Econ 201: Problem Set 3 Answers

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1 Econ 20: Problem Set 3 Ansers Instructor: Alexandre Sollaci T.A.: Ryan Hughes Winter 208 Question a) The firm s fixed cost is F C = a and variable costs are T V Cq) = 2 bq2. b) As seen in class, the optimal quantity produced by the firm is the quantity that equates the marginal cost to price. Hence, d 2 bq2) = p = q = p dq b. c) As seen above, the supply function for this firm is q = p/b. The elasticity of supply is thus ε S = d logqp)) d logp) = dqp) dp p qp) = b p p/b) =. d) It doesn t. This supply function is linear on prices, hich means that its price elasticity is constant. e) Since a is a fixed cost, it on t affect the quantity that the firm produce, given that the firm chooses to produce a positive amount. Fixed costs only affect the firm s production in the extensive margin that is, hether the firm chooses to produce or not. If a is too big, the firm might simply choose to leave the market. f) First, let us compute the firm s profit. From part b), e kno that q = p/b, hence πp) = pqp) cq) = p2 b a p ) 2 2 b p 2 = b 2 b a. Since the firm can recover the fixed cost if it shuts don, it ill only produce if it makes at least zero profits hich is the amount that it gets if it shuts don). Hence, πp) > 0 p 2 2 b a > 0 a < p 2 2 b. g) No that the costs are sunk, if the firm chooses to shut don, it ill have a loss of a or πp) = a. Therefore, the firm ill produce if πp) > a p 2 2 b a > a p 2 2 b > 0, hich is alays true, so the firm ill alays produce even if it ends up making negative profits). At first glance this might seem like eird result. Hoever, think about the options of the firm: if it shuts don, it is guaranteed to have a loss of a. If, instead, it starts to produce, it can only increase its profits or reduce its losses). Hence, the firm ill keep producing to cuts its losses, even if it makes negative profits.

2 Question 2 a) In the short-run, Bob ill minimize his costs given the amount of capital he has standard lanmoer). His cost of moing a lan are the cost of renting the lanmoer and his foregone ages at the grocery store remember that all costs are opportunity costs!). Let q be the number of lans that Bob mos and l the number of hours necessary to mo a lan. Bob s moing production function is therefore q = 2l. Thus, his cost function is the solution to C SR q) = min l l + 9 s.t. q = 2l This is a pretty simple problem to solve, and e can do so by simply substituting the constraint into the objective function. Given = $0, the cost function is C SR q) = 9+5q. b) In the short run, the fixed cost is 9 and the variable cost is 5q. c) We kno that the number of lans that Bob can mo is q = a p, hich means that the inverse demand for his services if given by pq) = a q. Hence, Bob solves The FOc yields max q pq)q C SR q) = max aq q 2 9 5q. q a 2q 5 = 0 = q = a 5 2 Plugging this into the demand function, e get that the price he ill charge is p = a q = a d) The same logic from part a) applies here. In this case, Bob s cost minimization problem is and his cost function is C SR q) = q C SR q) = min l l + 24 s.t. q = 4l e) The fixed cost is no 24 and the variable cost is 2.5q. f) Once again, Bob solves The FOC is and max q pq)q C SR q) = max aq q q/2. q a 2q 5 2 = 0 = q = 2a 5 4 p = a q = 2a g) In the long run, Bob can choose beteen using the standard or the turbo lanmoer. Minimizing beteen those to means that Bob s long run cost function is C LR q) = min{9 + 5q, q} here the first entry inside the min{} operator is the cost of using the standard lanmoer and the second entry is the cost of using the turbo lanmoer. Note that the cost of using the standard lanmoer gros at a faster rate than the cost of using the turbo lanmoer. This means that if Bob ants to mo a large number of lans, 2

