Simon Fraser University, Department of Economics, Econ 201, Prof. Karaivanov FINAL EXAM Answer key

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1 Simon Fraser University, Department of Economics, Econ 01, Prof. Karaivanov 017 FINAL EXAM Answer key I. TRUE or FALSE (5 pts each). [The answers below are just examples of correct answers, other possible correct answers can exist; use your judgement and award partial credit for partially correct answers]. 1. FALSE. If a consumer has monotonic preferences (more preferred to less) then her optimal consumption bundle would be on the budget line. If preferences are not monotonic then convexity just implies that the ICs are concave but the consumer could choose an optimal bundle within his budget set.. TRUE. For a competitive rm (which takes the output market price p `as given') total revenue T R(q) = pq where q is quantity sold, and so MR(q) = T R 0 (q) = p.. FALSE. Long-run prots are always weakly larger than short-run prots, LR SR since in the LR the rm can always choose the same x 1 ; and y as in the SR but has extra exibility if some input is xed at a sub-optimal level. For this result it doesn't matter if prots are positive or negative (in the short run). 4. FALSE. The tax introduces a wedge (dierence) between the price consumers pay vs. the price producers receive. The quantity bought and sold must be the same, otherwise the market is not in equilibrium (there would be excess supply or excess demand). 5. FALSE. An allocation is Pareto ecient if no consumer can be made better o without making someone else worse o. Problem 1 (5 pts) (a) (8 pts) Proceed in the usual way. Substitute c = 50 function and solve s by taking the derivative and setting it to zero: 10s s + 50 p s 10 s p = 0 or s = 10 p p s from the budget constraint into the utility This implies, c = 50 p s = 50 p (10 p ). Therefore c > 0 whenever 50 > p (10 p ). Otherwise (since Max cannot consume a negative amount), we have c = 0 and so s = m p = 50 p. (b) (5 pts) Call the bundle A. If p we have 50 > p (10 p ) = 5 and so Max consumes s A = 10 p = 5 sandwiches. His quantity demanded for coee is c A = 50 5(5) = 5. Max's demand curve for s is p = 10 so it is a straight line with intercept 10 and slope -1. (c) (6 pts) For m = 50 Max's demand function is s = 10 p. Thus, if there are 100 consumers identical to him, the market demand is q D (p ) = 100(10 p ) = p To nd the equilibrium price, set q D (p ) equal to the market supply, q S (p ) to obtain: the equilibrium quantity is q D (5) = q S (5) = p = 00p 1000 p = 5 1

2 (d) (6 pts) After the tax we have p D = ps + 4 and so the equilibrium prices solve p D = 00(p D 4) 1000 or p D = 8 and so p S = 4 the equil. quantity is q T = 00. Consumer's surplus is CS = (10 p D )qt = = 00; tax revenue is q T (4) = 800, the deadweight loss is (q q T )t (500 00)(4) = = 600 (e) (10 pts) We start from p = 5 and the price changes to p = 8. Call the resulting optimal bundle for Max, C. At p = 8 we still have p (10 p ) < m so s C = 10 p = and c C = 50 6() = 8. To nd the SE and IE, we need to nd the hypothetical income m 0 at which Max is just able to aord his original bundle (notice that this is bundle A from part (b)!) at the new prices. We have m 0 = c A + 8(s A ) = 5 + 8(5) = 65 At m 0 = 65 and p = 8 Max's optimal bundle (call it B) is s B = 10 8 = and c B = 65 6() = 5. This means that the SEs are: c B c A = +8 and s B s A = and the IEs are: c C c B = 5 and s C s B = 0. We see that Max needs m 0 m = 15 extra income. However, as we saw above, with income 65 he does not choose his original bundle (A). He instead chooses bundle C. Bundle A is feasible for Max after the tax if he receives a $15 lump-sum income subsidy, but not optimal. The higher sandwich price makes him consume more relatively cheaper coee and less sandwiches. Problem (5 pts) (Explain your answers!) (a) (9 pts) We proceed as class, the SR cost imization problem is 9 + subject to: + p = y Since x 1 = 9; the value is detered by the desired output level (nothing to imize), hence we obtain ^x SR (y) = (y ) and ^x SR 1 (y) = 9 The SR cost function is c SR (y) = w 1^x SR 1 (y) + w ^x SR (y) = 9 + (y ). The rm would shut down if p < AV C. We have V C(y) = (y ) = y 1y + 18 and so AV C(y) = y y. To nd AVC, take a derivative and set to zero: 18 y at y = and its imum is AV C() = 0. Therefore the rm never shuts down. (b) (8 pts) To nd the SR prot function and input demands, we solve: p( + p ) 9 = 0, so AVC is imized Take derivative and set to zero: p p = x SR (p) = p 16 and so the SR prot function is SR (p) = p( + p 4 ) 9 p 16 (c) (9 pts) The rm's LR cost imization problem is = p + p 8 9. x 1 + subject to p x 1 + p = y x 1 ;

