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::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::J:<6:\"\{\}-%%ROOTG6'-%)BOUNDS_XG6#$"#]!""-%)BOUNDS_YG6#$""!F*-%-BOUNDS_WIDTHG6#$"%+!)F*-%.BOUNDS_HEIGHTG6#$"$!pF*-%)CHILDRENG6" LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Cake Eating in Finite and Infinite Time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The following was implemented in Maple by Marcus Davidsson (2008) davidsson_marcus@hotmail.com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= and is based upon the work by Adamek (2006) The Cake-Eating Problem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH 1) Cake Eating in Finite Time JSFH JSFH JSFH IntroductionLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We assume that we have a cake that in the first period has a size of CS(1) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjQ1NGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1JI21uR0YkNiRRIjFGJy9GM1Enbm9ybWFsRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJ0Y+Rj5GQEY+

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print(); # input placeholder

LUkjQ1NHNiI2IyIiIg==

JSFH We now assume that we have five time periods. We assume that we consume a fraction of the cake in each time period denoted by C(1), C(2), C(3), C(4). Note that we assume that the cake is completly gone in the fifth time period which means that our consumption in period five is zero The equation of motion for the cake size in each periods are given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

We will now derive the expressions for consumption in the four periods LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Period-1 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The cake size in the last period is given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JI0NTRzYiNiMiIiYsLC1GJDYjIiIiRistSSJDR0YlRiohIiItRi02IyIiI0YuLUYtNiMiIiRGLi1GLTYjIiIlRi4=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= where C(1), C(2), C(3) and C(4) are the amount of consumtion in each period We again assume that all cake is gone in the last period which means that CS(5)=0. If we plug this into the previous equation then we get the expression for the terminal condition in period one 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyIiISwsLUkjQ1NHNiI2IyIiIkYpLUkiQ0dGJ0YoISIiLUYrNiMiIiNGLC1GKzYjIiIkRiwtRis2IyIiJUYs

We can now express this equation in C(1) terms only. 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Note that we discount consumption over time where B^t is a discount factor. Note that B=1/(1+r) where r is the interest rate. Note that since B^t < 1 it means that consumption will be decreasing over time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= If we plug the all the rewritten expressions into the previous equation we get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY4LUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUSNDU0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiRRIjFGJ0YvLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUrZXhlY3V0YWJsZUdGOC8lKXJlYWRvbmx5R0ZQLyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RidGL0YvLUYzNi1RKCZtaW51cztGJ0YvRjZGOUY7Rj1GP0ZBRkMvRkZRLDAuMjIyMjIyMmVtRicvRklGY28tRks2JVEiQ0YnRk5GUUZTRl9vLUklbXN1cEdGJDYlLUZLNiVRIkJGJ0ZORlEtRiM2JEZYRi8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYzNi1RJyZzZG90O0YnRi9GNkY5RjtGPUY/RkFGQy9GRlEmMC4wZW1GJy9GSUZncEZlby1GVDYkRl5wRi9GX28tRmlvNiVGW3AtRiM2JC1GLDYkUSIyRidGL0YvRmBwRmNwRmVvRmlwRl9vLUYjNictRmlvNiVGW3AtRiM2JC1GLDYkUSIzRidGL0YvRmBwRmNwRmVvRmlwRi8tRks2I1EhRidGaG5GLw==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyIiISwsLUkjQ1NHNiI2IyIiIkYpLUkiQ0dGJ0YoISIiKiZJIkJHRidGKUYqRilGLComKUYuIiIjRilGKkYpRiwqJilGLiIiJEYpRipGKUYs

We now solve for CS(1) 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JI0NTRzYiNiMiIiIsKi1JIkNHRiVGJkYnKiZJIkJHRiVGJ0YpRidGJyomKUYsIiIjRidGKUYnRicqJilGLCIiJEYnRilGJ0Yn

We now note that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEiI0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVFGIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjI0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHRjRGLw== We know that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JI0NTRzYiNiMiIiIsKi1JIkNHRiVGJkYnKiZJIkJHRiVGJ0YpRidGJyomKUYsIiIjRidGKUYnRicqJilGLCIiJEYnRilGJ0Yn

We now multiply both sides by B so we get 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyomSSJCRzYiIiIiLUkjQ1NHRiU2I0YmRiYqJkYkRiYsKi1JIkNHRiVGKUYmKiZGJEYmRixGJkYmKiYpRiQiIiNGJkYsRiZGJiomKUYkIiIkRiZGLEYmRiZGJg==