3 he ill prefer the turbo lanmoer. In particular, the threshold q at hich this happens is given by q = q = q = 6. We can therefore rite Bob s long run cost function as { C LR 9 + 5q if q 6 q) = q/2 if q > 6 Question 3 a) Since the firm is a price taker, the inverse) supply function if simply given by the marginal cost: pq) = c b) Given the supply function above, the only possible price of good in equilibrium is p = c c) The price that the producer receives continues to be p P = p = c any price loer than this and the producer ould shut don, any price higher and supply ould not be finite); hoever, consumers no pay p C = p + t = c + t. d) There is no change in producer surplus, since both before and after the tax is introduced the firm sells good for the same price, p, and makes zero profits so producer surplus is zero in both cases. It follos that consumers alone pay the tax. The intuition behind this result comes from the fact that the part producer or consumer) that is less price-elastic pays a higher share of the tax. Since supply is infinitely elastic, consumers are left ith the entire burden of the tax. Question 4 a) The firm solves Since F l) = l, the FOC is max pf l) l l [l] : pf l) = or p = It follos that the only case in hich the supply of good x and the demand for labor by the firm are ell defined is hen p =. If p >, the firm ill demand l = 24, so r = 0. But then the marginal utility of leisure ould tend to infinity, and the consumer ould not be illing to spend all of his time orking for any finite age. It follos that this cannot be an equilibrium. If p <, the firm shuts don, since it invariably makes negative profits if l > 0. In this case, x = 0, and e have the opposite case: the consumer s marginal utility for x ould tend to infinity, and therefore she ould be illing to trade a large amount of labor for a small quantity of x. It again follos that this cannot be an equilibrium. b) The consumer solves max x + r s.t. px + r = 24 + π. x,r 3

4 Note: the budget constraint above comes from the fact that Robinson makes money in to ays: by orking, hich yields l, and because he is the oner of the firm, hich yields profits π. We can rite Robinson s budget constraint as px = l + π. No using the fact that l = 24 r, the budget constraint can be reritten in the form shon in the maximization problem. The Lagrangian function of this problem is therefore L = x + r + λ24 + π px r) Taking the FOC s e get Using [x]/[r], e find or [x] : 2 x 2 = λp [r] : 2 r 2 = λ [λ] : px + r = 24 + π. r x) 2 = p p ) 2 r = x Plugging this into [λ], e get p ) 2 px + x = 24 + π or Similarly, xp, ) = 24 + π p + p 2 /. p ) 2 p ) π r = x = p + p 2 / rp, ) = 24 + π 2 /p + c) Using the results e got from part a), e have that p = and, as a consequence, π = 0. The demands above simplify to x = r = 24 + = 24 2 = = 24 2 = 2 l = 24 r = 2 Finally, normalizing the price of the final good to, e also have p = =. 4

5 d) The social planner s problem is to max x + r x,r s.t. x = F l) Substituting l = 24 r into the constraint, e find that x = 24 r. The planner s problem can therefore be ritten as max 24 r + r r The FOC is 2 24 r) 2 ) + 2 r 2 = 0 r 2 = 24 r) 2 r = 24 r r = 2. Since l = 24 r, e once more find l = 2 and x = F l) = F 2) = 2. This confirms that the planner s solution coincides ith the competitive equilibrium allocation. Question 5 a) The consumer solves The Lagrangian is FOC s: [C]/[R] gives us Plugging this into [λ], e find max C,R C R β s.t. pc + R = + r + π 00 L = C R β + λ + r + π ) 00 pc R [C] : C R β = λp [R] : βc R β = λ [λ] : pc + R = + r + π 00 R βc = p = R = βp C. pc + βp C = + r + π 00 Then C = + r + π/00 + β p R = βp C = βp + r + π/00 + β p. R = β + r + π/00 + β 5

6 b) The firm solves The FOC s are max kl rk l k,l [k] : [l] : l 2 k = r k 2 l = [k]/[l] yields k l = r. As seen in problem set 2, the profits for this firm are zero, since it has a CRS technology. c) First, e can use the normalization p = and the result that π = 0 to simplify C = β + r + r) and R = + β + β. We can also use the feasibility constraint L + R = to find L = R = β + r + β βr = + β). The market clearing condition in the market for capital is k = 00, since each consumer has one unit of capital and rents it to the firm note that K does not enter the consumer s utility function, so they don t consumer K). Using that the firm alays chooses k/l = /r, e can rite l = 00 r. The market clearing condition in the labor market is so After some algebra, e get that 00L = l 00L = 00 r ) βr 00 = 00 r + β). = r + 2β. Finally, the market clearing condition in the consumption good market is 00C = fk, l) = kl = r ) r 00 + r) = 00 + β 6

