Honors General Exam Solutions Harvard University April 201 PART 1: MICROECONOMICS
Question 1 The Cookie Monster gets a job as an analyst at Goldman Sachs. He used to like cookies, but now Cookie Monster only likes coffee and leisure. His utility function is u(c, l) = c 1/2 l 1/4 where c is cups of coffee and l is hours of leisure. Coffee costs $4 a cup. Cookie Monster has H hours to allocate between work and leisure, and he earns $40 for each hour he works. (a) Express Cookie Monster s maximization problem, including budget constraint. Solution: Cookie Monster must solve: max c 1 2 l 1 4 c,l s. t. 4c = 40(H l) The budget constraint shows that the amount he spends on coffee must equal the income he earns from work. He works H l hours, and earns $40 per hour, so his income is $40(H l). (b) Solve for the optimal consumption of coffee and leisure as a function of hours H. Solution: Start by substituting the budget constraint in for c in the utility function: max(10(h l)) 1 2 l 1 4 l You could proceed by solving this using the product rule, but that s a difficult calculation. An easier way to solve this is by taking the log of the argument, and then maximizing. 1 The maximizer of the log of the argument will also be the maximizer of the argument itself. the In general, this is a useful technique for solving nasty maximization problems (on the honors exam and otherwise). 1 A positive monotonic transformation of a function (e.g. a transformation that preserves ordinal relationships between points) also preserves extreme points. Thus, if a function f is maximized at some point x, and g is a positive monotonic transformation s.t. for any points a and b for which f(a) > f(b), it is also true that g(f(a)) > g(f(b)), then g(f( )) is also maximized at x. The log function is an example of a positive monotonic transformation.
So, we take the log of the function: max l max l max l log ((10(H l)) 1 2 l 1 4) log ((10(H l)) 1 2) + log (l 1 4) 1 2 log (10H 10l) + 1 log (l) 4 And now we can take the FOC of a much more manageable function: du = 1 ( 10) + 1 dl 2(10H 10l) 4l 10 + 1 = 0 2(10H 10l) 4l 1 = 10 4l 2(10H 10l) 1 = 1 4l 2H 2l 2H 2l = 4l 2H = 6l l = H Now, we return to the budget constraint to solve for c: 4c = 40(H l) c = 10(H l) c = 10(H H ) c = 20H So, cookie monster works for 2 H hours and enjoys 1 H leisure. He earns income $ 80 20 H which allows him to consume H cups of coffee. (c) Suppose Cookie Monster suddenly realizes he no longer needs sleep and H increases. What happens to his consumption of coffee? (Assume that the change in the need for sleep and any change in consumption of coffee are unrelated except through the change in H.) Is coffee a normal good, an inferior good, or neither?
Solution: As we calculated in part (b), Cookie Monster consumes 20 H cups of coffee. That is, his coffee consumption increases as H increases. Since an increase in H increases his income (as he ll now work more hours), we see that an increase in income leads to an increase in consumption of the good. Thus, coffee is a normal good. (d) Are coffee and leisure complements, substitutes, or neither? Explain. Solution: Neither. In order for a pair of goods to be complements or supplements, the demand for one good must change given a change in the price of the other. (In the case of complements, an increase in the price for one good reduces demand for the other; in the case of supplements, an increase in the price of one good increases demand for the other.) In this problem, however, a change in the price of leisure or coffee has no effect on the allocation of hours between work and leisure. Rather, only H affects consumption of coffee and leisure. Question 2 Cookie Monster decides to celebrate his Saturday night off by taking his girlfriend, Pie Monster, out for dessert. They can either go out for cookies or for pie. While Cookie Monster and Pie Monster are debating on the phone whether to meet at the cookie store or the pie store, Cookie Monster s cell phone battery dies. Thus, both must decide where to go without having agreed on where to meet. Both Cookie and Pie Monster know that the utility for each from each option is as follows: Pie Monster Cookie Monster Cookies Pie Cookies Pie 4,2 0,0 0,0 2,4 (a) Find all pure strategy Nash equilibria of this game. Solution: In a pure strategy Nash equilibrium, there must be no incentive for either player to deviate from their strategy. That is, the payoff derived from a player switching strategies must be less than the payoff he or she is
