1. Constant-elasticity-of-substitution (CES) or Dixit-Stiglitz aggregators. Consider the following function J: J(x) = a(j)x(j) ρ dj
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1 Macro II (UC3M, MA/PhD Econ) Professor: Matthias Kredler Problem Set 1 Due: 29 April 216 You are encouraged to work in groups; however, every student has to hand in his/her own version of the solution. 1. Constant-elasticity-of-substitution (CES) or Dixit-Stiglitz aggregators. Consider the following function J: ( 1 J(x) = ) 1/ρ a(j)x(j) ρ dj where each j represents one variety of a consumer good or an input into a production function and J(x) is either utility over the varieties or production of a final good. x : [, 1] R is from the space of continuous non-negative functions on the unit interval 1, and we assume ρ (, ) (, 1]. a : [, 1] R is a given continuous function of utility/productivity coefficients with a(j) j [, 1]. For the case ρ =, we define ( 1 ) J (x) = exp a(j)x(j)dj, which is called a Cobb-Douglas utility/production function. Also consider the following analogous utility/production functions for the finite-dimensional case. First, we have 1/ρ n J(x 1,..., x n ) = a j x ρ j, which is defined on the space of vectors in the positive cone {x R n : x j, j = 1,..., n}, where we have the same restriction on ρ as before and a j > for all j. For the case ρ = we define ( J n ) (x) = exp a j log x j = which also in this case is called Cobb-Douglas. (a) Contours. Use Matlab to plot the contours (indifference curves for utility, isoquants for production) of J when n = 2. Convince yourself that J converges to the perfect-complements/leontief case as ρ. Note that this convergence to Leontief also happens for the case n > 2 and the infinite-dimensional case J. (Hint: Have a look at the command contour.) (b) Homogeneity. Show that both J and J are homogeneous of degree one, i.e. J(λx) = λj(x) for all λ > and all x. Is there an economic interpretation of this property for the utility/production function? (c) Concavity. Show that both J and J are strictly concave if ρ < 1, i.e. J(λx + (1 λ) x) > λj(x) + (1 λ)j( x) for all x x and for all λ (, 1). 1 All results we will derive can be extended to the most-general case possible (the space of measurable functions); we merely assume continuity for technical convenience. n x aj j 1
2 (d) Gradient. i. Find the marginal effect of changing x into some direction h R n on J(x); specifically, find the w j (x) in the following expression: lim 1 [ ] J(x + h) J(x) = n J x j }{{} w j(x) Interpret w j (x) for both the utility and the production function economically. What happens to w j (x) when we change x j but keep the other components x i, i j, unchanged? ii. Now, carry out the equivalent operation in the infinite-dimensional case. Find the function w in lim 1 [ ] J(x + h) J(x) = 1 h j w(j)h(j)dj. Interpret the function w for the utility/production case. 2 (e) Inada condition if ρ < 1. Show that J satisfies an Inada condition if ρ < 1: for all x, x i > for all i j: lim w j(x) = x j Show that the same holds for w(j), some fixed j, in the case of J. (f) Inputs essential if ρ. For this part of the exercise, see J and J as a production function only. We will say that an input x j is essential for production if x j = J(x) =. Show that all inputs are essential in the CES aggregator J if and only if ρ. Show that the same holds for J in the following sense: Let inputs go to zero on some interval, i.e. for < δ < 1 calculate lim [ δ a(j)[x(j)] ρ dj + 1 δ a(j)x(j) ρ dj] 1/ρ. (g) First-order conditions of consumer s problem. Consider a consumer who chooses consumption of a vector of goods x subject to a budget constraint: max J(x) x s.t. 1 p(j)x(j)dj W for a given price function p > and given budget W >. Write the Lagrangian L and find that the gradient L x of L with respect to the function x (see Question 1d for how to do this). In the optimum, this gradient has to be equal to zero, i.e. L x (j) = for all j [, 1]. 3 From this first-order condition, find an expression 2 Note that w defines a linear functional on a function space that linearly approximates J, just as the gradient defines a linear functional approximating J in R n. The linear functional characterized by w is called the Frechet derivative of J at x (see Luenberger for a precise definition of this generalization of gradients.) 3 The Lagrange-multiplier theorem can be extended to infinite-dimensional spaces; see again Luenberger. 2