Which can be written as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRJ2V4cGFuZEYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIiVGJ0YvRjIvJStleGVjdXRhYmxlR0Y9RjlGOUZaRjk=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyomSSJCRzYiIiIiLUkjQ1NHRiU2I0YmRiYsKiomRiRGJi1JIkNHRiVGKUYmRiYqJilGJCIiI0YmRixGJkYmKiYpRiQiIiRGJkYsRiZGJiomKUYkIiIlRiZGLEYmRiY=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEiI0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVFGIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjI0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHRkwvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRi8= Now we know that X and Y are given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JI0NTRzYiNiMiIiIsKi1JIkNHRiVGJkYnKiZJIkJHRiVGJ0YpRidGJyomKUYsIiIjRidGKUYnRicqJilGLCIiJEYnRilGJ0Yn

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyomSSJCRzYiIiIiLUkjQ1NHRiU2I0YmRiYsKiomRiRGJi1JIkNHRiVGKUYmRiYqJilGJCIiI0YmRixGJkYmKiYpRiQiIiRGJkYsRiZGJiomKUYkIiIlRiZGLEYmRiY=

We now subtract BCS(initial) from both sides of CS(initial) so we get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbWlHRiQ2JVEkbGhzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEiWEYnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZALUkjbW9HRiQ2LVEoJm1pbnVzO0YnRkAvJSZmZW5jZUdGPy8lKnNlcGFyYXRvckdGPy8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZWRistRjY2JC1GIzYlLUYsNiVRIllGJ0YvRjJGPUZARkAtRkM2LVEiPUYnRkBGRkZIRkpGTEZORlBGUi9GVVEsMC4yNzc3Nzc4ZW1GJy9GWEZeby1GLDYlUSRyaHNGJ0YvRjJGNUZCRmBvRllGPUZA

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LywmLUkjQ1NHNiI2IyIiIkYoKiZJIkJHRiZGKEYkRighIiIsJi1JIkNHRiZGJ0YoKiYpRioiIiVGKEYtRihGKw==

JSFH Which can be written as 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyomLCYiIiJGJUkiQkc2IiEiIkYlLUkjQ1NHRic2I0YlRiUsJi1JIkNHRidGK0YlKiYpRiYiIiVGJUYtRiVGKA==

We now divide both sides by (1-B) so we get 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JI0NTRzYiNiMiIiIqJiwmLUkiQ0dGJUYmRicqJilJIkJHRiUiIiVGJ0YqRichIiJGJywmRidGJ0YuRjBGMA==

We can now solve for C(1) 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JIkNHNiI2IyIiIiooLUkjQ1NHRiVGJkYnLCZGJ0YnKiQpSSJCR0YlIiIlRichIiJGMCwmRidGJ0YuRjBGJw==

Which is the expression for the optimal consumtion in period one LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Period-2 Our terminal condition for period two is given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyIiISwqLUkjQ1NHNiI2IyIiIyIiIi1JIkNHRidGKCEiIi1GLDYjIiIkRi0tRiw2IyIiJUYt

We can now express this equation in C(2) terms only. 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= If we plug that into the previous equation we get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyIiISwqLUkjQ1NHNiI2IyIiIyIiIi1JIkNHRidGKCEiIiomSSJCR0YnRipGK0YqRi0qJilGL0YpRipGK0YqRi0=

We now solve for CS(2) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JI0NTRzYiNiMiIiMsKC1JIkNHRiVGJiIiIiomSSJCR0YlRitGKUYrRisqJilGLUYnRitGKUYrRis=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We now note that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEiI0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVFGIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjI0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHRjRGLw== We know that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JI0NTRzYiNiMiIiMsKC1JIkNHRiVGJiIiIiomSSJCR0YlRitGKUYrRisqJilGLUYnRitGKUYrRis=

We now multiply both sides by B so we get 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyomSSJCRzYiIiIiLUkjQ1NHRiU2IyIiI0YmKiZGJEYmLCgtSSJDR0YlRilGJiomRiRGJkYtRiZGJiomKUYkRipGJkYtRiZGJkYm