currently receiving. In this game, there are two such pure strategy Nash equilibria: the case in which they both go to the cookie store, and the case in which they both go to the pie store. (b) Are there any mixed strategy Nash equilibria? If so, find them. What is the expected payoff to each player in each mixed strategy Nash equilibrium (if there are any)? Solution: In a mixed strategy equilibrium, each player mixes his or her strategy such that the other player is indifferent between his or her strategies. To find the mixed strategy equilibrium in this problem, let p be the probability that Cookie Monster picks cookies and q the probability that Pie Monster picks cookies. For Cookie Monster to be indifferent between choosing the cookie store and choosing the pie store, the expected outcomes of each must be equal. Thus, 4(q) + 0(1 q) = 0(q) + 2(1 q) 6q = 2 q = 1 Similarly, for Pie Monster to be indifferent between choosing the cookie store and choosing the pie store, the expected outcomes of each must be equal. Thus, 2(p) + 0(1 p) = 0(p) + 4(1 p) 6p = 4 p = 2 Thus, in equilibrium, Cookie Monster will choose the cookie store with probability 2 and the pie store with probability 1, while Pie Monster will choose the cookie store with probability 1 and the pie store with probability 2. The payoff to each is 4. (To see this, you can select either player and either strategy, and given the mixing probabilities we calculated above, the expected payoff will always be 4.) (c) Cookie Monster and Pie Monster both remember an agreement they made long ago: if they got in a fight that they could not resolve, Cookie Monster would win if the lights on top of 0 Rockefeller Plaza are blue, and Pie Monster would win if they are red. Both decide independently that if the
lights are blue, they will go to the cookie store and if the lights are red they will go to the pie store. Before they look at 0 Rock, they know that the probability of the lights being blue is 1 2 and the probability of red is 1 2. What is the expected payoff of this coordination strategy? Solution: Under this scenario, the only outcomes are (cookie, cookie) and (pie, pie), since they will both choose the same outcome, depending on the color of the lights. Given the probabilities of the lights being blue and red, they will pick (cookie, cookie) with probability 1 and (pie, pie) with 2 probability 1. The expected payoff is (, ). (Since the other player is also 2 playing this coordinated strategy, neither player has any incentive to deviate, as doing so would result in a payoff of 0 for both players.) (d) If a mixed strategy equilibrium exists, is the payoff in (c) larger, smaller, or the same as the payoff in (b)? Provide intuition why. Solution: It is larger. The coordination device (the lights) eliminates the outcome that Pie Monster and Cookie Monster end up going to different places, when the outcome (0,0) is a possibility. With the coordination device, the only outcomes possible are (2,4) and (4,2), resulting in higher expected payoffs. Question Cookie Monster decides he no longer wants to work in investment banking, so he gets a job as the assistant to the CEO of a private equity firm. The CEO is a very busy person with a lot of money. She has 5 hours of leisure per week and $5000 to spend on consumption. Cookie Monster, on the other hand, has no money and lots of leisure time. He is endowed with $10 and 20 hours of leisure per week. The utility function of the CEO is U CEO (c, l) = (l 5) 1 2c 1 2 and the utility of Cookie Monster is now U CM (c, l) = 8l 2 5c 5, where l is hours of leisure per week and c is consumption. (a) Suppose Cookie Monster gets paid wage w. Solve for the CEO s demand for hours of labor from Cookie Monster as a function of w. Solution: Let h be the hours of labor demanded from Cookie Monster by the CEO. The CEO solves:
max(l 5) 1 2 c 1 2 c,l s. t. c = 5000 hw l = 5 + h The budget constraint shows that the CEO s consumption is equal to her initial endowment of $5,000 minus anything she pays Cookie Monster for his labor. The last equation shows that her leisure will be equal to 5 plus any hours that Cookie Monster works for her. Now, substituting gives max h (h) 1 2(5000 hw) 1 2 As in Question 1, you could solve this using product rule, but that will lead to some unpleasant calculations and manipulations. Instead, take the log of the argument and maximize that function. (The value of h that maximizes the log of the function also maximizes the function itself.) max h max h log ((h) 1 2(5000 hw) 1 2) log ((h) 1 2) + log ((5000 hw) 1 2) 1 + 1 ( w) = 0 2h 2(5000 hw) 2hw = 2(5000 hw) hw = 5000 hw 2hw = 5000 h = 2500 w (b) Solve for the supply of hours of labor by Cookie Monster as a function of w. Solution: Let s be the hours of labor supplied by Cookie Monster. Cookie Monster solves the following maximization problem: max 8l 2 5c 5 c,l s. t. c = 10 + sw l = 20 s The budget constraint shows that Cookie Monster s consumption equals his initial endowment of $10 plus any income he earns working for the CEO.
