ADVANCED MACROECONOMICS I

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1 Name: Students ID: ADVANCED MACROECONOMICS I I. Short Questions (21/2 points each) Mark the following statements as True (T) or False (F) and give a brief explanation of your answer in each case The Malthusian theory of population dynamics implied that long-run living standards were bound to stagnate even in the face of technological progress in agricultural production. Piketty identifies top income tax rates as a key factor underlying changes in inequality. In the Lucas Supply Function y = b(p E[p]), the coefficient b depends on the extent of macroeoconomic instability in the economy. In a Keynesian AS-AD model with a linear accelerationist Phillips curve, a linear aggregate demand function and with endogenous monetary policy, the dynamic stability of the model may depend on the size of exogenous shocks.

2 1. True. In Malthusian theory, technological progress increases the population that can be fed with existing resources, but by virtue of diminishing returns to labor, living standards eventually fall back to the subsistence level. 2. True. Piketty finds empirically that the level of top income tax rates is inversely correlated with inequality. High marginal tax rates, he argues, discourage rent seeking behaviour by top income earners. 3. True. b reflects the signal extraction problem producers face in the presence of macroeconomic volatility. Higher aggregate volatility translates into a weaker response of the representative producer to any given price signal she receives from the market. 4. True. An exogenous shock large enough to render monetary policy ineffective due to a binding Zero Lower Bound on the nominal interest rate can bring about a deflationary trap with destabilizing dynamics. For small shocks, monetary policy retains traction and thereby ensures dynamic stability.

3 II. 3 Problems (30 Points) Problem 1 [12 Points: (a) 3P, (b) 5P, (c) 4P] Consider the Ramsey-Cass-Koopmans model of economic growth with zerodepreciation (δ = 0) and a flat tax rate τ on capital income. T t are lump-sum transfers from the government to the representative household. Technology A t and population N t grow at the exogenous rates g and n, respectively. Labor is supplied inelastically. The representative household s behaviour is given by (1) (2) ċ t = (1 τ)r t ρ θg c t θ ẋ t = (1 τ)r t x t + w t c t (n + g)x t + T t where x t denotes household s wealth and c t denotes household s consumption, both in per-effective worker units at time t. r t represents the pre-tax real rate of return (net of depreciation) on capital. w t is the real wage. The firms employ capital and labor to produce aggregate output Y t according to: (3) F(A t L t, K t ) = (A t L t ) 1 α K α t α, ρ and θ are given structural parameters. (a) State the firms optimization problem and set up the first-order conditions. Solution: With W t = A t w t : real wage per worker, FOCs: max K,L Π t := (A t L t ) 1 α K α t W t L t r t K t Π t = α(a t L t ) 1 α Kt α 1 r t = 0 K t Π t = (1 α)a 1 α t L α t Kt α W t = 0. L t (b) Combine your results from (a) with the household s optimality conditions (1) and (2) to derive the equilibrium dynamics for consumption c t and capital per-effective worker k t, taking into account that markets clear (k t = x t ). Taxes are completely redistributed to the households via lump-sum transfers T t. Solution: Note first that in terms of the per-effective worker notation (and because of δ = 0), the FOCS are: αk α 1 t = r t (1 α)k α t = w t

4 Hence, we can rewrite the Euler-equation as: ċ t c t = (1 τ)αkα 1 t θ ρ θg The evolution of the household s wealth can be used to obtain the resource constraint by inserting r t, w t and T t. Moreover, redistribution of tax revenues implies T t = τr t x t. Finally, with x t = k t and ẋ t = k t, (2) gets: k t = (1 τ)αk α 1 t k t + (1 α)k α t c t (n + g)k t + ταk α 1 t k t = k α t c t (n + g)k t (c) Draw a phase diagram to depict the dynamics of the model. How does τ affect the long-run steady state values of per-capita consumption and the saving rate? The steady state consumption is c = k α (n + g)k. Hence, the steady state saving rate is 1 k c α = (n + g)k 1 α. Evidently, the fall in k SS from k SS (τ) to k SS (τ ) also implies a fall in the saving rate.

