDYtUSFGJ0Y+RkdGSkZMRk5GUEZSRlRGVkZZRj4=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImFHNiI2IyIiIiwmKiYmRiQ2IyIiIUYnLCZGJ0YnSSJyR0YlRidGJ0YnJkkiY0dGJUYrISIi

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJImFHNiI2IyIiIywmKiYmRiQ2IyIiIkYsLCZGLEYsSSJyR0YlRixGLEYsJkkiY0dGJUYrISIi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== From the equations above we can see that there is a pattern emerging LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== For example if we have a ten period economy then consumption from c_9 to c_0 should evolve according to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

KiYsJiIiIkYkSSJyRzYiRiRGJCZJImFHRiY2IyIiJ0Yk

KigsJiIiIkYkSSJyRzYiRiRGJCZJImFHRiY2IyIiJkYkLCZGJEYkSSJCR0YmRiQhIiI=

KigsJiIiIkYkSSJyRzYiRiRGJCZJImFHRiY2IyIiJUYkLChGJEYkSSJCR0YmRiQqJClGLCIiI0YkRiQhIiI=

KigsJiIiIkYkSSJyRzYiRiRGJCZJImFHRiY2IyIiJEYkLCpGJEYkSSJCR0YmRiQqJClGLCIiI0YkRiQqJClGLEYqRiRGJCEiIg==

KigsJiIiIkYkSSJyRzYiRiRGJCZJImFHRiY2IyIiI0YkLCxGJEYkSSJCR0YmRiQqJClGLEYqRiRGJCokKUYsIiIkRiRGJCokKUYsIiIlRiRGJCEiIg==

KigsJiIiIkYkSSJyRzYiRiRGJCZJImFHRiY2I0YkRiQsLkYkRiRJIkJHRiZGJCokKUYrIiIjRiRGJCokKUYrIiIkRiRGJCokKUYrIiIlRiRGJCokKUYrIiImRiRGJCEiIg==

KigsJiIiIkYkSSJyRzYiRiRGJCZJImFHRiY2IyIiIUYkLDBGJEYkSSJCR0YmRiQqJClGLCIiI0YkRiQqJClGLCIiJEYkRiQqJClGLCIiJUYkRiQqJClGLCIiJkYkRiQqJClGLCIiJ0YkRiQhIiI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtRiw2LVEiOkYnRi9GMkY1RjdGOUY7Rj1GP0ZBRkQvJStleGVjdXRhYmxlR0Y0Ri8= %; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8= and assets from a_1 to a_7 should evolves according to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== print(); # input placeholder 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCYqJiwmIiIiRiVJInJHNiJGJUYlJkkiYUdGJzYjIiIhRiVGJSZJImNHRidGKiEiIg==

LCYqJiwmIiIiRiVJInJHNiJGJUYlLCYqJkYkRiUmSSJhR0YnNiMiIiFGJUYlJkkiY0dGJ0YsISIiRiVGJSZGLzYjRiVGMA==

LCYqJiwmIiIiRiVJInJHNiJGJUYlLCYqJkYkRiUsJiomRiRGJSZJImFHRic2IyIiIUYlRiUmSSJjR0YnRi4hIiJGJUYlJkYxNiNGJUYyRiVGJSZGMTYjIiIjRjI=

LCYqJiwmIiIiRiVJInJHNiJGJUYlLCYqJkYkRiUsJiomRiRGJSwmKiZGJEYlJkkiYUdGJzYjIiIhRiVGJSZJImNHRidGMCEiIkYlRiUmRjM2I0YlRjRGJUYlJkYzNiMiIiNGNEYlRiUmRjM2IyIiJEY0

LCYqJiwmIiIiRiVJInJHNiJGJUYlLCYqJkYkRiUsJiomRiRGJSwmKiZGJEYlLCYqJkYkRiUmSSJhR0YnNiMiIiFGJUYlJkkiY0dGJ0YyISIiRiVGJSZGNTYjRiVGNkYlRiUmRjU2IyIiI0Y2RiVGJSZGNTYjIiIkRjZGJUYlJkY1NiMiIiVGNg==

