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PS5 KEY1

# PS5 KEY1 - Econ 100A Intermediate Microeconomics Dr...

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Econ 100A: Intermediate Microeconomics, Dr. Famulari Problem Set #5 I. Cross-Price Effects Question 1. Toni minimizes her expenditures subject to achieving utility level __ U . Suppose Toni’s utility function can be represented as 2 10 xy U = A. What is the Lagrangian for this problem? ) 10 ( 2 xy U y p x p L y x - + + = λ B. What are the first order conditions? 0 10 ) 3 ( 0 20 ) 2 ( 0 10 ) 1 ( 2 2 = - = = - = = - = xy U L xy p y L y p x L y x λ λ λ C. Solve for Toni’s (income) compensated demand functions X C (Px, Py, __ U ) and Y C (Px, Py, __ U ) F.O.C (1) implies 2 10 y p x λ = F.O.C (2) implies xy p y 20 λ = Dividing these two expressions we have the tangency condition y x y x y x p xp y x y p p xy y p p 2 2 1 20 10 2 = = = substitute this value of Y into the utility constraint 2 ) 2 ( 10 y x p xp x U = and solve for x. 2 3 ) 2 ( 10 y x p p x U = and so 3 2 10 ) 2 ( x U p p x y = and finally 3 1 3 2 ) 10 ( ) 2 ( U p p x x y C = . y C = 3 1 3 2 ) 10 ( ) 2 )( ( 2 U p p p p x y y x and so 3 1 3 1 ) 10 ( ) 2 ( U p p y y x C = 1

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D. What are Toni’s minimum expenditures, E*, to achieve a given level of utility at prices Px and Py (the expenditure function)? 3 1 3 2 ) 10 ( ) 2 ( ) , , ( * U p p p U p p E x y x y x = + 3 1 3 1 ) 10 ( ) 2 ( U p p p y x y 3 1 3 2 3 1 3 1 3 1 3 2 3 1 3 2 ) 10 ( 2 ) 10 ( ) 2 1 ( ) , , ( * U p p U p p U p p E y x y x y x + = 3 1 3 2 3 1 3 2 3 1 3 2 3 1 3 1 3 2 ) 10 ( ) 2 1 )( 3 ( ) , , ( * ) 10 ( ) 2 ( ) 2 1 ( ) , , ( * U p p U p p E U p p U p p E y x y x y x y x = + = E. What is the definition of net complements? Are X and Y net complements? Prove your answer. Net complements are goods where 0 < y c P X . 0 ) 2 1 ( ) 10 ( ) 2 ( 3 2 3 1 3 1 = - x x y y c p U p p p x . Therefore, Y and X are net substitutes and not net complements Question 2: A consumer gets utility from two goods, X and Y. X and Y are normal goods and are gross substitutes for the consumer. Using budge lines and indifference curves, Explain and Illustrate the income and substitution effects of an increase in the price of Y. 2
The initial equilibrium is at point A: (U A ,X A and Y A ). With the price increase, the consumer is at lower level of utility, U B , and the new utility maximizing quantities of X and Y at equilibrium point B (X B and Y B ) Because X and Y are gross substitutes, you must also show that the consumption of X INCREASES as a result of the increase in the price of Y. So, X B must be greater than X A . The SUBSTITUION EFFECT: When Y is relatively more expensive, holding utility constant, the consumer substitutes away from consuming Y and towards consuming X. This is illustrated by the movement from A to A’ in the graph above. How did I find A’? Hold utility constant at the initial level, U A . Draw a line parallel to the new budget line (the blue line – the one after the price increase) but that is tangent to U A . This is the red line in the graph above. This tangency gives you A’. The INCOME EFFECT: The price increase also reduces the budget set. This reduction in income is measured from (A’ to B). How will a decline in income effect the consumer’s consumption of X and Y? As stated in the problem, both goods are normal. Therefore, I show both X and Y falling (X B <X A’ and Y B <Y A’ ).

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