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::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::J:<5:\"\{\}-%%ROOTG6'-%)BOUNDS_XG6#$"#]!""-%)BOUNDS_YG6#$""!F*-%-BOUNDS_WIDTHG6#$"%+!)F*-%.BOUNDS_HEIGHTG6#$"$!pF*-%)CHILDRENG6" LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Advanced Microeconomics: Slutsky Equation, Roy\342\200\231s Identity and Shephard's Lemma JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The following was implemented in Maple by Marcus Davidsson (2008) davidsson_marcus@hotmail.com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 1) Marshallian Demand LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== JSFH We assume that we have two goods: good one and good two. We now note that our utility function U is a function of how much we consume of good one LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and how much we consume of good two LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ . Where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy are utility elasticities for good one and good two. Our utility function is therefore given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

KiYpJkkicUc2IjYjIiIiSSZhbHBoYUdGJkYoKSZGJTYjIiIjSSViZXRhR0YmRig=

We now note that the price of good one is denoted by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJ0Y+ and the price of good two is denoted by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ Our budget constraint is therefore given by the equation below which says that our income LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5MjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy must be equal to the cost of spending from good one (p1*q1) plus the cost of spending from good two (p2*q2) 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0klSW90YUc2IiwmKiYmSSJwR0YkNiMiIiJGKiZJInFHRiRGKUYqRioqJiZGKDYjIiIjRiomRixGL0YqRio=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH In order to set up the Lagrange we have to write our budget constraint in an equal to zero form. The above equation can be written as 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LywmSSVJb3RhRzYiIiIiKiYmSSJwR0YlNiNGJkYmJkkicUdGJUYqRiYhIiIqJiZGKTYjIiIjRiYmRixGMEYm

or as 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LChJJUlvdGFHNiIiIiIqJiZJInBHRiQ2I0YlRiUmSSJxR0YkRilGJSEiIiomJkYoNiMiIiNGJSZGK0YvRiVGLA==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

Our Lagrangian where our Lagrange multiplier is given by LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEmJlBzaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn is therefore given by JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCYqJikmSSJxRzYiNiMiIiJJJmFscGhhR0YnRikpJkYmNiMiIiNJJWJldGFHRidGKUYpKiZJJFBzaUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJ0YpLChJJUlvdGFHRidGKSomJkkicEdGJ0YoRilGJUYpISIiKiYmRjlGLUYpRixGKUY6RilGKQ==

We now maximize our utility function subject to the budget constraint by setting the first order conditions of the Lagrangian with respect to the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJ0Y+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIn5GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlhGPg==equal to zero in order to solve for the Marshallian demands JSFH The first order conditions (foc's) are given by JSFH JSFH Foc 1 JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LywmKiopJkkicUc2IjYjIiIiSSZhbHBoYUdGKEYqRitGKkYmISIiKSZGJzYjIiIjSSViZXRhR0YoRipGKiomSSRQc2lHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRihGKiZJInBHRihGKUYqRiwiIiE=

We add 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 on both sides to get the equality 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyoqKSZJInFHNiI2IyIiIkkmYWxwaGFHRidGKUYqRilGJSEiIikmRiY2IyIiI0klYmV0YUdGJ0YpKiZJJFBzaUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJ0YpJkkicEdGJ0YoRik=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJInBHNiI2IyIiIiosKSZJInFHRiVGJkkmYWxwaGFHRiVGJ0YsRidGKiEiIikmRis2IyIiI0klYmV0YUdGJUYnSSRQc2lHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiVGLQ==

JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Foc 2 JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LywmKiopJkkicUc2IjYjIiIiSSZhbHBoYUdGKEYqKSZGJzYjIiIjSSViZXRhR0YoRipGMEYqRi0hIiJGKiomSSRQc2lHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRihGKiZJInBHRihGLkYqRjEiIiE=

We add 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 on both sides to get the equality 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyoqKSZJInFHNiI2IyIiIkkmYWxwaGFHRidGKSkmRiY2IyIiI0klYmV0YUdGJ0YpRi9GKUYsISIiKiZJJFBzaUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJ0YpJkkicEdGJ0YtRik=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJInBHNiI2IyIiIyosKSZJInFHRiU2IyIiIkkmYWxwaGFHRiVGLSkmRitGJkklYmV0YUdGJUYtRjFGLUYwISIiSSRQc2lHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiVGMg==

Foc 3 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVElZGlmZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRIkxGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRJyYjOTM2O0YnL0YwRkVGQS8lK2V4ZWN1dGFibGVHRkVGQUZBRmZuRkE=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LChJJUlvdGFHNiIiIiIqJiZJInBHRiQ2I0YlRiUmSSJxR0YkRilGJSEiIiomJkYoNiMiIiNGJSZGK0YvRiVGLA==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRKGlzb2xhdGVGJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2KC1GLDYlUSIlRidGL0YyLUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRicvRk5RLDAuMzMzMzMzM2VtRictRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSEZobi9GTkZpbi1GLDYlUScmIzkyMTtGJy9GMEY9RjkvJStleGVjdXRhYmxlR0Y9RjlGOUZkb0Y5

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0klSW90YUc2IiwmKiYmSSJwR0YkNiMiIiJGKiZJInFHRiRGKUYqRioqJiZGKDYjIiIjRiomRixGL0YqRio=

Which is our initial budget constraint JSFH Manipulation of Equations We now note that we can divide A by B and still leave the mathematical logic intact. So we get JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyomJkkicEc2IjYjIiIiRigmRiU2IyIiIyEiIioqSSZhbHBoYUdGJkYoJkkicUdGJkYnRixJJWJldGFHRiZGLCZGMEYqRig=

cross multiply 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyooJkkicEc2IjYjIiIiRigmSSJxR0YmRidGKEklYmV0YUdGJkYoKigmRiU2IyIiI0YoSSZhbHBoYUdGJkYoJkYqRi5GKA==