LCYqJiwmIiIiRiVJInJHNiJGJUYlLCYqJkYkRiUsJiomRiRGJSwmKiZGJEYlLCYqJkYkRiUsJiomRiRGJSZJImFHRic2IyIiIUYlRiUmSSJjR0YnRjQhIiJGJUYlJkY3NiNGJUY4RiVGJSZGNzYjIiIjRjhGJUYlJkY3NiMiIiRGOEYlRiUmRjc2IyIiJUY4RiVGJSZGNzYjIiImRjg=

LCYqJiwmIiIiRiVJInJHNiJGJUYlLCYqJkYkRiUsJiomRiRGJSwmKiZGJEYlLCYqJkYkRiUsJiomRiRGJSwmKiZGJEYlJkkiYUdGJzYjIiIhRiVGJSZJImNHRidGNiEiIkYlRiUmRjk2I0YlRjpGJUYlJkY5NiMiIiNGOkYlRiUmRjk2IyIiJEY6RiVGJSZGOTYjIiIlRjpGJUYlJkY5NiMiIiZGOkYlRiUmRjk2IyIiJ0Y6

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== We can now set up our general model. For consumption we get: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== print(); # input placeholder 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCQqJiQiIzYhIiIiIiImSSJhRzYiNiMiIidGJ0Yn

LCQqJiQiK2laIT53JiEjNSIiIiZJImFHNiI2IyIiJkYnRic=

LCQqJiQiK08hWzYtJSEjNSIiIiZJImFHNiI2IyIiJUYnRic=

LCQqJiQiK1AhM1o6JCEjNSIiIiZJImFHNiI2IyIiJEYnRic=

LCQqJiQiKzNbKHpqIyEjNSIiIiZJImFHNiI2IyIiI0YnRic=

LCQqJiQiKy9RMidIIyEjNSIiIiZJImFHNiI2I0YnRidGJw==

JCIrKCpcMGE/ISIo

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8= so for assets we get: 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIrLl0lZiUqKSEiKA==

JCIrMVhbJ3koISIo

JCIrZ3oyNmwhIig=

JCIrZjI4M14hIig=

JCIrUSkpKVtjJCEiKA==

JCIrREZLbj0hIig=

JCIiIUYj

We can plot the asset trajectory as follows: 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

%;

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

which means that consumption must be given by: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkobWZlbmNlZEdGJDYmLUYjNictSSNtaUdGJDYmUSRzZXFGJy8lJXNpemVHUSMxNEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYlLUYjNi4tSSVtc3ViR0YkNiUtRjE2JlEiY0YnRjRGN0Y6LUYjNigtRjE2JlEiaUYnRjRGN0Y6RjRGNy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnRjovJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi5RIixGJ0Y0L0Y7USdub3JtYWxGJy8lJmZlbmNlR0ZOLyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRk4vJSpzeW1tZXRyaWNHRk4vJShsYXJnZW9wR0ZOLyUubW92YWJsZWxpbWl0c0dGTi8lJ2FjY2VudEdGTi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnRkktRlY2LlEiPUYnRjRGWUZlbi9GaG5GTkZpbkZbb0Zdb0Zfb0Zhby9GZG9RLDAuMjc3Nzc3OGVtRicvRmdvRl5wLUkjbW5HRiQ2JUZURjRGWS1GVjYuUSMuLkYnRjRGWUZlbkZccEZpbkZbb0Zdb0Zfb0Zhby9GZG9RLDAuMjIyMjIyMmVtRicvRmdvRmVvLUYxNiZRIm5GJ0Y0RjdGOi1GVjYuUSgmbWludXM7RidGNEZZRmVuRlxwRmluRltvRl1vRl9vRmFvRmZwL0Znb0ZncC1GYXA2JVEiMUYnRjRGWUY0RkxGWUY0RllGNEZMRllGWS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJy1GVjYtUSFGJ0ZZRmVuRlxwRmluRltvRl1vRl9vRmFvRl1wRl9wLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHRmVvLyUmZGVwdGhHRmFyLyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUZdcjYmRl9yRmJyRmRyL0ZnclElYXV0b0YnRmlxRkxGWQ==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NykkIisoKlwwYT8hIihGI0YjRiNGI0YjJCIrKSpcMGE/RiU=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== We can plot the consumption trajectory as follows: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJSVzaXplR1EjMTRGJy8lK2V4ZWN1dGFibGVHRjRGLw== 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