Which can be written as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRJ2V4cGFuZEYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIiVGJ0YvRjIvJStleGVjdXRhYmxlR0Y9RjlGOUZaRjk=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyomSSJCRzYiIiIiLUkjQ1NHRiU2IyIiI0YmLCgqJkYkRiYtSSJDR0YlRilGJkYmKiYpRiRGKkYmRi1GJkYmKiYpRiQiIiRGJkYtRiZGJg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEiI0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVFGIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjI0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHRkwvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRi8= Now we know that X and Y are given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JI0NTRzYiNiMiIiMsKC1JIkNHRiVGJiIiIiomSSJCR0YlRitGKUYrRisqJilGLUYnRitGKUYrRis=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyomSSJCRzYiIiIiLUkjQ1NHRiU2IyIiI0YmLCgqJkYkRiYtSSJDR0YlRilGJkYmKiYpRiRGKkYmRi1GJkYmKiYpRiQiIiRGJkYtRiZGJg==

We now subtract BCS(2) from both sides of CS(2) so we get 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LywmLUkjQ1NHNiI2IyIiIyIiIiomSSJCR0YmRilGJEYpISIiLCYtSSJDR0YmRidGKSomKUYrIiIkRilGLkYpRiw=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Which can be written as 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyomLCYiIiJGJUkiQkc2IiEiIkYlLUkjQ1NHRic2IyIiI0YlLCYtSSJDR0YnRitGJSomKUYmIiIkRiVGLkYlRig=

We now divide both sides by (1-B) so we get 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JI0NTRzYiNiMiIiMqJiwmLUkiQ0dGJUYmIiIiKiYpSSJCR0YlIiIkRixGKkYsISIiRiwsJkYsRixGL0YxRjE=

We can now solve for C(2) 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JIkNHNiI2IyIiIyooLUkjQ1NHRiVGJiIiIiwmRitGKyokKUkiQkdGJSIiJEYrISIiRjEsJkYrRitGL0YxRis=

Which is the expression for the optimal consumtion in period two LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Period-3 Our terminal condition for period three is given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyIiISwqLUkjQ1NHNiI2IyIiIyIiIi1JIkNHRidGKCEiIi1GLDYjIiIkRi0tRiw2IyIiJUYt

We can now express this equation in C(3) terms only.

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

JSFH If we plug that into the previous equation we get JSFHJSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyIiISwoLUkjQ1NHNiI2IyIiJCIiIi1JIkNHRidGKCEiIiomSSJCR0YnRipGK0YqRi0=

We now solve for CS(3) JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JI0NTRzYiNiMiIiQsJi1JIkNHRiVGJiIiIiomSSJCR0YlRitGKUYrRis=

JSFH We now note that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEiI0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVFGIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjI0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHRjRGLw== We know that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JI0NTRzYiNiMiIiQsJi1JIkNHRiVGJiIiIiomSSJCR0YlRitGKUYrRis=

We now multiply both sides by B so we get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEnJnNkb3Q7RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSS8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZYLUYvNiVRJGxoc0YnRjJGNS1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYvNiVRIiVGJ0YyRjUvJStleGVjdXRhYmxlR0ZJRj5GPi1GRDYtUSI9RidGPkZHRkpGTEZORlBGUkZUL0ZXUSwwLjI3Nzc3NzhlbUYnL0ZaRmZvRitGQy1GLzYlUSRyaHNGJ0YyRjVGaG5GYG9GPg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyomSSJCRzYiIiIiLUkjQ1NHRiU2IyIiJEYmKiZGJEYmLCYtSSJDR0YlRilGJiomRiRGJkYtRiZGJkYm

Which can be written as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRJ2V4cGFuZEYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIiVGJ0YvRjIvJStleGVjdXRhYmxlR0Y9RjlGOUZaRjk=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyomSSJCRzYiIiIiLUkjQ1NHRiU2IyIiJEYmLCYqJkYkRiYtSSJDR0YlRilGJkYmKiYpRiQiIiNGJkYtRiZGJg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEiI0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVFGIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjI0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHRkwvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRi8= Now we know that X and Y are given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JI0NTRzYiNiMiIiQsJi1JIkNHRiVGJiIiIiomSSJCR0YlRitGKUYrRis=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyomSSJCRzYiIiIiLUkjQ1NHRiU2IyIiJEYmLCYqJkYkRiYtSSJDR0YlRilGJkYmKiYpRiQiIiNGJkYtRiZGJg==