The last equation shows that Cookie Monster s leisure is equal to 20 hours minus any he supplies working for the CEO. Substituting in gives max 8(20 s) 2 5(10 + sw) 5 s Again, we take the log of the function and maximize. max s log (8(20 s) 2 5(10 + sw) 5)) 2 + 5(20 s) w 5(10+sw) = 0 w = 2 5(10+sw) 5(20 s) 2(10 + sw) = w(20 s) 20 + 2sw = 60w sw 5sw = 60w 20 s = 12 4 w (c) Solve for the equilibrium wage rate by equating Cookie Monster s supply function with the CEO s demand. How many hours will Cookie Monster supply to the CEO at this wage? How much money will Cookie Monster earn? Solution: Setting labor supply and demand equal gives: l = s = 12 4 w w = 626 2500 Plugging w into either the supply or demand function yields the equilibrium number of hours: w And Cookie Monster will earn s = h = 12 4 w = 12 4 626 11.98 hours
25 ($ 626 /hour) (11.98 hours) = $2500 (d) Sketch an Edgeworth box in consumption-leisure space. Indicate the initial endowments and the competitive solution. Sketch indifference curves for both the CEO and Cookie Monster through the initial endowments and shade the region in which there are gains from trade. Solution: See below. Note that while Cookie Monster s indifference curve, passing through the initial endowment point, Point A, is a normal, wellbehaved curve, the CEO s is not. Given her initial endowment of 5 hours of leisure and $5000 consumption, her utility is actually zero. Her utility function, U CEO (c, l) = (l 5) 1 2c 1 2, requires her to have l > 5 and c > 0 in order for her to have positive utility. With her initial endowment of l = 5, she has utility zero. Since a utility curve shows all the bundles that provide the same utility, that initial indifferent curve must show all the points that provide zero utility. Those points are all those along l CEO = 5 and c CEO = 0. The shaded area represents all of the Pareto-improving points (and includes the border). The final, competitive solution, Point B, is a Pareto-optimal point, as it is not possible to make either individual better off without making the other worse off. This point lies on the contract curve (not shown). Cookie Monster
Leisure (Hours) B ICCM ICCEO A CEO Consumption ($) 5010 (e) After beginning work for the CEO, Cookie Monster discovers that the CEO s mood is also a function of the performance of her investment portfolio. Her utility is given by U CEO (c, l) = (l 5) 1/2 c 1/2 + s&p 1000 where s&p is the S&P 500 at the start of business. How does this change the answers to sections (a) through (d)? Solution: This will have no effect on the supply or demand of labor and thus no effect on the answers in (a) through (d), since the S&P 500 term is just a constant, and will drop out when the CEO s FOC is derived. It will not affect the optimal bundles of leisure and consumption for either the CEO or Cookie Monster. Question 4 In his newly found free time, Cookie Monster decides to start a business making caffeinated cookies to sell to his friends who work at Goldman Sachs. The total cost of making q cookies is TC = 50 + 0.5q 2. The demand for caffeinated cookies is p = 100 q. (a) Graph the demand curve for caffeinated cookies. Label both axes. Solution:
Price ($/cookie) 100 Demand for Caffeinated Cookies Quantity (Cookies) 100 (b) Compute the equilibrium price, quantity, and Cookie Monster s profits. Solution: We find equilibrium quantity by setting marginal cost equal to marginal revenue. First, we calculate revenue: Then, we find marginal revenue: Next, we find marginal cost: MR = drev dq Rev = pq = (100 q)q = 100q q 2 = 100 2q MC = dtc dq = q We set marginal revenue and marginal cost equal and solve for q: MR = MC 100 2q = q q = 100 j