%;

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== and discounted utility from consumption must be given by: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIrdGcpXEsmISIq

JCIrSWsqMyVbISIq

JCIrT1wiM1MlISIq

JCIrXjN1K1MhIio=

JCIrUXIuUE8hIio=

JCIrKVIoUjFMISIq

JCIrO2UiZSskISIq

We can plot the discounted utility from consumption as follows: 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

%;

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

The sum of the discounted utility form consumption over time is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEkYWRkRicvJSVzaXplR1EjMTRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiUtRiM2Li1JJW1zdWJHRiQ2JS1GLDYmUSNkdUYnRi9GMkY1LUYjNiYtRiw2JlEiaUYnRi9GMkY1Ri9GMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYuUSIsRidGLy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGUy8lKnN5bW1ldHJpY0dGUy8lKGxhcmdlb3BHRlMvJS5tb3ZhYmxlbGltaXRzR0ZTLyUnYWNjZW50R0ZTLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRkw2LlEifkYnRi9GT0ZRL0ZVRlNGVkZYRlpGZm5GaG5Gam4vRl5vRlxvRkUtRkw2LlEiPUYnRi9GT0ZRRmNvRlZGWEZaRmZuRmhuL0Zbb1EsMC4yNzc3Nzc4ZW1GJy9GXm9GaW8tSSNtbkdGJDYlRkpGL0ZPLUZMNi5RIy4uRidGL0ZPRlFGY29GVkZYRlpGZm5GaG4vRltvUSwwLjIyMjIyMjJlbUYnRmRvLUYsNiZRIm5GJ0YvRjJGNS1GTDYuUSgmbWludXM7RidGL0ZPRlFGY29GVkZYRlpGZm5GaG5GYXAvRl5vRmJwLUZccDYlUSIxRidGL0ZPRi9GT0YvRk8tRkw2LVEiO0YnRk9GUUZURlZGWEZaRmZuRmhuRmpuRmpvRk8=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIrbClvOyZHISIp

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8= To summarize we not that in this example we got: The sum of the discounted utility form consumption over time is 28,51 and consumption in all periods are given by 205.4 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== 3.1 ) The Most Straight Forward and Simplest Way of Doing Things Again we can solve the above problem much easier by using Dynamic Programming as follows: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NykvJkkiQ0c2IjYjIiIhJCIrKCpcMGE/ISIoLyZGJTYjIiIiRikvJkYlNiMiIiMkIisnKlwwYT9GKy8mRiU2IyIiJEYpLyZGJTYjIiIlJCIrKSpcMGE/RisvJkYlNiMiIiZGKS8mRiU2IyIiJ0Y+

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== or we can solve it with Lagrange as follows LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== 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print(); # input placeholder

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

PCovSSdsYW1iZGFHNiIkIisuYjRbRiEjNy8mSSJjR0YlNiMiIiEkIisoKlwwYT8hIigvJkYrNiMiIiJGLi8mRis2IyIiI0YuLyZGKzYjIiIkRi4vJkYrNiMiIiVGLi8mRis2IyIiJkYuLyZGKzYjIiInRi4=

NyQkIihwOyZHISImNykvJkkiY0c2IjYjIiIhJCIoYlMwIyEiJS8mRik2IyIiIkYtLyZGKTYjIiIjRi0vJkYpNiMiIiRGLS8mRik2IyIiJSQiKGFTMCNGLy8mRik2IyIiJkYtLyZGKTYjIiInRi0=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8= Which is the same results as we have had previously The sum of the discounted utility form consumption over time is 28.5 and consumption in all periods are given by 205.4 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8= JSFH JSFH JSFH JSFH JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkNGLw==