UNIVERSITY OF WISCONSIN-MADISON DEPARTMENT OF ECONOMICS ECONOMICS 301 - 002 FALL 2021 Problem Set #2 (to be submitted online by 11:59 p.m. CDT Friday, October 15) 1 . A utility maximizing consumer has the following utility function: U (X, Y) = min [ 2X , Y ] a. Derive the consumer's optimal demands X*(Px, Py,I) and Y*(Px,Py,I), assuming he faces the standard, linear budget constraint: PxX + PYY =I. Show the optimum in a well-labeled graph. 1 . This consumer's optimal demand for X is: a. X* = 2* [I/ (Px + PY) ] b . X* = 1/2 * [ I/ (Px + PY) ] X* = I/ (Px + 2*Py) ] x* = 1/ (2*Px + PY) ] 11. This consumer's optimal demand for Y is: Y* = 4* [ I/ (Px + PY ) ] Y* = [I/ (Px + PY ) ] Y* = 2* [ I/ (Px + 2*Py ) ] Y* = 2* [I/ (2*Px + Py ) ] b. Derive an expression for this consumer's income consumption curve. Derive an expression for this consumer's price consumption curve for Px . Comment on their similarities or differences. i. This consumer's income consumption curve is: Y* = X* Y* = 2X* Y* = 1 X* Y* = 3/2 X*ii. This consumer's price consumption curve for Px is: a. Y* = X* b. Y* = 2 X* C. Y* = 1/2X* Y* = 3/2 X* iii. This consumer's price consumption curve for Px and their income consumption curve are , because any change in Px results in a. Different ; a stronger substitution effect than income effect Different ; no income effect The same ; a stronger substitution effect than income effect The same ; no substitution effect 2. A utility maximizing consumer has the following utility function: U (X, Y) = 2X1/2 + Y a. Are these preferences strictly monotonic? Strictly convex? Explain. The preferences represented by this utility function are monotonic and convex. a. Weakly ; weakly b . Weakly ; strictly C. Strictly ; weakly Strictly ; strictly b. Assuming our utility maximizer faces a linear budget constraint of the form PxX + PyY = 1, derive his/her optimal demands X*( Px , Py , I ) and Y*( Px , Py , I ) . You may use any method you prefer, but please show your work. If these demands are piecewise functions, then state each piece clearly (namely, is a corner solution possible, and, if so, what would X* and Y* be then?) i. If I > , then this consumer's optimal demand for X is and his optimal demand for Y is Px / Pv ; I / Px ; 0 (Py / Px ) ; I /Px ; 0 Px / Py ; Py/ Px ; I /Py ( PY )? / Px ; (Py/ Px ) ; (I/Py) - (Py/Px )ii. If I is less than or equal to then the optimum is a corner solution, and he buys a. Px / Py ; all X and no Y. b. ( Py / Px ) ; all Y and no X. C . ( PY )' / Px ; all X and no Y. d. ( PY )' / Px ; all Y and no X. C. Assuming an interior optimum, what is the own-price elasticity of demand for X? The cross price elasticity of demand for X with respect to Py? i. At any interior optimum here, the own-price elasticity of demand for X is: C. d. -1/2 ii. At any interior optimum here, the cross-price elasticity of the demand for X with respect to Py is implying that X and Y are a. 1 ; substitutes b. 2 ; substitutes C . 1 ; complements 2 ; complements d. Graph the Engel curve for Y. Label intercept and slope values carefully. i. Which of the following graphs (located at the end of these questions!) correctly depicts the Engel curve for Y? ae. Suppose, initially, that I = 100, Px= 2 and Py =2. Graph this consumer's initial optimum point (A) in a well-labeled graph. Your indifference curve should show whether strict convexity holds and whether it can hit an axis (or both). i. At these initial price and income levels, this consumer maximizes utility at by consuming units of X and units of Y. a. 1 ; 50 b. 2: 49 1 : 49 d. 2: 50 f. The price of X falls to 1. Find the consumer's new optimum point (C) and label it on the same graph as in (e). Draw another indifference curve to show this new optimum. i. As a result of this fall in the price of X, this consumer now maximizes utility by consuming units of X and units of Y. 4 ; 46 b. 2: 46 2 : 48 4 : 48 Find the coordinates of the substitution effect point (B) of this price change. Label it and draw the compensated budget line tangent to the original indifference curve through point A. [Hint: What do we know about good X for interior solutions here?] i . The equation of this consumer's initial, utility-maximizing indifference curve is: Uo = 51 jollies Uo = 50 jollies C. Uo = 49 jollies Uo = 48 jollies ii. Because for interior solutions, X is a(n) good, point B lies of point C Inferior ; due West Normal ; due West C. Neuter ; due South Normal ; due South [Note: "West" and "South" denote compass directions here.]iii. The coordinates of point B (substitution effect point) are: a XB = 4 ; YB = 50 b . XB = 4 ; YB = 47 C . XB = 1 ; YB = 48 d. XB = 1 ; YB =47 3. In a two-good world, a utility maximizer with strictly convex preferences has a perfectly inelastic demand curve for good X. Derive this demand curve in a simple indifference curve /demand curve diagram (one right under the other) with a standard linear budget constraint. What kind of good must X be? Why? i. X is a(n) good in this case, and the substitution effect of a change in the price of X the income effect of that price change. a. Normal ; outweighs b . Inferior ; outweighs C. Neuter ; equals Inferior ; equals ii. If the demand for X is indeed perfectly inelastic, any decrease in the price of X will a. Not change the consumer's total expenditure on good X. b. Increase the consumer's total expenditure on good Y. Both (a) and (b) are true. Neither (a) nor (b) is true. 4. Larry consumes only two goods, watches (X) and coffee (Y). He is a utility maximizer with strictly convex preferences and faces a linear budget constraint in prices and income [ Px*X + Py*Y = I ]. Furthermore, we have observed that Larry's income elasticity of coffee demand is negative [ O In Y / 0 In I