Which means that we can write the previous equation as JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyomJkkicEc2IjYjIiIiRigmSSJxR0YmRidGKCoqJkYlNiMiIiNGKEkmYWxwaGFHRiZGKCZGKkYtRihJJWJldGFHRiYhIiI=

or as 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyomJkkicEc2IjYjIiIjIiIiJkkicUdGJkYnRikqKiZGJTYjRilGKSZGK0YuRilJJWJldGFHRiZGKUkmYWxwaGFHRiYhIiI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We know since previously that C is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0klSW90YUc2IiwmKiYmSSJwR0YkNiMiIiJGKiZJInFHRiRGKUYqRioqJiZGKDYjIiIjRiomRixGL0YqRio=

If we plug in AA in C we get 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0klSW90YUc2IiwmKiomSSJwR0YkNiMiIiMiIiJJJmFscGhhR0YkRismSSJxR0YkRilGK0klYmV0YUdGJCEiIkYrKiZGJ0YrRi1GK0Yr

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpc2ltcGxpZnlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSIlRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkBGPUZA

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0klSW90YUc2IioqJkkicEdGJDYjIiIjIiIiJkkicUdGJEYoRiosJkkmYWxwaGFHRiRGKkklYmV0YUdGJEYqRipGLyEiIg==

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJInFHNiI2IyIiIyoqSSVJb3RhR0YlIiIiJkkicEdGJUYmISIiLCZJJmFscGhhR0YlRipJJWJldGFHRiVGKkYtRjBGKg==

This is the expression for the Marshallian demand for good two. If we plug in BB in C we get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVElc3Vic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiNRIUYnLUYjNiotRiM2Ki1JJW1zdWJHRiQ2JS1GLDYlUSJwRidGL0YyLUYjNigtSSNtbkdGJDYkUSIyRicvRjNRJ25vcm1hbEYnLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSlyZWFkb25seUdGMS8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnRk0vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJ0ZNLyUmZmVuY2VHRlQvJSpzZXBhcmF0b3JHRlQvJSlzdHJldGNoeUdGVC8lKnN5bW1ldHJpY0dGVC8lKGxhcmdlb3BHRlQvJS5tb3ZhYmxlbGltaXRzR0ZULyUnYWNjZW50R0ZULyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGW3AtRkI2JS1GLDYlUSJxRidGL0YyRkdGWkZPRlJGVUZXRk0tRmhuNi1RIj1GJ0ZNRltvRl1vRl9vRmFvRmNvRmVvRmdvL0Zqb1EsMC4yNzc3Nzc4ZW1GJy9GXXBGZ3AtSSZtZnJhY0dGJDYoLUYjNiwtRkI2JUZELUYjNigtRko2JFEiMUYnRk1GT0ZSRlVGV0ZNRlpGZ24tRkI2JUZgcEZgcUZaRmduLUYsNiVRJyZiZXRhO0YnL0YwRlRGTUZPRlJGVUZXRk0tRiM2KC1GLDYlUScmIzk0NTtGJ0ZqcUZNRk9GUkZVRldGTS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZXIvJSliZXZlbGxlZEdGVEZPRlJGVUZXRk0tRmhuNi1RIixGJ0ZNRltvL0Zeb0YxRl9vRmFvRmNvRmVvRmdvRmlvL0ZdcFEsMC4zMzMzMzMzZW1GJy1GaG42LVEifkYnRk1GW29GXW9GX29GYW9GY29GZW9GZ29GaW9GXHAtRiw2JVEiQ0YnRi9GMkZNRk1GUkZN

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0klSW90YUc2IiwmKiYmSSJwR0YkNiMiIiJGKiZJInFHRiRGKUYqRioqKkYnRipGK0YqSSViZXRhR0YkRipJJmFscGhhR0YkISIiRio=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpc2ltcGxpZnlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSIlRidGL0YyL0YzUSdub3JtYWxGJ0Y9LyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGPQ==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0klSW90YUc2IioqJkkicEdGJDYjIiIiRikmSSJxR0YkRihGKSwmSSZhbHBoYUdGJEYpSSViZXRhR0YkRilGKUYtISIi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlgtRi82JVEmc29sdmVGJ0YyRjUtSShtZmVuY2VkR0YkNiQtRiM2Ji1GLzYlUSIlRidGMkY1LUZENi1RIixGJ0Y+RkcvRktGNEZMRk5GUEZSRlQvRldRJjAuMGVtRicvRlpRLDAuMzMzMzMzM2VtRictRiw2JUYuLUYjNiRGOkY+RkBGPkY+LyUrZXhlY3V0YWJsZUdGSUY+

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJInFHNiI2IyIiIioqSSVJb3RhR0YlRicmSSJwR0YlRiYhIiIsJkkmYWxwaGFHRiVGJ0klYmV0YUdGJUYnRixGLkYn

This is the expression for the Marshallian demand for good one. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We can now derive our indirect utility function for this Marshallian demand example. We plugin in the expressions for the Marshallian demand for good one and good two into our expenditure function 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

KiYpKipJJUlvdGFHNiIiIiJJJmFscGhhR0YmRicsJiZJInBHRiY2I0YnRicmSSJ0R0YmRixGJyEiIiwmRihGJ0klYmV0YUdGJkYnRi9GKEYnKSoqRiVGJ0YxRicmRis2IyIiI0YvRjBGL0YxRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= This is the expression for the indirect utility function for the Marshallian Demand. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We can now plot these two Marshallian demand functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIn5GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlhGPg==as follows. JSFH JSFH We assume that income is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5MjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRJDEwMEYnRjJGMg== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NDU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRJDAuM0YnRjJGMg== which means that 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 since we have constant return to scale JSFHJSFH JSFH The Marshallian demand for good one is given by JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

%;

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We can see that when the price for good two LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRidGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRkBGQ0Y+ increases the Marshallian demand for good two 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 decreases JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 2) Hicksian Demand JSFHJSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== JSFH JSFH We assume that we have two goods: good one and good two. We now note that our utility function U is a function of how much we consume of good one LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and how much we consume of good two LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ . Where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy are utility elasticities for good one and good two. Our utility function is therefore given by 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print(); # input placeholder