5 Problem 2 [8 Points: (a) 6P, (b) 2P] Consider the RBC model with Cobb-Douglas production function as in equation (3) from Problem 1. Households derive utility from consumption and leisure and rent out capital and labour to the representative firm. Technology is assumed to follow an AR(1) process given by (4) log A t = ρ A log A t 1 + ε A,t where ε A,t is white-noise and ρ A (0, 1). Suppose the economy is in its long run equilibrium and a negative shock to technology hits the economy. (a) Describe the responses of C t, Y t, K t, as well as L t and explain the role of income and substitution effects in shaping these responses. Solution: The shock amounts to a sudden, though transient, reduction in A. As a consequence, the marginal productivity of labor falls as well. Since standard RBC models assume that intratemporal and intertemporal substitution effects dominate the income effect on labor supply for a temporary shock, labor supply is reduced. Aggregate output thus falls on account of both lower technology and lower labor input. Since the shock is temporary, consumers smooth consumption and reduce their consumption by a smaller amount than output (negative income effect plus intertemporal substitution effect) so that the saving rate is reduced and capital formation is slowed down as a consequence. It is only in limiting special cases, such as the Brock-Mirman model, that labor supply and the saving rate remain unchanged. (b) How did working hours per-capita change in the industrialized countries over the past 100 years and what does this observation tell you about income and substitution effects? Solution: Working hours fell strongly over the past century. This observation tells us that the income effect on labor supply is stronger in the long run than the substitution effect. Of course, long-run permanent advances in technology do not create incentives for intertemporal substitution. As labor is made ever more productive by technological progress, agents use their higher incomes to increase both their leisure and their consumption of goods and services.

6 Problem 3 [10 Points: (a) 3P, (b) 5P, (c) 2P] Consider the following version of the Clarida-Galí-Gertler (1999) model, with the central bank pursuing an optimal discretionary monetary policy: (5) (6) (7) (8) (9) L = αx 2 (π π T ) 2 x : output gap π = λx + π e + u x e : expected output gap x = d 0 ϕ[i π e ] + x e + g π : inflation u = ρu 1 + û π e : inflation expectations g = µg 1 + ĝ i : nominal interest rate x, π, x e, π e and i are endogenously determined. u and g represent cost push shocks and demand shocks, respectively. û and ĝ are white noise. α, λ, ϕ, ρ and µ are given parameters. Autonomous demand d 0 > 0 and the inflation target π T are exogenously given. The notation u s is equivalent to u t s. (a) What are the long-run equilibrium values of x, π, and i (in the absence of exogenous shocks), and how are these equilibrium values affected by an increase in d 0? Solution: In the zero-shock long-run equilibrium, û = ĝ = u = g = 0. Moreover, π e = π = π T and x e = x = 0 (the central bank realizes its lowest possible loss). (7) can be rewritten to obtain the long-run equilibrium nominal interest rate i: i = d 0 ϕ + πt An increase in d 0 increases the nominal and real interest rates, leaving equilibrium x and π unchanged. (b) Assuming an optimal response of discretionary monetary policy, how do x, π, and i respond to (i) a shock û < 0? (ii) a shock ĝ < 0? Solution: (i) A shock û < 0 is a deflationary cost-push shock. Optimal monetary policy will balance the impacts on inflation and output by lowering the nominal and the real interest rate. Hence: x, π, i, [(i π e ) ]. (ii) A shock ĝ < 0 is a negative demand shock which can be neutralized by strong enough a response in i. In particular, to keep π and x constant, it is sufficient to lower nominal interest rates by ĝ/ϕ. Hence: x, π, i. For the case of shocks large enough to imply i = 0 (ZLB) monetary policy can become ineffective in both cases so that the responses in i are weaker.

7 (c) What role does the size of α play in the two cases considered in (b)? Solution: α is the relative weight on the output gap in the central bank s loss function. For high values of α, the response of x to a supply shock is less pronounced whereas the response in π is more pronounced. The nominal interest rate i therefore reacts less strongly for high values of α, too. In the case of a demand shock, the size of α does not matter.