3 for the ratios x (j) x (i), j i, under the optimal demand schedule. Show that the demand schedule exhibits constant elasticity of substitution, i.e. d ln [ x(j)/x(i)] d ln[p(j)/p(i)] = d[x(j)/x(i)] x(j)/x(i) d[p(j)/p(i)] p(j)/p(i) = σ for some constant σ. Find an expression for σ. What is the economic interpretation of σ? What happens as ρ 1, ρ and ρ? (h) Demand function, price elasticity, expenditure shares. Express all x (i), i [, 1], as a function of one quantity x (j) (fixing j) and use the budget constraint at equality (why can we do that?) to obtain a demand function of the form ( ) α x (j) = W P, P p(j) where P is a price index (a function of all prices {p(i)} 1 ) and α is a number to be determined by you. What is the price elasticity of demand, d ln x (j)/d ln p(j)? Show that the expenditure shares x (j)p(j)/w are decreasing in p(j) if ρ >, constant in the Cobb-Douglas case and increasing in p(j) if ρ <. Argue with the demand elasticity to give an intuition for this result. (i) Monopolistic competition, mark-ups. Suppose that each good j [, 1] is produced by a monopolist who produces at a constant marginal cost c(j). Find the optimal price the monopolist should set, taking demand x (j) as given. (Note that the monopolist is so small we say atomistic that he does not influence the price index P, which is an important simplification with respect to the finitedimensional case.) What is the mark-up p (j)/c(j)? Show that the monopolist s problem does not have a solution when ρ <. In economic terms, why is this the case? Comment: CES demand systems and monopolistic competition are important workhorses in new-keynesian models (where price-setting is an important aspect of equilibrium), endogenous-growth models (where monopoly rents drive the invention of new varieties j, entailing increases in aggregate productivity) and international-trade models (where consumers love of variety is essential to generate the observed high levels of trade between similar countries). 2. Efficient allocation and flexible-price benchmark in the New-Keynesian model. (a) Planner s problem. Consider a planner who faces the physical environment of the basic New-Keynesian model that we saw in class. Note that the planner s resource-allocation problem is static since there is no investment in capital. We will split the problem into two parts, similar to how we proceeded when solving the household s problem in the decentralized problem. i. Show that in order to produce a given level of output c, the planner should equalize labor inputs across varieties and n (i) = n = c/a for all i. Show that this occurs in the decentralized equilibrium if and only if all prices are equalized, i.e. iff P t (i) = P t for all i. 3
4 ii. Find an equation that characterizes the optimal aggregate consumption c and labor choice n. Compare this equation to its equivalent in the deterministic steady state of the decentralized economy. Is labor supply systematically too high or too low with respect to the efficient benchmark in the decentralized equilibrium? What is the source of inefficiency? (b) Flexible-price benchmark. We used some results from the flexible-price benchmark (θ = ) when deriving the New-Keynesian Phillips Curve in class. We will now see where these come from. We will denote variables in this flexible-price benchmark with an n-superscript (where n stands for natural ). We already showed in class that under flexible prices, all firms i [, 1] set Pt n (i) = MCt n, (1) 1 }{{} M where M 1 is the so-called frictionless mark-up and MCt n is marginal cost (under flexible prices) at t. You may use the equilibrium equations that we derived in class for the general case, i.e. θ. i. Show that the real marginal cost, MCt n /Pt n, is constant over time. ii. Show that the real wage, W r,n t = Wt n /Pt n, is a fixed fraction of worker productivity: W r,n t = 1 A t. (2) iii. Show that output/consumption is directly proportional to hours