KiYpJkkicUc2IjYjIiIiSSZhbHBoYUdGJkYoKSZGJTYjIiIjSSViZXRhR0YmRig=

We now note that the price of good one is denoted by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJ0Y+ and the price of good two is denoted by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ Our budget constraint is therefore given by the equation below which says that our income LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5MjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy must be equal to the cost of spending from good one (p1*q1) plus the cost of spending from good two (p2*q2) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JVEnJiM5MjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJW1zdWJHRiQ2JS1GLDYlUSJwRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictRiM2JC1JI21uR0YkNiRRIjFGJ0YyRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GNjYtUScmc2RvdDtGJ0YyRjlGO0Y9Rj9GQUZDRkUvRkhRJjAuMGVtRicvRktGXW8tRk02JS1GIzYkLUZNNiUtRiw2JVEicUYnRlJGVEZWRmZuRjItRiM2JC1GLDYjUSFGJ0YyRmZuLUY2Ni1RIitGJ0YyRjlGO0Y9Rj9GQUZDRkUvRkhRLDAuMjIyMjIyMmVtRicvRktGYXAtRk02JUZPLUYjNiQtRlk2JFEiMkYnRjJGMkZmbkZpbi1GTTYlRmVvRmVwRmZuLyUrZXhlY3V0YWJsZUdGMUYy

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0klSW90YUc2IiwmKiYmSSJwR0YkNiMiIiJGKiZJInFHRiRGKUYqRioqJiZGKDYjIiIjRiomRixGL0YqRio=

We now note that instead of maximizing utility for a given income as in the previous example with Marshallian demands we will now minimize the amount of income spent for given a target utility level let say LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. This gives us the Hicksian demands for good one and good two . JSFH JSFH Our Lagrangian where our Lagrange multiplier is given by LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEmJlBzaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn is therefore given by JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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The first order conditions (foc's) are given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Foc 1 JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJInBHNiI2IyIiIiosSSRQc2lHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiVGJykmSSJxR0YlRiZJJmFscGhhR0YlRidGMEYnRi4hIiIpJkYvNiMiIiNJJWJldGFHRiVGJw==

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print(); # input placeholder

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JSFH Foc 2 JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVElZGlmZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRIkxGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUklbXN1YkdGJDYlLUYsNiVRInFGJ0YvRjItRiM2JS1JI21uR0YkNiRRIjJGJ0ZBRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLyUrZXhlY3V0YWJsZUdGRUZBRkEtRj42LVEiPUYnRkFGQy9GR0ZFRkhGSkZMRk5GUC9GU1EsMC4yNzc3Nzc4ZW1GJy9GVkZoby1GW282JFEiMEYnRkFGYW9GQQ==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJInBHNiI2IyIiIyosSSRQc2lHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiUiIiIpJkkicUdGJTYjRi1JJmFscGhhR0YlRi0pJkYwRiZJJWJldGFHRiVGLUY1Ri1GNCEiIg==

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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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Which is our initial budget constraint LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Manipulation of Equations JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEqdW5wcm90ZWN0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEiREYnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRj5GP0Y/LUkjbW9HRiQ2LVEiOkYnRj8vJSZmZW5jZUdGPi8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXRkFGPw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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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

LyZJInFHNiI2IyIiIikqLEkidkdGJUYnKSZJInBHRiVGJkklYmV0YUdGJSEiIilJJmFscGhhR0YlLCRGLkYvRi8pJkYtNiMiIiNGMkYvKUYuRi5GLyokLCZGMUYnRi5GJ0Yv

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= This is the expression for the Hicksian demand for good one LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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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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This is the expression for the Hicksian demand for good two. We can now derive our indirect utility function for this Hicksian example. We plugin in the expressions for the Hicksian demand for good one and good two into our expenditure function 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 JSFH 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print(); # input placeholder

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print(); # input placeholder

L0kidkc2IiosKSomSSVJb3RhR0YkIiIiLCYmSSJwR0YkNiNGKUYpKiYmRiw2IyIiI0YpKSoqKUYrLCZJJmFscGhhR0YkRilJJWJldGFHRiRGKUYpKUY2LCZGNiEiIkY3RjpGKSlGL0Y5RikpRjdGNUYpKiRGNUY6RilGKUY6RjVGKSlGK0Y3RikpRjYsJEY3RjpGKSlGL0ZARikpRjdGN0Yp

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print(); # input placeholder

L0kidkc2IiouKUklSW90YUdGJCwmSSZhbHBoYUdGJCIiIkklYmV0YUdGJEYqRiopRilGKUYqKSZJInBHRiQ2I0YqLCRGKSEiIkYqKUYoLCZGKUYyRitGMkYqKSZGLzYjIiIjLCRGK0YyRiopRitGK0Yq