7 Using the equation e found ith the labor market clearing condition, e can rerite this expression as r + 2β ) r + r = + β 2r + β ) = + β r +2β + 2β and finally, e get that the equilibrium rental rate of capital is r = 2 + 2β No all e need to do is go back to all of the expressions e found before and plug in the value for r. Starting ith ages, e get that = r + 2β = + 2β 2 + 2β Labor is and therefore Leisure can be found as Finally, consumption is = 2 + 2β L = r L = l = + 2β β R = L R = 2β + 2β.. C = + β + r ) = C = + 2β. + β + β 2r For the sake of completeness, e can also include K =, k = 00 and p = as part of our solution. Profits, as mentioned before, are π = 0. 7

8 Question 6 a) A firm hose manager has productivity ill choose labor to maximize profit: The FOC is π) = max n nθ n [n] : θn θ = = n) = ) θ θ. The firm s labor demand is n), hich depends of as shon above. Furthermore, managers ith higher managerial productivity hire more orkers. This model is commonly referred to as a span of control model, here more productive managers are able to control manage) more orkers. b) The firm s profit is π) = [n)] θ n) = ) θ θ θ θ ) θ π) = θ θ θ θ θ θ) θ ) θ π) = θ θ θ θ) ) θ c) Agents ill choose to become hichever occupation pays them more. Since the manager is paid the firm s profit and the orker is paid a age, agents ill compare these to quantities. Note that profits are increasing in managerial productivity, hile ages are fixed. Hence, agents ith high managerial productivity ill choose to become managers and agents ith lo managerial productivity ill choose to become orkers. The cutoff ill be defined by: π) = ) θ θ θ θ θ) = θ θ θ θ θ) = θ = θ θ θ) θ. d) First of all, note that determines the number of orkers and firms in the economy. Since e kno the probability density function of, f), the share of agents ith productivity belo is ) β F ) = fa)da =, hich is increasing on. Conversely, the share of firms managers) is fa)da = ) β = F ). No note that if e increase ages, e ill also increase as shon in part c)) because more agents ill ant to become orkers. Furthermore, if increases, so does F ), the share of orkers in the economy. Given that the population is fixed, by increasing ages, 8

9 e ill decrease the share of managers in this economy, therefore decreasing the number of firms. The intuition is simple: because agents self select into becoming orkers or managers, an increase in ages ill make becoming a orker more attractive to the loer productivity managers. In addition, it ill also decrease profits for all managers, making it even more attractive to become a orkers. Some managers ill therefore shut don their firms and get a job orking for someone else, hich decreases the amount of firms in the economy. e) The total amount of orkers in the economy is and the total demand for labor is N N fa)da na)fa)da. In the first expression, e re just integrating over all agents ith managerial productivity smaller than i.e. orkers. In the second expression, e re integrating over the number of managers multiplied by their labor demand, hich gives us the aggregate demand in the economy. The labor market condition is N fa)da = N Substituting values, ) β = β ) ) θ = β ) ) θ θ = β ) ) θ θ = β ) na)fa)da. ) aθ θ a β da θ a β θ) θ da [ ] θ a 2 θ β θ) θ 2 θ β θ) θ β θ) + θ 22 θ β θ) here the last step follos from the fact that β > 2 θ, hich makes 2 θ β θ) < 0. No plug in = θ θ θ) θ θ θ θ) θ to get ) β = β ) ) θ θ θ θ β θ) + θ 2 β θ θ θ) θ) β β ) θ)θ θ = β θ) + θ 2 After some algebra, e get β θ θβ ) θ) θ)β ) = Thus, in equilibrium, the age is θ θ θ) θ θ θ θ) θ) β θ)+θ 2 θ ) 2 θ β θ) θ β θβ ) β θ) + θ 2 β θ θβ ) θ) θ)β ) = β θθβ ) θ) θ)β ) β 2) β θ) + θ 2 = θ θβ ) θ) θ)β ) β 2) β θ) + θ 2 9 ) β..