We now subtract BCS(3) from both sides of CS(3) so we get 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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JSFH Which can be written as 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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We now divide both sides by (1-B) so we get 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JI0NTRzYiNiMiIiQqJiwmLUkiQ0dGJUYmIiIiKiYpSSJCR0YlIiIjRixGKkYsISIiRiwsJkYsRixGL0YxRjE=

We can now solve for C(3) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoaXNvbGF0ZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYpLUYsNiVRIiVGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRIkNGJ0YvRjItRjY2JC1GIzYlLUkjbW5HRiQ2JFEiM0YnRkEvJStleGVjdXRhYmxlR0ZFRkFGQS1GLDYjUSFGJ0Zdb0ZBRkFGXW9GQQ==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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Which is the expression for the optimal consumtion in period three JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Period-4 Our terminal condition for period three is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUSNDU0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiRRIjRGJ0YvLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUrZXhlY3V0YWJsZUdGOC8lKXJlYWRvbmx5R0ZQLyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RidGL0YvLUYzNi1RKCZtaW51cztGJ0YvRjZGOUY7Rj1GP0ZBRkMvRkZRLDAuMjIyMjIyMmVtRicvRklGY28tRks2JVEiQ0YnRk5GUUZTRmVuRmhuRmpuRlxvRi8=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyIiISwmLUkjQ1NHNiI2IyIiJSIiIi1JIkNHRidGKCEiIg==

The above equation is already express in only C(4) terms which means that we can solve for consumption directly

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JIkNHNiI2IyIiJS1JI0NTR0YlRiY=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Which is the expression for the optimal consumtion in period four Consumtion Dynamics Over Time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== We can now visualize the dynamics of consumtion over time We first note that the equations of motion for the cake size in period two and three are given by JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCYtSSNDU0c2IjYjIiIiRictSSJDR0YlRiYhIiI=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCgtSSNDU0c2IjYjIiIiRictSSJDR0YlRiYhIiItRik2IyIiI0Yq

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCotSSNDU0c2IjYjIiIiRictSSJDR0YlRiYhIiItRik2IyIiI0YqLUYpNiMiIiRGKg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

This means that the equation of motions can be written as 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JIkNHNiI2IyIiIiooLUkjQ1NHRiVGJkYnLCZGJ0YnSSJCR0YlISIiRicsJkYnRicqJClGLCIiJUYnRi1GLQ==

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LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JIkNHNiI2IyIiIyooLCYtSSNDU0dGJTYjIiIiRi0tRiRGLCEiIkYtLCZGLUYtSSJCR0YlRi9GLSwmRi1GLSokKUYxIiIkRi1GL0Yv

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JIkNHNiI2IyIiJCooLCgtSSNDU0dGJTYjIiIiRi0tRiRGLCEiIi1GJDYjIiIjRi9GLSwmRi1GLUkiQkdGJUYvRi0sJkYtRi0qJClGNEYyRi1GL0Yv

JSFH

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiNEYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVLUYsNiVRI0NTRidGL0YyRjUvJStleGVjdXRhYmxlR0ZGRj4=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JIkNHNiI2IyIiJSwqLUkjQ1NHRiU2IyIiIkYsLUYkRishIiItRiQ2IyIiI0YuLUYkNiMiIiRGLg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We now assume that 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEkMC45RidGOS1GNjYtUSI6RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS8lK2V4ZWN1dGFibGVHRj1GOQ== Which means that the amount of consupmtion in each period are given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIrVD8jeSFIISIp

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIrUClScWgjISIp

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print(); # input placeholder

JCIrYWVMYkIhIik=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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print(); # input placeholder

JCIrb0EhKT5AISIp

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We first make sure that all cake is consumed 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print(); # input placeholder

JCIrKysrKzUhIig=

We can now plot the consumption over time 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

%;