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= This is the expression for the indirect utility function for the Hicksian Demand. We can now plot these two Hicksian demand functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We assume that income is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRJDEwMEYnRjlGOQ== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NDU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRJDAuM0YnRjJGMg== which means that 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 since we have constant return to scale LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The Hicksian demand for good one is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZCLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGVS8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GNjYtUSJ+RidGOUY7L0Y/Rj1GQEZCRkRGRkZIRkovRk5GTC1GLDYlUSJ2RidGL0YyLUY2Ni1RKiZjb2xvbmVxO0YnRjlGO0Zbb0ZARkJGREZGRkgvRktGT0ZNLUkjbW5HRiQ2JFEjMjBGJ0Y5LUY2Ni1RIjpGJ0Y5RjtGW29GQEZCRkRGRkZIRmNvRk1GaG5GaG4tRiw2JVEnJiM5NDU7RicvRjBGPUY5RmBvLUZlbzYkUSQwLjNGJ0Y5RmhvRmhuRmhuLUYsNiVRJyYjOTQ2O0YnRl5wRjlGYG8tRmVvNiRRJDAuN0YnRjlGaG9GaG5GaG4tSSVtc3ViR0YkNiUtRiw2JVEicEYnRi9GMi1GIzYkLUZlbzYkUSIyRidGOUY5LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGYG8tRmVvNiRRIjVGJ0Y5RmhvRlAtRlE2JkZTRlZGWS9GZm5RJWF1dG9GJy1GLDYlUSVwbG90RidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNj0tSSVtc3VwR0YkNiUtRmFyNiQtSSZtZnJhY0dGJDYoLUYjNiRGXW9GOS1GIzYqLUkobXN1YnN1cEdGJDYnRltxLUYjNiQtRmVvNiRRIjFGJ0Y5RjktRiw2JVEnJmJldGE7RidGXnBGOS8lMXN1cGVyc2NyaXB0c2hpZnRHRmVxRmNxLUY2Ni1RMSZJbnZpc2libGVUaW1lcztGJ0Y5RjtGW29GQEZCRkRGRkZIRkpGXG8tRmZyNiUtRiw2JVEoJmFscGhhO0YnRl5wRjktRiM2JS1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGW29GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORlx1RmlzRjlGXHRGXnQtRmJzNidGW3FGXnEtRmFyNiRGZnRGOUZcdEZjcUZedC1GZnI2JUZpc0Zpc0ZcdEY5LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZpdS8lKWJldmVsbGVkR0Y9RjktRltzNihGZnMtRiM2JkZjdC1GNjYtUSIrRidGOUY7RltvRkBGQkZERkZGSEZbdUZddUZpc0Y5RmR1Rmd1Rmp1Rlx2Rlx0RmhuRmhuRmhuLUY2Ni1RIixGJ0Y5RjtGPkZARkJGREZGRkhGSi9GTlEsMC4zMzMzMzMzZW1GJ0Zobi1GaXA2JUZbcUZkc0ZjcS1GNjYtUSI9RidGOUY7RltvRkBGQkZERkZGSEZjb0ZNRmZzLUY2Ni1RIy4uRidGOUY7RltvRkBGQkZERkZGSEZbdUZcby1GZW82JFEjMTBGJ0Y5RmV2RmhuLUYsNiVRJ2xhYmVsc0YnRi9GMkZcdy1GYXI2Ji1GIzYoRmp2RmV2LUYsNiVRIkhGJ0YvRjJGaG4tRmlwNiUtRiw2JVEicUYnRi9GMkZkc0ZjcUY5RjkvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGZXZGaG4tRiw2JVEmY29sb3JGJ0YvRjJGXHctRiw2JVEkcmVkRidGL0YyRmV2RmhuLUYsNiVRKnRoaWNrbmVzc0YnRi9GMkZcdy1GZW82JFEiM0YnRjlGOUY5LyUrZXhlY3V0YWJsZUdGPUY5

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

%;

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

We can see that when the price for good one LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ increases the Hicksian demand for good one LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1JJW1zdWJHRiQ2JS1GLDYlUSJxRidGL0YyLUYjNiUtSSNtbkdGJDYkUSIxRidGOUYvRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y5 decreases LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFH JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The Hicksian demand for good two is given by 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We can see that when the price for good two LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ increases the Hicksian demand for good two LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1JJW1zdWJHRiQ2JS1GLDYlUSJxRidGL0YyLUYjNiUtSSNtbkdGJDYkUSIyRidGOUYvRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y5 decreases LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 3) The Relationship Between Marshallian and Hicksian Demand: The Slutsky Equation We start by comparing the two demand curves for the Marshallian and Hicksiand demand for the two goods For good one we get: 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We are now going to discuss the relationship between the Marshallian demand and the Hicksian demand. through the Slutsky equation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The Slutsky equation relates the changes in Marshallian demand to changes in Hicksian demand. The general formula for Slutsky equation is given by 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 which says that the partial derivative of the marshillian demand for good i with respect to the price of good i is equal to partial derivative of the Hicksian demand for good i with respect to the price of good i minus the partial derivative of the marshillian demand for good i with respect to income multiplied by marshillian demand for good i. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= For good one we get: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC8lK2V4ZWN1dGFibGVHRj1GOQ== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Expression on left hand side of the Slutsky Equation JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCQqKkklSW90YUc2IiIiIkkmYWxwaGFHRiVGJikmSSJwR0YlNiNGJiIiIyEiIiwmRidGJkklYmV0YUdGJUYmRi1GLQ==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The first expression on the right hand side on the Slutsky Equation The expression for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ is given by JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCQqLilJInZHNiIqJCwmSSZhbHBoYUdGJiIiIkklYmV0YUdGJkYqISIiRiopJkkicEdGJjYjRiosJComLCYqJiIiI0YqRitGKkYqRilGKkYqRihGLEYsRiopRikqJkYrRipGKEYsRiopJkYvNiNGNUY3RiopRisqJkYpRipGKEYsRipGKEYsRiw=

We now substitute in the expression for the indirect utility function for the Hicksian demand LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCQqLikqLilJJUlvdGFHNiIsJkkmYWxwaGFHRigiIiJJJWJldGFHRihGK0YrKUYqRipGKykmSSJwR0YoNiNGKywkRiohIiJGKylGKSwmRipGM0YsRjNGKykmRjA2IyIiIywkRixGM0YrKUYsRixGKyokRilGM0YrKUYvLCQqJiwmKiZGOUYrRixGK0YrRipGK0YrRilGM0YzRispRioqJkYsRitGKUYzRispRjdGQ0YrKUYsKiZGKkYrRilGM0YrRilGM0Yz

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCQqLEklSW90YUc2IiIiIkkmYWxwaGFHRiVGJikmSSJwR0YlNiNGJiIiIyEiIiksJkYnRiZJJWJldGFHRiVGJkYsRi1GMEYmRi0=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The second expression on the right hand side on the Slutsky Equation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

KiopSSZhbHBoYUc2IiIiIyIiIikmSSJwR0YlNiNGJ0YmISIiKSwmRiRGJ0klYmV0YUdGJUYnRiZGLEklSW90YUdGJUYn

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We can now finish up by subtracting L2 from L1 and verify that this is indeed the same expression as the partial derivative of the marshillian demand for good one with respect to the price of good one LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpc2ltcGxpZnlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSNMMUYnRi9GMi1JI21vR0YkNi1RKCZtaW51cztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGVC1GLDYlUSNMMkYnRi9GMi8lK2V4ZWN1dGFibGVHRkVGQUZBRlpGQQ==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCQqKkklSW90YUc2IiIiIkkmYWxwaGFHRiVGJiwmRidGJkklYmV0YUdGJUYmISIiKSZJInBHRiU2I0YmIiIjRipGKg==

Which indeed is the same expression we had before for the partial derivative of the marshillian demand for good one with respect to the price of good one which was given by: 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCQqKkklSW90YUc2IiIiIkkmYWxwaGFHRiVGJiwmRidGJkklYmV0YUdGJUYmISIiKSZJInBHRiU2I0YmIiIjRipGKg==