worked (not only to a first-order approximation, but precisely): Y n t = C n t = A t N n t. (3) iv. Show that hours worked depend only on TFP and are given by ( 1 ) 1 Nt n σ+ψ = A 1 σ ψ+σ t. Do hours work increase or decrease when a positive TFP shock hits? Give an intuition. v. Derive the following expression, which we used in class in order to determine the output gap y t y n t : ( MC n ) ln t Pt n = (σ + ψ) ln Yt n (1 + ψ) ln A t. vi. Finally, show that the real interest rate, rt n following expression: = i n t E t [π n t+1], is given by the r n t = ρ + σ 1 + ψ σ + ψ E t[ a t+1 ], where we define ρ ln β and a t+1 a t+1 a t. Note that the real natural rate is needed to derive the Dynamic IS Curve. vii. Suppose that the government disposes of the following instrument: It can pay a proportional employment subsidy τ to firms for the outlays that 4
5 they have for paying workers. This subsidy is financed by lump-sum taxes. Show that for the right choice of τ, the flexible-price equilibrium attains the efficient allocation. What is the efficient τ? 4 3. Cost-push shocks in the New-Keynesian Model. Consider the New-Keynesian model given by: IS : x t = φ[i t E t π t+1 ] + E t x t+1 + g t, NKP C : π t = λx t + βe t π t+1 + µ t, where x t is the output gap and π t is the inflation rate. g t can be interpreted as a demand shock and µ t is a cost-push shock (capturing all elements other than demand that might affect the marginal cost). These shocks follow g t = ρ g g t 1 + ĝ t µ t = ρ u µ t 1 + ˆµ t (a) Iterate the IS equation forward to show that current aggregate economic activity depends on beliefs about future policy. (b) Iterate the New-Keynesian Phillips curve forward to show that current inflation depends on firms beliefs about future marginal cost. (c) Suppose that the central bank has the following objective function: [ max 1 ] 2 E t β i [αx 2 t+i + πt+i] 2 i= The central bank cannot commit to a specific behaviour nor can it affect expectations. In each period, the central bank chooses {x t, π t } to maximize the objective function subject to the NKPC, taking inflation expectations as given. It then decides what interest rate {i t } it should set to achieve its goal. Find the optimality condition of the central bank. Interpret. (d) Guess that, due to rational expectations, we have E t [π t+1 ] = ρ u π t. Find the expression for inflation and output gap and verify that your guess was correct. How does the presence of cost-push shocks affect the trade-off between inflation and output gap? Under which conditions is extreme inflation targeting optimal? (e) Plug in the results on the IS equation and find how the optimal policy can be implemented. How should monetary policy respond to cost-push and demand shocks? 4. Simulation of the New-Keynesian Model with cost-push shocks. We are going to use Dynare to simulate the New-Keynesian model given in the previous exercise. The variance of the shocks is.1. Consider the following calibration: φ =.25, λ =.2, β =.98, ρ g = ρ u =.9 and α =.5. Consider two alternative policies: the optimal one given in 3c, and a simple Taylor Rule of the type (4) i t = 1.5π t +.5x t. (5) Note that it is easier to include the two different models (optimal policy and interest rate rule) in the same.mod-file in two different blocks. 4 It is common in New-Keynesian models to assume that such a subsidy is in place. The idea is that we don t want positive effects of monetary policy to be due to fighting the pricing power of monopolies. There are other institutions than the Central Bank that should aim to eliminate these inefficiencies (Antitrust agency, government etc.). 5
6 (a) Plot the response of the inflation rate, output gap and the interest rate to a 1% cost-push shock for the two policies. Is the Taylor Rule qualitatively close to optimal? Why/why not? (b) Plot the response of the inflation rate, output gap and the interest rate to a 1% demand shock for the two policies. Is the Taylor Rule qualitatively close to optimal? Why/why not? (c) Can you plot the variances of inflation and output gap as a function of α in response to a cost-push shock (only for the optimal policy)? 6
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