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We can now plot the cake size over time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVElcGxvdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzY2LUY2NiYtRiM2LS1JI21uR0YkNiRRIjFGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGQi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZKLyUqc3ltbWV0cmljR0ZKLyUobGFyZ2VvcEdGSi8lLm1vdmFibGVsaW1pdHNHRkovJSdhY2NlbnRHRkovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GPzYkUSIyRidGQkZELUY/NiRRIjNGJ0ZCRkQtRj82JFEiNEYnRkJGRC1GPzYkUSI1RidGQi8lK2V4ZWN1dGFibGVHRkpGQkZCLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRkQtRjY2Ji1GIzYxLUYsNiVRI0NTRidGL0YyLUY2NiQtRiM2JUY+RmNvRkJGQkZERl9wLUY2NiQtRiM2JUZnbkZjb0ZCRkJGREZfcC1GNjYkLUYjNiVGam5GY29GQkZCRkRGX3AtRjY2JC1GIzYlRl1vRmNvRkJGQkZELUY/NiRRIjBGJ0ZCRmNvRkJGQkZlb0Zob0ZELUZFNi1RIn5GJ0ZCRkgvRkxGSkZNRk9GUUZTRlVGVy9GZW5GWS1GLDYlUSZjb2xvckYnRi9GMi1GRTYtUSI9RidGQkZIRmhxRk1GT0ZRRlNGVS9GWFEsMC4yNzc3Nzc4ZW1GJy9GZW5GYXItRiw2JVEmYmxhY2tGJ0YvRjJGREZlcS1GLDYlUSp0aGlja25lc3NGJ0YvRjJGXXJGam5GREZlcS1GLDYlUSdsYWJlbHNGJ0YvRjJGXXItRjY2Ji1GIzYoLUYsNiVRJXRpbWVGJ0YvRjJGREZlcS1GLDYlUSljYWtlc2l6ZUYnRi9GMkZjb0ZCRkJGZW9GaG9GY29GQkZCRmNvRkI=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

%;

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

Alternative Formulation Note that we could have solved the above problem in a much more straight forward way LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We assume that our utility from cake consumption at time t is given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JInVHNiI2IyZJImNHRiU2I0kidEdGJS1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlRiY=

We now note that we discount utility from cake 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

We now note that 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= If we plug in the previous expression in this equation we get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyomSSRtYXhHJSpwcm90ZWN0ZWRHIiIiLUkiVUc2IjYjJkkiY0dGKTYjSSJ0R0YpRiYtSSRTdW1HNiRGJUkoX3N5c2xpYkdGKTYjKiYpSSJCR0YpRi5GJi1JI2xuR0YxRipGJg==

JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= In our case our objective function Ob is therefor given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCoqJkkiQkc2IiIiIi1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMmSSJDR0YlNiNGJkYmRiYqJilGJCIiI0YmLUYoNiMmRi42I0YyRiZGJiomKUYkIiIkRiYtRig2IyZGLjYjRjlGJkYmKiYpRiQiIiVGJi1GKDYjJkYuNiNGQEYmRiY=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyMvLCwmSSNDU0c2IjYjIiIiIiIiJkkiQ0c2IjYjIiIiISIiJkkiQ0c2IjYjIiIjISIiJkkiQ0c2IjYjIiIkISIiJkkiQ0c2IjYjIiIlISIiIiIh

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We again assume that 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEkMC45RidGOS1GNjYtUSI6RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS8lK2V4ZWN1dGFibGVHRj1GOQ== JSFH This gives us JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRLU9wdGltaXphdGlvbkYnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZALUkjbW9HRiQ2LVEiOkYnRkAvJSZmZW5jZUdGPy8lKnNlcGFyYXRvckdGPy8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZWRj1GQA== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEpTWF4aW1pemVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2LC1GLDYlUSNPYkYnRi9GMi1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEkY29uRidGL0YyRj0tRj42LVEifkYnRkFGQy9GR0ZFRkhGSkZMRk5GUEZSL0ZWRlQtRiw2JVEvaXRlcmF0aW9ubGltaXRGJ0YvRjItRj42LVEiPUYnRkFGQ0ZobkZIRkpGTEZORlAvRlNRLDAuMjc3Nzc3OGVtRicvRlZGYW8tSSNtbkdGJDYkUSUxMDAwRidGQS8lK2V4ZWN1dGFibGVHRkVGQUZBLUY+Ni1RIjtGJ0ZBRkNGRkZIRkpGTEZORlBGUkZib0Znb0ZB

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyQkIjMnbz5mIzNjNiUpKiohIzw3Ji8mSSJDRzYiNiMiIiIkIjMqKUgkPTYvQXkhSCEjOy8mRik2IyIiIyQiM3UocDpvJClScWgjRi8vJkYpNiMiIiQkIjNfKFx3USZlTGJCRi8vJkYpNiMiIiUkIjNedSUqPW9BISk+QEYv