JSFH For good two we get: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH Expression on left hand side of the Slutsky Equation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCQqKkklSW90YUc2IiIiIkklYmV0YUdGJUYmKSZJInBHRiU2IyIiI0YsISIiLCZJJmFscGhhR0YlRiZGJ0YmRi1GLQ==

JSFH The first expression on the right hand side on the Slutsky Equation The expression for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCQqLilJInZHNiIqJCwmSSZhbHBoYUdGJiIiIkklYmV0YUdGJkYqISIiRiopJkkicEdGJjYjRioqJkYpRipGKEYsRiopRikqJkYrRipGKEYsRiopJkYvNiMiIiMsJComLCYqJkY3RipGKUYqRipGK0YqRipGKEYsRixGKilGK0YxRipGKEYsRiw=

We now substitute in the expression for the indirect utility function for the Hicksian demand JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCQqLikqLilJJUlvdGFHNiIsJkkmYWxwaGFHRigiIiJJJWJldGFHRihGK0YrKUYqRipGKykmSSJwR0YoNiNGKywkRiohIiJGKylGKSwmRipGM0YsRjNGKykmRjA2IyIiIywkRixGM0YrKUYsRixGKyokRilGM0YrKUYvKiZGKkYrRilGM0YrKUYqKiZGLEYrRilGM0YrKUY3LCQqJiwmKiZGOUYrRipGK0YrRixGK0YrRilGM0YzRispRixGPkYrRilGM0Yz

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCQqLEklSW90YUc2IiIiIkkmYWxwaGFHRiVGJiksJkYnRiZJJWJldGFHRiVGJiIiIyEiIikmSSJwR0YlNiNGK0YrRixGKkYmRiw=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The second expression on the right hand side on the Slutsky Equation JSFH JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

KiopSSViZXRhRzYiIiIjIiIiKSZJInBHRiU2I0YmRiYhIiIpLCZJJmFscGhhR0YlRidGJEYnRiZGLEklSW90YUdGJUYn

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH We can now finish up by subtracting L2 from L1 and verify that this is indeed the same expression as the partial derivative of the marshillian demand for good one with respect to the price of good one JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpc2ltcGxpZnlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSNMMUYnRi9GMi1JI21vR0YkNi1RKCZtaW51cztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGVC1GLDYlUSNMMkYnRi9GMi8lK2V4ZWN1dGFibGVHRkVGQUZBRlpGQQ==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCQqKkklSW90YUc2IiIiIkklYmV0YUdGJUYmLCZJJmFscGhhR0YlRiZGJ0YmISIiKSZJInBHRiU2IyIiI0YvRipGKg==

Which indeed is the same expression we had before for the partial derivative of the marshillian demand for good two with respect to the price of good two which was given by: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCQqKkklSW90YUc2IiIiIkklYmV0YUdGJUYmLCZJJmFscGhhR0YlRiZGJ0YmISIiKSZJInBHRiU2IyIiI0YvRipGKg==

JSFHJSFH JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 4) Roy\342\200\231s Identity and Marshallian Demands LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJStleGVjdXRhYmxlR0Y9Rjk= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We can also estimate the Marshallian demands by using Roys Identity which starts from the indirect utility function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5for the Marshallian demand and derives the corresponding Marshallian demand functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+. The general formula for Roys Identity is given byJSFH JSFH JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

which says that the Marshillian demand for good i is equal to the partial derivative of the indirect utility function for the Marshallian demand with respect to the price of good i divided by the partial derivative of the indirect utility function for the Marshallian demand with respect to the income. We know since previously that our indirect utility function for the Marshallian demand is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

KiYpKipJJUlvdGFHNiIiIiJJJmFscGhhR0YmRicmSSJwR0YmNiNGJyEiIiwmRihGJ0klYmV0YUdGJkYnRixGKEYnKSoqRiVGJ0YuRicmRio2IyIiI0YsRi1GLEYuRic=

For good one we get: This means that according to Roys Identity the marsihilian demand for good one is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0kjcTFHNiIqLCkqKkklSW90YUdGJCIiIkkmYWxwaGFHRiRGKSZJInBHRiQ2I0YpISIiLCZGKkYpSSViZXRhR0YkRilGLkYqRilGKkYpRitGLikqKkYoRilGMEYpJkYsNiMiIiNGLkYvRi5GMEYpLCYqKkYmRilGKkYpRihGLkYxRilGKSoqRiZGKUYxRilGMEYpRihGLkYpRi4=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpc2ltcGxpZnlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSIlRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkBGPUZA

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0kjcTFHNiIqKkklSW90YUdGJCIiIkkmYWxwaGFHRiRGJyZJInBHRiQ2I0YnISIiLCZGKEYnSSViZXRhR0YkRidGLA==

Which is the same expressions that we had before for the marsihilian demand for good one which was given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJInFHNiI2IyIiIioqSSVJb3RhR0YlRidJJmFscGhhR0YlRicmSSJwR0YlRiYhIiIsJkYqRidJJWJldGFHRiVGJ0Yt

For good two we get: This means that according to Roys Identity the marsihilian demand for good two is given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0kjcTJHNiIqLCkqKkklSW90YUdGJCIiIkkmYWxwaGFHRiRGKSZJInBHRiQ2I0YpISIiLCZGKkYpSSViZXRhR0YkRilGLkYqRikpKipGKEYpRjBGKSZGLDYjIiIjRi5GL0YuRjBGKUYwRilGM0YuLCYqKkYmRilGKkYpRihGLkYxRilGKSoqRiZGKUYxRilGMEYpRihGLkYpRi4=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpc2ltcGxpZnlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSIlRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkBGPUZA

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0kjcTJHNiIqKkklSW90YUdGJCIiIkklYmV0YUdGJEYnJkkicEdGJDYjIiIjISIiLCZJJmFscGhhR0YkRidGKEYnRi0=

