Econ 4021B - Winter 2016 Dana Galizia, Carleton University Assignment 2 Due: Thursday, March 2, 2017 (in class) 1. (10 marks) In Lecture Note 1, we argued that an increase in the capital stock ¯k has a similar qualitative effect on the equilibrium of the

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Econ 4021B - Winter 2016 Dana Galizia, Carleton University Assignment 2 Due: Thursday, March 2, 2017 (in class) 1. (10 marks) In Lecture Note 1, we argued that an increase in the capital stock ¯k has a similar qualitative effect on the equilibrium of the economy as an increase in z, but we did not formally derive this effect. Do this now. Specifically, find expressions for dl/d¯k and dc/d¯k (analogous to the expressions we found for dl/dz and dc/dz in Section 4.2.1 of Lecture Note 1), and determine whether the signs of these expressions are positive, negative, or ambiguous. 2. (40 marks total) Consider the following modification of the model of Lecture Note1. In particular, in that model we assumed that firms owned the capital stock. Suppose instead that households own the fixed level of the capital stock; that is, households begin with ¯k units of capital, while firms don’t have any. Instead, in order to produce (using the same production function as before, y = zf(n, k)), firms must not only hire labour from the labour market at nominal wage W, but must also rent capital in a capital market at nominal rental rate R. Let r = R/P denote the real rental rate. The model is otherwise identical to the one we studied in class. Assume throughout that none of the NNC’s ever bind. (a) (3 marks) Write down the household’s nominal budget constraint, and use it to derive the real budget constraint. You can assume throughout that this budget constraint always holds with equality. (b) (6 marks) Write down the household’s optimization problem, set up the Lagrangian, and use it to obtain the FOC’s with respect to c and l. Combine these FOC’s to eliminate the Lagrange multiplier, yielding a single optimality condition relating c and l. (c) (5 marks) Write down the firm’s optimization problem, and find its FOC’s characterizing its demand for n and k. (d) (4 marks) Substitute the firm’s FOC’s into the expression for its real profits π. Given that f is CRS (which implies that fn(n, k)n + fk(n, k)k = f(n, k)), what does π equal? (e) (1 mark) Write down the government budget constraint. (f) (3 marks) Write down the market-clearing conditions for the labour, capital, and goods markets. (g) (6 marks) Reduce the set of equations characterizing the equilibrium of this economy to two equations in two endogenous variables, c and l. How do these equilibrium conditions compare to the ones we derived when firms owned the capital stock? How, if at all, will the equilibrium values of c and l differ? 1 Econ 4021B - Winter 2016 Dana Galizia, Carleton University (h) (4 marks) Given the equilibrium levels of capital rental payments and firm profits, is the consumer’s budget constraint any different here than it was when the firm owned the capital stock? (i) (4 marks) Set up the social planner’s problem (but don’t worry about solving it). How, if at all, is its problem different from the one we had when the firm owned the capital stock? (j) (4 marks) Given your answers above, in this model does it matter whether households or firms own the capital stock? If so, how? 3. (50 marks total) Consider a modification of the search and matching model of the labour market discussed in class. First, we assume that there is no unemployment insurance (i.e., b = 0). Second, we assume that g(h) = 0 for every household h. Finally, we assume that M(V, H) = V µH1−µ , where µ is a parameter satisfying 0 < µ < 1. In every other respect, the model set-up is the same (including the assumption that wages are determined by Nash bargaining with the worker’s share given by β). (a) (4 marks) Consider a matched firm-worker pair. Assuming the two come to an agreement over a wage w, write down the firm’s surplus, the worker’s surplus, and the total surplus. Using these expressions, find the Nash wage. (b) (4 marks) Using the functional form for M given above, find expressions for the probability that a worker finds a match, p(θ), and the probability that a firm finds a match, q(θ). (c) (6 marks) Given the Nash wage you determined in (a) and the expression for q(θ) you found in (b), write down an expression for the firm’s expected profit from searching for a worker, R, as a function of θ and exogenous variables only. Use this expression to solve for the equilibrium level of tightness, θ ∗ , as an explicit function of exogenous variables. Use this expression to argue that we must have θ ∗ > 0. (d) (6 marks) Given the Nash wage and the expression for p(θ) you found above, write down an expression for a household’s expected benefit of searching, J, as a function of θ and exogenous variables only. Using this expression, argue that, in equilibrium, every household will search for work (i.e., we will have H∗ = N). (e) (8 marks) Compute the equilibrium matching probabilities p(θ ∗ ) and q(θ ∗ ), number of vacancies V , number matches m, total output y, labour force participation rate η, number of unemployed workers U, and unemployment rate u, expressed in terms of exogenous variables only. (f) (22 marks total) Suppose a social planner (SP) wishes to maximize total output net of 2 Econ 4021B - Winter 2016 Dana Galizia, Carleton University firms’ search costs. Assume throughout part (f) that, as above, every household will search for work (i.e., H = N). i. (4 marks) Let f denote total output net of firms’ search costs. Find an expression for f as a function of V and exogenous variables only. ii. (5 marks) Suppose the SP can choose V . Find the value of V it would choose and denote it VSP . Under what condition(s) would VSP = V ∗ (i.e., when would the de-centralized equilibrium V you found in (e) coincide with the optimal V )? iii. (5 marks) Suppose a government (without dictatorial powers) would like to implement VSP as the de-centralized equilibrium outcome. The only tool it has at its disposal is a tax on firm search. That is, letting τ be the amount of the tax, the firm’s search cost is (1 + τ )k instead of k. What should the government set τ equal to? (HINT: You should be able to find the resulting equilibrium V without needing to fully re-derive the equilibrium with the tax in place.) iv. (5 marks) Under what conditions is the result from (iii) an actual tax (i.e., when is τ > 0) and when is it a subsidy (i.e., when is τ < 0)? Interpret your answer. (HINT: Consider whether V ∗ is greater or less than VSP without the tax/subsidy, and then think about what the government would need to do to incentivize firms to make the socially optimal choice) v. (3 marks) As the above should make clear, the de-centralized equilibrium of this search model is not a competitive equilibrium: if it were, then the First Welfare Theorem would apply, in which case we would always have V ∗ = VSP , but we don’t. It turns out there are two externalities at play here that result in the FWT not applying. See if you can identify any one of them, and explain how you think it would tend to affect how many firms end up entering. (HINT: Does a firm bear all of the costs associated with its decision to search? Does it receive all of the benefits?) 

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