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Which is the same values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIn5GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlhGPg==, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiNEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ we had before LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFH 2) Cake Eating in infinite Time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== The cake size in the end for a n period example is given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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We again assume that all cake is gone in the last period which means that CS(n+1)=0. If we plug this into the previous equation then we get the expression for the terminal condition in period one 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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We can now express this equation in C(1) terms only. 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 JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

JSFHJSFH JSFH If we plug the all the rewritten expressions into the previous equation we get JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY4LUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUSNDU0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiRRIjFGJ0YvLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUrZXhlY3V0YWJsZUdGOC8lKXJlYWRvbmx5R0ZQLyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RidGL0YvLUYzNi1RKCZtaW51cztGJ0YvRjZGOUY7Rj1GP0ZBRkMvRkZRLDAuMjIyMjIyMmVtRicvRklGY28tRks2JVEiQ0YnRk5GUUZTRl9vLUklbXN1cEdGJDYlLUZLNiVRIkJGJ0ZORlEtRiM2JEZYRi8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYzNi1RJyZzZG90O0YnRi9GNkY5RjtGPUY/RkFGQy9GRlEmMC4wZW1GJy9GSUZncEZlby1GVDYkRl5wRi9GX28tRmlvNiVGW3AtRiM2JC1GLDYkUSIyRidGL0YvRmBwRmNwRmVvRmlwRl9vLUYjNictRmlvNiVGW3AtRiM2JC1GSzYlUSJuRidGTkZRRi9GYHBGY3BGZW9GaXBGLy1GSzYjUSFGJ0ZobkYv

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyIiISwsLUkjQ1NHNiI2IyIiIkYpLUkiQ0dGJ0YoISIiKiZJIkJHRidGKUYqRilGLComKUYuIiIjRilGKkYpRiwqJilGLkkibkdGJ0YpRipGKUYs

which can be written as 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JI0NTRzYiNiMiIiIsKi1JIkNHRiVGJkYnKiZJIkJHRiVGJ0YpRidGJyomKUYsIiIjRidGKUYnRicqJilGLEkibkdGJUYnRilGJ0Yn

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JI0NTRzYiNiMiIiIqJiwqRidGJ0kiQkdGJUYnKiQpRioiIiNGJ0YnKUYqSSJuR0YlRidGJy1JIkNHRiVGJkYn

We now note that ########################################## if we assume that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEkMC45RidGOS1GNjYtUSI6RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS8lK2V4ZWN1dGFibGVHRj1GOQ== then the sum 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0kiWEc2IiQiJ2YmbyUhIiY=

can be approximated by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0kiWEc2IiQiKysrZiZvJSEiKg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== ########################################## The sum of 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 can therefor be approximated by the expression 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

KiYsJiIiIkYkKUkiQkc2IiwmSSJuR0YnRiRGJEYkISIiRiQsJkYkRiRGJkYqRio=

which means that we can write the above expression as 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JI0NTRzYiNiMiIiIqJjcjKiYsJkYnRicpSSJCR0YlLCZJIm5HRiVGJ0YnRichIiJGJywmRidGJ0YtRjBGMEYnLUkiQ0dGJUYmRic=

which gives us 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JIkNHNiI2IyIiIiomLUkjQ1NHRiVGJkYnNyMqJiwmRidGJylJIkJHRiUsJkkibkdGJUYnRidGJyEiIkYnLCZGJ0YnRi9GMkYyRjI=

which means that that consumption is given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JIkNHNiI2IyIiIiomLUkjQ1NHRiVGJkYnNyMqJiwmRidGJ0kiQkdGJSEiIkYnLCZGJ0YnKUYuLCZJIm5HRiVGJ0YnRidGL0YvRic=

We now note that ########################################## when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZzcmFycjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmaW5maW47RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUY5 then 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 We can show this by noting that 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

########################################## This means that our previous equation is reduced to 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JIkNHNiI2IyIiIiomLUkjQ1NHRiVGJkYnNyMkRichIiJGJw==

We can now visualize the consumtion and cake size over time JSFH JSFH JSFH Consumption vector 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

%;

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print(); # input placeholder

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Legal Notice: The copyright for this application is owned by the authors. Neither Maplesoft nor the authors are responsible for any errors contained within and are not liable for any damages resulting from the use of this material.. This application is intended for non-commercial, non-profit use only. Contact the authors for permission if you wish to use this application in for-profit activities. JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=