Which is the same expressions that we had before for the marsihilian demand for good two which was given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJInFHNiI2IyIiIyoqSSVJb3RhR0YlIiIiSSViZXRhR0YlRiomSSJwR0YlRiYhIiIsJkkmYWxwaGFHRiVGKkYrRipGLg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 5) Roy\342\200\231s Identity and the Lagrange Multiplier Note that the expression in the denominator for Roy's Identity 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 where again Roys Identity is given by 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 is equal to the Lagrange multiplier in the consumer optimization example. The Lagrange multiplier and the expression 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 is the consumers Marginal Utility of Income which measure how much extra utility the consumer get if we increase the constraint in form of income with one unit LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEiI0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVFIIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRi8=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We can first prove that the Lagrange multiplier measure how much extra utility we get if we increase income with one unit Step-1) The maximized utility and optimal quantities of good one and two when \316\231=1000 are given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC8lK2V4ZWN1dGFibGVHRj1GOQ== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 print(); # input placeholder LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEjb2JGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JKG1zdWJzdXBHRiQ2Jy1GLDYlUSJxRidGL0YyLUYjNigtSSNtbkdGJDYkUSIxRidGOS8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJy8lK2V4ZWN1dGFibGVHRj0vJSlyZWFkb25seUdGMS8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnRjktRiw2JVEoJmFscGhhO0YnL0YwRj1GOS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGZW8tRjY2LVExJkludmlzaWJsZVRpbWVzO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZccC1GUDYnRlItRiM2KC1GWDYkUSIyRidGOUZlbkZobkZqbkZcb0Y5LUYsNiVRJyZiZXRhO0YnRmJvRjlGY29GZm8tRjY2LVEiOkYnRjlGO0Y+RkBGQkZERkZGSEZKRk1GaG5GOQ== print(); # input placeholder 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 print(); # input placeholder LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRLU9wdGltaXphdGlvbkYnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZALUkjbW9HRiQ2LVEiOkYnRkAvJSZmZW5jZUdGPy8lKnNlcGFyYXRvckdGPy8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZWRj1GQA== 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyQkIjNRWiZcYFI+TlMjISM6NyQvJkkicUc2IjYjIiIiJCIzdSlRLysrKysrIkYlLyZGKTYjIiIjJCIzUTtNKioqKioqKioqXCRGJQ==

The maximize utility when \316\231=1000 is therefore given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJInVHNiI2IyIiIiQiM1FaJlxgUj5OUyMhIzo=

and the optimal quantities of good one and two are given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJInFHNiI2IyIiIiQiM3UpUS8rKysrKyIhIzo=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSS8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZYLUY7NiRRNDM0OS45OTk5OTk5OTkzNDE2MzhGJ0Y+Rj4=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJInFHNiI2IyIiIyQiM1E7TSoqKioqKioqKlwkISM6

Step-2) For our consumer optimization example the Lagrange was given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC8lK2V4ZWN1dGFibGVHRj1GOQ== 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCYqJikmSSJxRzYiNiMiIiJJJmFscGhhR0YnRikpJkYmNiMiIiNJJWJldGFHRidGKUYpKiZJJFBzaUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJ0YpLChJJUlvdGFHRidGKSomJkkicEdGJ0YoRilGJUYpISIiKiYmRjlGLUYpRixGKUY6RilGKQ==

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LywmKiopJkkicUc2IjYjIiIiSSZhbHBoYUdGKEYqRitGKkYmISIiKSZGJzYjIiIjSSViZXRhR0YoRipGKiomSSRQc2lHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRihGKiZJInBHRihGKUYqRiwiIiE=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LywmLUkjaHNHNiI2Iy8sJioqKSZJInFHRiY2IyIiIkkmYWxwaGFHRiZGL0YwRi9GLCEiIikmRi02IyIiI0klYmV0YUdGJkYvRi8qJkkkUHNpRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmRi8mSSJwR0YmRi5GL0YxIiIhRi9GN0YvRjc=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0kkUHNpRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiKiwpJkkicUdGJzYjIiIiSSZhbHBoYUdGJ0YtRi5GLSkmRis2IyIiI0klYmV0YUdGJ0YtJkkicEdGJ0YsISIiRipGNg==

if we now plug in the optimal values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and the parameter values we get 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0kkUHNpRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiJCIrJ1I+TlMjISM1

Which means that according to our theory the difference in maximized utility when \316\231=1000 and when \316\231=1001 should be 0.2403 Step-3) We now increase income with one unit from 1000 to 1001 so we get JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY9LUkjbWlHRiQ2JVEnJiM5NDU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFEkMC4zRidGMi1GNjYtUSI6RidGMkY5RjtGPUY/RkFGQ0ZFRkdGSi1GNjYtUSJ+RidGMkY5RjtGPUY/RkFGQ0ZFL0ZIUSYwLjBlbUYnL0ZLRldGUy1GLDYlUScmIzk0NjtGJ0YvRjJGNS1GTTYkUSQwLjdGJ0YyRlBGUy1JJW1zdWJHRiQ2JS1GLDYlUSJwRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictRiM2JC1GTTYkUSIxRidGMkYyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNS1GTTYkUSIzRidGMkZQRlMtRmpuNiVGXG8tRiM2JC1GTTYkUSIyRidGMkYyRmhvRjVGYnBGUEZTLUYsNiVRJyYjOTIxO0YnRi9GMkY1LUZNNiRRJTEwMDFGJ0YyRlBGU0Yy print(); # input placeholder 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 print(); # input placeholder 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 print(); # input placeholder LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRLU9wdGltaXphdGlvbkYnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZALUkjbW9HRiQ2LVEiOkYnRkAvJSZmZW5jZUdGPy8lKnNlcGFyYXRvckdGPy8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZWRj1GQA== 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyQkIjM4KCpbdTlII2ZTIyEjOjckLyZJInFHNiI2IyIiIiQiM2UiUS8rKys1KyJGJS8mRik2IyIiIyQiM1VGTSoqKioqKlwuTkYl

The maximize utility when \316\231=1001 is therefore given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJInVHNiI2IyIiIyQiMzgoKlt1OUgjZlMjISM6