3 substitute for x 1 from the isoquant to obtain the simpler problem: + (y p x ) or + y y p + take derivative and set to zero: y p x = 0 and so the LR conditional demand for input is ^ (y) = y 9 and hence, ^x 1 = (y function is c(y) = ^x 1 + ^ = y y ) = 4y 9. The LR cost This implies MC(y) = 4 y which is always upward sloping, and AC(y) = y which has imum at 0. Therefore the rm's supply function solves p = MC(y), or p = 4 y which yields y (p) = 4 p. (d) (9 pts) Finally, for the LR prot imization we need to solve: x 1 ; p( p x 1 + p ) x 1 take the partial derivatives with respect to x 1 and and set to zero: p x 1 = 1 p = which gives the long-run input demands x p 1 (p) = 4 and x p (p) = 16. The LR prot function is p( p + p 4 ) p 4 p 16 = p 8. At p = 6 the SR and LR prots are equalizes since x 1 = 9 = x 1 (6) { that is, when p = 6; the rm already uses the optimal quantity of input 1 in the SR and so it does not change anything in the LR. Problem (5 pts) (a) ( pts) Since AC(y) = c(y) y, the rms' cost function is c(y) = y 9y + 9y. This implies MC(y) = c 0 (y) = 9y 18y + 9 = 9(y 1) (b) (5 pts) The rms will produce y > 0 as long as p AC. Take the derivative of AC(y) and set to zero: 6y 9 = 0 so AC is achieved at y = and equals = 9 4 = :5. For any p > 9 4 the rm would produce a positive amount. (c) (10 pts) First we need to nd the rm's supply f-n. Notice that MC is upward sloping for y > 1 and so, y (p) = 0 for p < 9=4 and y (p) solves p = MC(y) otherwise, or, p = 9(y 1) which implies y (p) = p p + 1

