UNIVERSITY OF WISCONSIN-MADISON DEPARTMENT OF ECONOMICS ECONOMICS 301 - 002 FALL 2021 Problem Set #3 (to be submitted online by 11:59 p.m. Friday, October 8th) 1 . Your preferences are summarized by the utility function U ( N , Y ) = N*Y3 where Y is a composite good and N is the number of leisure hours you consume each day. Thus, 24-N =L where L is the number of labor hours supplied each day. Also assume that you have some non-wage income Io . a. Given that the price of Y is Py and your nominal wage (in $ / hour) is w . derive your optimal demands for leisure and the composite good: N*( . ) & Y*(. ) and therefore L*( . ). Don't forget the 24 hour constraint, and comment on whether your optimal N*, Y*, LS* make economic sense in their arguments....namely, "as w goes down, LS* ..." 1 . Your utility-maximizing demand for leisure ( N ) here is given by the following function: a N* = [24*w + Io ]* [w /PY ] b. N* = [ 24*w + 10 ] / [ w /PY ] C . N* = [ 24*w- + 10 ] / PY N* = [ 24*w + 10 ]/ [ 4*w ] ii. Your utility-maximizing demand for the composite good ( Y ) here is given by the following function: a. Y* = 34 * [ 24*w + Io ]/ PY ] b. Y* = 34 * [ 24*w + Io ]/ 4*w ] C. Y* = 34 * [ 24*w + Io ] Y* = 34 * [ 24*w + Io ]/ [ 4*PY ]in. Your optimal labor supply ( LS ] here is given by the following function: a. L3 = 1s w* Io b. LS = 24 or" In c. L3 = 18 (In 4*w) d. LS = 24 ( In 4*w ) b. Sketch your labor supply curve from (a) and comment on its shape. i. Which diagram correctly sketches the labor supplyr curve here: a. A b. B c. C d. D ii We can infer from that sketch that whenever the wage changes and the substitution effect and the income effect on leisure are both non-zero, a. The substitution effect dominates the income effect. b. The income effect dominates the substitution effect. c. The substitution effect just balances the income effect. :1. Not enough information to tell. 6. Suppose the wage rate is $20fbour. What level of non-wage income will induce you NOT to work? i. At a wage rate of $20 fbour, if non-wage income is , you will maximize utility by NOT working and taking hours of leisure. a. $1920; 12 1). $1440 ; 24 c $1920 ; 24 11 $1440 ; 12 Explain why this comer-type solution is possible in (c). Sketch your budget constraint I in ( N , Y ) space ] given the non-wage income you found in (c) and show your optimal consumption point. i. Which diagram represents the corner-type solution in (c)? A s. b. B c. C d. D (1. Income ; leisure all day In the late [9905, the US. passed sweeping legislation reforming the function and practice of providing welfare benets to the poor. Though simplied, this question asks you to examine the potential effects of this law on the behavior of Billy and Charlie (two representative welfare recipients) based on theories we've used in our unit on consumer theory. Suppose, as indicated in the diagrams at the end of this problem set, Billy and Charlie face the usual n-adcoffs between leisure and all other goods [AOG's), and they view both of those goods as normal goods. In addition, each is able to use welfare benets when not working to purchase some subsistence level of all other goods. Assume, initially, both of our representative welfare recipients optimize at point \"A\" in the diagram, choosing to consume, for example, $100 worth of all other goods and 24 hours of leisure daily. Assume the initial wage is $10 I hour, and that the price of all other goods is $1 3' unit of A00. Reproduce a SINGLE COPY of the diagram above and show either consumer's optimization at point \"A\". You may assume heishe has the usual \"bowed in\" indifference curve shape, and that preferences are strictly convex and strictly monotonic. [s it possible for this consumer's MRS (out of AUG and into leisure) to be different from the real wage here? Why or why not? i. Which diagram shows Billy's or Charlie's optimization at point \"A" where hefshe is consuming IOO units of AUG and 24 hours of leisure daily. a. A b. B c. C b