Step-4) If we now subtract 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from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EidUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSp1bmRlcmxpbmVHRjcvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNictSSNtbkdGJDYmUSIyRidGMkY4L0Y7USdub3JtYWxGJ0YyRjVGOEY6LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGMkY4RkM=. This gives us LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JFE0MjQwLjU5MjI5MTQ3NDQ4OTcxM0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKCZtaW51cztGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkctRiw2JFE0MjQwLjM1MTkzOTUzNDk1NDczOEYnRi8vJStleGVjdXRhYmxlR0Y4Ri8=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIoP05TIyEiKA==

Which is the same value as the Lagrange multiplier had LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEiI0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVFIIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRi8=JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We can now prove that 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 the same as the Lagrange multiplier in the consumer optimization example LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJStleGVjdXRhYmxlR0Y9Rjk= # We now since previously that our indirect utility function for the Marshallian example is given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

KiYpKipJJUlvdGFHNiIiIiJJJmFscGhhR0YmRicmSSJwR0YmNiNGJyEiIiwmRihGJ0klYmV0YUdGJkYnRixGKEYnKSoqRiVGJ0YuRicmRio2IyIiI0YsRi1GLEYuRic=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCYqKikqKkklSW90YUc2IiIiIkkmYWxwaGFHRidGKCZJInBHRic2I0YoISIiLCZGKUYoSSViZXRhR0YnRihGLUYpRihGKUYoRiZGLSkqKkYmRihGL0YoJkYrNiMiIiNGLUYuRi1GL0YoRigqKkYkRihGMEYoRi9GKEYmRi1GKA==

We again assume that 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 Which means that V is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVErbXVsdGlwbGllckYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIrJ1I+TlMjISM1

Which again is the same value as the Lagrange multiplier had previously LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEiI0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVFIIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRi8=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 6) Shephard's Lemma: Hicksian Demand and the Expenditure Function JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJStleGVjdXRhYmxlR0Y9Rjk= JSFHJSFH We can also estimate the Hicksian demands by using Shephard's lemma which stats that the partial derivative of the expenditure function \316\231 with respect to the price i is equal to the Hicksian demand for good i. The general formula for Shephards lemma is given byLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

For good one we get: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We know since previously that the Hicksian demand for good one is given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJInFHNiI2IyIiIikqLEkidkdGJUYnKSZJInBHRiVGJkklYmV0YUdGJSEiIilJJmFscGhhR0YlLCRGLkYvRi8pJkYtNiMiIiNGMkYvKUYuRi5GLyokLCZGMUYnRi5GJ0Yv

Which can be written as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJInFHNiI2IyIiIiosKUkidkdGJSokLCZJJmFscGhhR0YlRidJJWJldGFHRiVGJyEiIkYnKSZJInBHRiVGJiwkKiZGLkYnRixGL0YvRicpRi1GNEYnKSZGMjYjIiIjRjRGJylGLkYzRic=

This means in order for Shephards lemma to work the partial derivative of the expenditure function \316\231 with respect to the price of good one must be equal to this expression. Our expenditure function is given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0klSW90YUc2IiwmKiYmSSJwR0YkNiMiIiJGKiZJInFHRiRGKUYqRioqJiZGKDYjIiIjRiomRixGL0YqRio=

If we substitute in the expressions for the Hicksian demand for good 1 and good 2 we get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEnJklvdGE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJW1zdWJHRiQ2JS1GLDYlUSJwRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictRiM2Iy1JI21uR0YkNiRRIjFGJ0YyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRjY2LVExJkludmlzaWJsZVRpbWVzO0YnRjJGOUY7Rj1GP0ZBRkNGRS9GSFEmMC4wZW1GJy9GS0Zdby1JJW1zdXBHRiQ2JS1JKG1mZW5jZWRHRiQ2JC1JJm1mcmFjR0YkNigtRiM2Iy1GLDYlUSJ2RidGUkZULUYjNiktSShtc3Vic3VwR0YkNidGT0ZWLUYsNiVRJyZiZXRhO0YnRi9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHRmhuRmZuRmluLUZgbzYlLUYsNiVRKCZhbHBoYTtGJ0YvRjItRiM2JC1GNjYtUSomdW1pbnVzMDtGJ0YyRjlGO0Y9Rj9GQUZDRkUvRkhRLDAuMjIyMjIyMmVtRicvRktGYnFGYnBGZXBGaW4tRmBwNidGTy1GIzYjLUZZNiRRIjJGJ0YyLUZjbzYkRlxxRjJGZXBGZm5GaW4tRmBvNiVGYnBGYnBGZXAvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmRyLyUpYmV2ZWxsZWRHRjFGMi1GZm82KEZYLUYjNiUtRiw2I1EhRictRiM2JUZpcC1GNjYtUSIrRidGMkY5RjtGPUY/RkFGQ0ZFRmFxRmNxRmJwRl1zRl9yRmJyRmVyRmdyRmVwRmJzLUZNNiVGT0ZmcUZmbkZpbi1GYG82JS1GY282JC1GZm82KEZoby1GIzYpLUZgcDYnRk9GVi1GY282JC1GIzYkRl5xRmlwRjJGZXBGZm5GaW4tRmBvNiVGaXBGaXBGZXBGaW4tRmBwNidGT0ZmcUZpcEZlcEZmbkZpbi1GYG82JUZicEZjdEZlcEZfckZickZlckZnckYyRmlyRmVw

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0klSW90YUc2IiwmKiYmSSJwR0YkNiMiIiJGKikqLEkidkdGJEYqKUYnSSViZXRhR0YkISIiKUkmYWxwaGFHRiQsJEYvRjBGMCkmRig2IyIiI0YzRjApRi9GL0YwKiQsJkYyRipGL0YqRjBGKkYqKiZGNUYqKSosRi1GKilGJywkRjJGMEYwKUYyRjJGMClGNUYyRjApRi9GP0YwRjlGKkYq

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The partial derivative of the above function with respect to the price of good one is therefore given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCgpKixJInZHNiIiIiIpJkkicEdGJjYjRidJJWJldGFHRiYhIiIpSSZhbHBoYUdGJiwkRixGLUYtKSZGKjYjIiIjRjBGLSlGLEYsRi0qJCwmRi9GJ0YsRidGLUYnKihGI0YnRjdGLUYsRidGLSosRjJGJykqLEYlRicpRiksJEYvRi1GLSlGL0YvRi0pRjJGL0YtKUYsRj1GLUY2RidGN0YtRilGLUYvRidGJw==