10 Question 7 a) Simply by using the definition of Pareto dominance, if {ˆx i } I i=, {ŷj } J j= ) Pareto dominates {x i } I i=, {yj } J j= ) then u i x i ) u i ˆx i ) for all i and u i x i ) < u i ˆx i ) for at least one i b) We ill have that p x i p ˆx i for all i. Furthermore, the inequality is strict for every i for hich u i x i ) < u i ˆx i ). This claim can also be proven by contradiction: suppose that p x i > p ˆx i. This means that consumer i could afford the bundle ˆx i and ould still have some money left: m i = p x i > p ˆx i. She could therefore use the extra money to purchase more goods; for example, she could purchase the bundle here ε 0 and is such that x i = x i,..., x i n) = ˆx i + ε,..., ˆx i n + ε n ) = ˆx i + ε p x i = m i or p ε = m i p ˆx i. Finally, note that, because consumer i s utility function is strictly increasing in all of its coordinates and x i ˆx i, e ould have that u i x i ) > u i ˆx i ) u i x i ). But since x i is affordable by consumer i, it can t be true that x i maximizes her utility subject to her budget constraint and therefore the assumption that p x i > p ˆx i has lead us to a contradiction ith the fact that x i is a competitive equilibrium allocation. It follos that this assumption cannot be true, and therefore e must have that p x i p ˆx i. To prove that p x i < p ˆx i henever u i x i ) < u i ˆx i ), e follo a similar argument. We already kno that p x i p ˆx i for all i, so no e only need to sho that it can t be the case that p x i = p ˆx i for all i. Again, e proceed by contradiction: e assume that p x i = p ˆx i for all i. But in this case, the bundles {ˆx i } I i= are affordable for every consumer. Take one consumer, say consumer n, such that u n x n ) < u n ˆx n ). We kno that this consumer exists because {ˆx i } I i= Pareto dominates {x i } I i= by our first assumption in the question). But then e have again found a bundle that can be afforded by consumer n and that gives her a higher utility than x n, hich is once more a contradiction ith the fact that x n is part of a competitive equilibrium allocation i.e. maximizes consumer n s utility subject to her budget constraint). If follos that it cannot be the case that p x i = p ˆx i for all i; i fact, it must necessarily be true that p x i < p ˆx i for every consumer i for hich u i x i ) < u i ˆx i ). 0

11 c) In part b) e found that p x i p ˆx i for all i, ith strict inequality for at least some i. Summing over i, e get I I p x i < p ˆx i. i= Since both {ˆx i } I i=, {ŷj } J j= ) and {xi } I i=, {yj } J j= ) are feasible allocations, it must the case that I J I J x i = y j and ˆx i = ŷ j. But this implies that i= j= p J y j < p j= i= i= J ŷ j. d) The inequality from part c) means that p y j < p ŷ j for at least one firm j convince yourself of that if it is not immediate). But recall that, because y j includes all inputs and outputs of the firm, p y j is the firm s profit. It follos that, since j= p y j < p ŷ j for some firm j, the competitive equilibrium allocation does not maximize firm j s profit ŷ j is another feasible in the sense that firm j can produce it ith its current technology) allocation for firm j that yields higher profits. e) The result from part d) contradicts that fact that {x i } I i=, {yj } J j= ) is a competitive equilibrium because e found at least one firm j ho is not maximizing profits ith that allocation. Since all firm maximize profits in a competitive equilibrium, it follos that {x i } I i=, {yj } J j= ) could not be a CE allocation. No lets take a step back to think about hat e just did. We stated that {x i } I i=, {yj } J j= ) is a competitive equilibrium allocation ith prices p. Then e made the assumption that there exists another allocation {ˆx i } I i=, {ŷj } J j= ) that is feasible and that Pareto dominates our initial competitive equilibrium allocation. We then proceeded to ork out hat are the logical implications of that assumption, and e found that, if our assumption is true, then {x i } I i=, {yj } J j= ) cannot be a competitive equilibrium allocation. But this contradicts the very foundation of our argument! What can e conclude then? We can conclude that our assumption that there exists some feasible allocation that Pareto dominates the competitive equilibrium is necessarily false. Since that assumption is false, it follos that there is no other feasible allocation that Pareto dominates the competitive equilibrium allocation. Put differently, the competitive equilibrium allocation is Pareto efficient. j=