4 (note that this is always >1 and so MC is upward sloping at y (p)). With 0 identical rms in the industry, market supply, for p 9=4 is: p p q S (p) = 0( + 1) = 10p p + 0 Market supply is 0 for p < 9=4. To nd the equilibrium, set supply = demand: 10 p p + 0 = p p which yields p = 6 (which is > 9=4 indeed). The equilibrium quantity is q = 10 p p + 0 = 90. Each rm makes a prot since they produce y (6) = and at y = their average costs are AC() = 9 < 6 = p. Since p > AC the rm makes a prot because each unit sold earns strictly more revenue, p than it costs on average, AC(y ). (d) (7 pts) In a LR industry equilibrium the price received by producers must equal their AC (otherwise prots exist and more rms will enter). Thus, p S = 9=4 = :5 and p D = p S + 1:75 = 4. At p S = 9=4 each rm produces, y ( 9 4 ) =. Market demand in the LR industry equilibrium is q D (p D ) = p 4 = 10 and so the equilibrium number of rms is n = qd (p D ) y (p S ) = 10 = = 60 Problem 4 (0 pts) (a) (8 pts) Indeed, for perfect complements preferences represented by fx 1 ; g, we know that at optimum the consumer would choose x 1 = (otherwise income is wasted), and so, from the budget constraint p 1 x 1 + p = m, using x 1 =, the demand functions are: x 1 = x = m p 1 + p For p 1 = 1 and m = m B this is exactly B 0 s given demand function for good 1. We are also given B's demand curve (inverse demand function) for good, from where, expressing x B in terms of p we have x B = m 1+p { the demand function for perfect complements again. A's preferences are for perfect substitutes and can be represented by the utility function u A (x 1 ; ) = x 1 +. Since one unit of good 1 always yields the same utility as one unit of good for A, his utility is imized when he can buy the most of the cheaper good. If both goods have the same price (which happens if p = 1), any quantities that satisfy his budget constraint, x 1 + = m A imize his utility (the IC coincides with his budget line). Thus, A's demand functions are: 1 = 0 and x = ma p if p < 1 1 = m A and x = 0 if p > 1 any 1 ; with 1 + = m A if p = 1 (b) (4 pts) The box has size 6-by-4. The endowment point is in the bottom-right corner. A's IC through W is a straight line with slope -1 (perfect substitutes). B's IC through W is an L-shaped line with kink at B's origin (why? B's preferences are for perfect complements and note that his utility at W equals f0; 4g = 0 = f0; 0g) (c) ( pts) As in class, m A = p 1 w A 1 + p w A = 6 and mb = p 1 w B 1 + p w B = 4p 4

5 (d) (6 pts) A competitive equilibrium, CE is prices (1; p ) and a feasible consumption allocation ( 1 ; xa ; xb 1 ; xb ) such that: (i) for each person i = A; B, the consumption bundle (x i 1 ; xi ) is optimal given the person's preferences, income, m i and the prices (1; p ) [that is the bundle (xi 1 ; xi ) is what i demands at (p 1; p ; mi )] (ii) at the prices (1; p ) both markets clear, that is total demand = total supply. [Alternatively students can depict the CE graphically as a budget line through W and the two ICs just touching at some point on this line] To solve for CE, note that from B's preferences we must have x B 1 = x B = mb 1+p = 4p 1+p. That is, the CE allocation must lie on the 45-degree line. Similarly, from A's preferences, his optimal choice is a corner solution for p > 1 or p < 1 { so cannot be on the 45-degree line. Thus, we must have p = 1 [to see that, you can plot a picture at which A's IC is just touching to B's IC]. Using p = 1, we therefore obtain the CE allocation: xb 1 = 4 = = xb and so, by feasibility, 1 = w1 A + wb 1 x B 1 = 4 and = w A + wb x B =. (e) (10 pts) We will argue that if A and B start at allocation M then no feasible mutually benecial trades exist. That is, there is no way to make one person better o without making the other worse o, hence M is Pareto ecient. Indeed, when B consumes (1; 1), then by feasibility A 0 s consumption bundle is (5; ). Plot A's IC through M { it is a straight line with slope -1. How about B's IC through M [now looking from B's origin] { it is an L-shaped curve with its right-angled kink at (1; 1). Note that A's and B's indierence curves through M that you just plotted are just touching { all points on B's IC except M are below and to the left of A's IC. Therefore, there do not exist feasible allocations that make A better o without making B worse o and vice versa. This means that M is Pareto ecient. Allocation N is not Pareto ecient since B's indierence curve (L-shaped line) through his consumption bundle (5; ) [from B's origin] crosses A's IC (straight line with slope 1). Therefore there exist feasible allocations for which A and/or B can be made better o starting from N, without making anyone worse o. For example, the allocation at which A consumes (; 1) and B consumes (; ) makes A strictly better o while B is indierent compared to N. This implies that allocation N is not Pareto ecient. 5