Which can be written as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiPUYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkkvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGWC1GLzYlUSlzaW1wbGlmeUYnRjJGNS1JKG1mZW5jZWRHRiQ2JC1GIzYqLUZENi1RIn5GJ0Y+RkdGSkZMRk5GUEZSRlQvRldRJjAuMGVtRicvRlpGYW8tRi82JVEiJUYnRjJGNS1GRDYtUSIsRidGPkZHL0ZLRjRGTEZORlBGUkZURmBvL0ZaUSwwLjMzMzMzMzNlbUYnRl1vLUZENi1RIidGJ0Y+RkdGSkZMRk5GUEZSRlQvRldRLDAuMTExMTExMWVtRidGYm8tRi82JVEpc3ltYm9saWNGJ0YyRjVGXHBGXW9GPg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJInFHNiI2IyIiIiosKUkidkdGJSokLCZJJmFscGhhR0YlRidJJWJldGFHRiVGJyEiIkYnKSZJInBHRiVGJiwkKiZGLkYnRixGL0YvRicpRi1GNEYnKSZGMjYjIiIjRjRGJylGLkYzRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Which we can see is the same expression that we had previously for the Hicksian demand for good one given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJInFHNiI2IyIiIiosKUkidkdGJSokLCZJJmFscGhhR0YlRidJJWJldGFHRiVGJyEiIkYnKSZJInBHRiVGJiwkKiZGLkYnRixGL0YvRicpRi1GNEYnKSZGMjYjIiIjRjRGJylGLkYzRic=

. Which means that Shephards Lemma seams to work for good one. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= For good two we get: We know since previously that the Hicksian demand for good two is given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJInFHNiI2IyIiIykqLEkidkdGJSIiIikmSSJwR0YlNiNGKywkSSZhbHBoYUdGJSEiIkYyKUYxRjFGMikmRi5GJkYxRjIpSSViZXRhR0YlRjBGMiokLCZGMUYrRjdGK0Yy

Which can be written as JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJInFHNiI2IyIiIyosKUkidkdGJSokLCZJJmFscGhhR0YlIiIiSSViZXRhR0YlRi4hIiJGLikmSSJwR0YlNiNGLiomRi1GLkYsRjBGLilGLSwkRjVGMEYuKSZGM0YmRjdGLilGL0Y1Ri4=

This means in order for Shephards lemma to work the partial derivative of the expenditure function \316\231 with respect to the price of good two must be equal to this expression. Our expenditure function is given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0klSW90YUc2IiwmKiYmSSJwR0YkNiMiIiJGKiZJInFHRiRGKUYqRioqJiZGKDYjIiIjRiomRixGL0YqRio=

If we substitute in the expressions for the Hicksian demand for good 1 and good 2 we get 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0klSW90YUc2IiwmKiYmSSJwR0YkNiMiIiJGKikqLEkidkdGJEYqKUYnSSViZXRhR0YkISIiKUkmYWxwaGFHRiQsJEYvRjBGMCkmRig2IyIiI0YzRjApRi9GL0YwKiQsJkYyRipGL0YqRjBGKkYqKiZGNUYqKSosRi1GKilGJywkRjJGMEYwKUYyRjJGMClGNUYyRjApRi9GP0YwRjlGKkYq

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The partial derivative of the above function with respect to the price of good two is therefore given by 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCgqLCZJInBHNiI2IyIiIkYoKSosSSJ2R0YmRigpRiRJJWJldGFHRiYhIiIpSSZhbHBoYUdGJiwkRi1GLkYuKSZGJTYjIiIjRjFGLilGLUYtRi4qJCwmRjBGKEYtRihGLkYoRjhGLkYtRihGM0YuRigpKixGK0YoKUYkLCRGMEYuRi4pRjBGMEYuKUYzRjBGLilGLUY8Ri5GN0YoKihGOUYoRjhGLkYwRihGLg==

Which can be written as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiPUYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkkvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGWC1GLzYlUSlzaW1wbGlmeUYnRjJGNS1JKG1mZW5jZWRHRiQ2JC1GIzYqLUZENi1RIn5GJ0Y+RkdGSkZMRk5GUEZSRlQvRldRJjAuMGVtRicvRlpGYW8tRi82JVEiJUYnRjJGNS1GRDYtUSIsRidGPkZHL0ZLRjRGTEZORlBGUkZURmBvL0ZaUSwwLjMzMzMzMzNlbUYnRl1vLUZENi1RIidGJ0Y+RkdGSkZMRk5GUEZSRlQvRldRLDAuMTExMTExMWVtRidGYm8tRi82JVEpc3ltYm9saWNGJ0YyRjVGXHBGXW9GPg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJInFHNiI2IyIiIyosKUkidkdGJSokLCZJJmFscGhhR0YlIiIiSSViZXRhR0YlRi4hIiJGLikmSSJwR0YlNiNGLiomRi1GLkYsRjBGLilGLSwkRjVGMEYuKSZGM0YmRjdGLilGL0Y1Ri4=

Which we can see is the same expression that we had previously for the Hicksian demand for good two given by. 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyZJInFHNiI2IyIiIyosKUkidkdGJSokLCZJJmFscGhhR0YlIiIiSSViZXRhR0YlRi4hIiJGLikmSSJwR0YlNiNGLiomRi1GLkYsRjBGLilGLSwkRjVGMEYuKSZGM0YmRjdGLilGL0Y1Ri4=

Which means that Shephards lemma seams to work for good two as well. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2KFEkVGhlRicvJSVzaXplR1EjMjhGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHRjQvJSxtYXRodmFyaWFudEdRLGJvbGQtaXRhbGljRicvJStmb250d2VpZ2h0R1ElYm9sZEYnLUkjbW9HRiQ2MFEifkYnRi9GMi9GOEY8RjovJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1GLDYoUSRFbmRGJ0YvRjJGNUY3RjpGPS1GPjYwUSIhRidGL0YyRkFGOkZCRkVGR0ZJRktGTUZPL0ZSUSwwLjExMTExMTFlbUYnL0ZVRmduRllGWUZZLyUrZXhlY3V0YWJsZUdGRC9GOFEnbm9ybWFsRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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