To find price, we plug q into the demand function: Profit is: p = 100 q p = 100 100 p = 200 π = Revenue Cost = (( 200 2 ) (100 )) (50 + 0.5 (100 ) ) = 20,000 (50 + 0.5 ( 10,000 )) 9 9 = 20,000 9 = 14550 9 450 9 5,000 9 $1,616.67 (c) The Cookie Monster finds cheaper caffeine powder to use in his cookies. His total cost function is now TC = 50 + 0.25q 2. Redo (b) with this new cost function. How is Cookie Monster s profit affected by this change in costs? Solution: Marginal revenue is unchanged. Marginal cost is now q 2. As before, we set marginal revenue and marginal cost equal to find equilibrium quantity: MR = MC 100 2q = q 2 q = 40 Using the demand function, we solve for price: p = 100 q = 100 40 = 60 Profit, then, is: π = Revenue Cost = (60)(40) (50 + 0.25(40) 2 ) = 2400 (50 + 400)
= 2400 450 = $1950 Cookie Monster s profit has increased as a result of the cheaper caffeine powder. (d) To improve public health, Mayor Michael Bloomberg imposes a $25 tax on each caffeinated cookie sold in New York City. Solve for the new equilibrium price and quantity of cookies using the cost function from parts (a) and (b). How does this tax affect Cookie Monster s profits? What is the tax revenue for the city from Cookie Monster s cookie business? What is the deadweight loss from the tax? Solution: The tax will drive a wedge between the price paid by consumers and the price received by Cookie Monster. The new demand function will be: Revenue is: And marginal revenue is: p + 25 = 100 q 6p = 75 q Revenue = (75 q)q = 75q q 2 MR = 75 2q Marginal cost is q (as calculated in part (b)). To find equilibrium quantity, we set marginal revenue equal to marginal cost: MR = MC 75 2q = q q = 25 We then find price using the new demand function: Next, we solve for profit: p = 75 q = 50
π = Revenue Cost = (50)(25) 50 0.5(25) 2 = $887.50 So, relative to part (b), Cookie Monster s profits have fallen by $729.17 as a result of the tax. The city receives tax revenue of: Government Revenue = qt = (25)($25) = $625 In the graph below, the light blue area represents this government revenue. Finally, it is easiest to calculate deadweight loss graphically. (Note that the firm is a monopolist, so there is DWL to start with. Here, we re calculating the additional DWL resulting from the cookie tax.) In the graph below, the blue lines represent the demand and marginal revenue curves before the tax, and the red lines represent them with the tax. The tax results in a lower equilibrium quantity, and the DWL is the gray shaded area shown below. Why? Prior to the tax, this area was consumer surplus (the triangle at the top of the area) and producer surplus (the remaining hexagon). However, with the tax, the reduction in demand for cookies at the higher price eliminates this surplus. To calculate deadweight loss, we need to find the dimensions of the grey shaded hexagon, and calculate its area. Its height is: 100 25 = 25 The length of the base of the hexagon can be found by plugging q = 25 into the marginal cost function and the original demand function: 75 25 = 50 Similarly, the length of the top of the hexagon can be found by plugging q = 100 into the marginal cost function and the original demand function: 200 100 = 100 Thus, the area of the hexagon, and the deadweight loss created by the Mayor s tax is:
Price ($/cookie) = 1 2 (b 1 + b 2 )(h) = 1 100 (50 + 2 )(25) $47.22 Market for Caffeinated Cookies 100 75 200 50 Gov. Revenue DWL from tax MC 25 MR MR D D 25 100 50 75 100 Quantity (Cookies)