Macroeconomics II Problem Set 1 Solutions PDF

Problem Set I Macroeconomics II Solutions Stefano Maria Corbellini 1 Solution: CIES/CRRA Utility Functions Suppose that a household lives for two periods...

Problem Set I Macroeconomics II Solutions Stefano Maria Corbellini 1 Solution: CIES/CRRA Utility Functions Suppose that a household lives for two periods (t = 0, 1) and maximizes lifetime utility given by: c1−σ u(c0 ) + βu(c1 ) with u(c) = σ > 0 and σ 6= 1 (1) 1−σ 00 1. The coefficient of relative risk aversion is defined as − uu0 (c (ct ) t) ct. Derive the coefficient of relative risk aversion. Solution: The measure of relative risk aversion (RRA) is defined as u00 (ct )ct RRA = −. u0 (ct ) For the utility function given in this task, the first and second derivatives are given by 1 u0 (ct ) = (1 − σ)c−σ t = c−σ t 1−σ u00 (ct ) = −σc−σ−1 t The RRA is u00 (ct )ct −σc−σ−1 t ct RRA = − 0 = − −σ = σ. u (ct ) ct Thus, the function given in this exercise features constant relative risk aversion (CRRA). c1−σ −1 2. Now use the more general CRRA-utility function: u(c) = 1−σ. Show that limσ→1 u(c) = ln(c). f (x) f 0 (x) Hint: Use l’Hôpital’s rule: If lim f (x) = lim g(x) = 0 or ±∞, then lim = lim 0 x→c x→c x→c g(x) x→c g (x) Solution: With σ → 1, both the numerator and the denominator go to zero. Therefore, we can apply l’Hôpital’s rule: If f (x) f 0 (x) lim f (x) = lim g(x) = 0, then lim = lim 0 x→x0 x→x0 x→x0 g(x) x→x0 g (x) Applying this rule we have c1−σ t −1 ln ct c1−σ t (−1) lim = lim = ln(ct ). σ→1 1 − σ σ→1 −1 Macroeconomics II Problem Set I 3. Suppose the household receives some income w0 at date 0 and receives no income afterwards. The intertemporal budget constraint is then given by: c1 c0 + = w0 w0 > 0, R > 1 R Write down the Lagrangian for the households’  ∗  optimization problem and solve for the optimal c1 consumption at date 1 relative to date 0. c∗0 Solution: The Lagrangian is given by:  c1  L = u(c0 ) + βu(c1 ) + λ w0 − c0 − R The first-order conditions are given by: ∂L = u0 (c0 ) − λ = 0 (2) ∂c0 ∂L 1 = βu0 (c1 ) − λ = 0 (3) ∂c1 R 1 0 From (2) we get λ = u0 (c0 ). Inserting this into (3) gives βu0 (c1 ) − u (c0 ) = 0. Hence optimal R consumption levels satisfy: σ u0 (c∗0 ) c∗1 c∗1  1 = = βR ⇒ = (βR) σ (4) u0 (c∗1 ) c∗0 ∗ c0 4. Derive the intertemporal elasticity of substitution. Hints: The intertemporal elasticity of substitu- c∗ tion denotes the elasticity of optimal consumption ∗1 with respect to changes in the interest rate R. c0 Recall that the elasticity of a function f (x) with respect to x equals: x 0 εx = f (x) f (x) Solution: We can make use of the definition given in the question, i.e. use the definition of the elasticity x 0 εx = f (x) f (x). First, we need to find the components of our function. Recall that we are interested in the response of the consumption path to a change in R. The consumption path is given by the 1 solution from (4). It follows that f (x) = (βx) σ , where we replaced the interest rate R with x to emphasize that this is the variable of interest. The IES is then given by x 0 εx = f (x) f (x) x 1 1 = σ −1 1 β(βx) (βx) σ σ βx 1 1 = σ −1 1 (βx) (βx) σ σ 1 = σ 2 Solution: Constant Returns to Scale Production Functions Suppose firms employ a production function f (K, L) with constant returns to scale. 2/6 Macroeconomics II Problem Set I 1. Denote w and r as the market wage and capital rental rate respectively. Suppose factor markets are competitive, such that the wage equals the marginal product of labor (w = fL (K, L))1 and the capital rental rate equals the marginal product of capital (r = fK (K, L)). Show that constant returns to scale combined with competitive factor markets imply that firms make zero profits. Solution: A production function with constant returns to scale has the property that when all input factors are multiplied by α then the output is multiplied by α, too. I.e. the function f is homogeneous of degree 1. f (αK, αL) = αf (K, L) In a competitive factor market, capital and labour are rewarded according to the value of their marginal product. We can show this from the maximization problem of the firm: max f (K, L) − rK − wL K,L ⇒ f1 (K, L) = r (5) f2 (K, L) = w (6) Making use of equation (5) and (6), we can write the profits π of the firm in the following form: π = f (K, L) − f1 (K, L)K − f2 (K, L)L (7) From the properties of a CRTS function we know that αf (K, L) = f (αK, αL) (8) holds for all α. Differentiating equation (8) with respect to α yields f (K, L) = f1 (αK, αL)K + f2 (αK, αL)L (9) = f1 (K, L)K + f2 (K, L)L. (10) Where we find equation (10) by evaluating equation (9) for α = 1. Using the result from equation (10) we find that the profits from equation (7) must be zero. K 2. Write output per worker as a function of the capital-labor ratio k = L Solution: Y K We use L = y and L = k as the notation for output per worker and capital per worker. Using 1 the properties of a CRTS function and choosing α = L we can write the oput per worker in the following form:   Y K y= =f , 1 = f (k, 1) (11) L L 3. Write both the wage rate and the capital rental rate as functions of k (assuming that w and r equal the marginal product of the respective production factor). Solution: 1f x (x, y) denotes the partial derivative of f (x, y) with respect to x. Equivalently, we will use f1 (x, y) to denote the partial derivative of f (x, y) with respect to its first argument. 3/6  Macroeconomics II Problem Set I To find the marginal products we multiply equation (11) for L and take derivatives with respect to K and L:   K yL = f (K, L) = f ,1 L L   K 1 f1 (K, L) = fK (K, L) = f1 , 1 L = f1 (k, 1) (12) L L       K K K f2 (K, L) = fL (K, L) = f , 1 + f1 , 1 L − 2 = f (k, 1) − f1 (k, 1)k (13) L L L (14) From (5) and (6) we know that the rental rate r is given by (12) and the wage w is given by (13). Example: Cobb-Douglas Production function f (K, L) = K α L1−α  α f (K, L) K y= = = k α = f (k) L L f1 (K, L) = αK α−1 L1−α = αk α−1 = f1 (k, 1) f2 (K, L) = (1 − α)K α L−α = (1 − α)k α = f (k, 1) − f1 (k, 1)k 3 Solution: Competitive Equilibrium in a Production Economy We now consider a static economy with production. There are N households. Each is endowed with k̄ units of capital. Households work lh hours and consume c consumption goods. Their utility is U = u(c) − v(lh ) where u(·) is increasing and strictly concave and v(·) is increasing and weakly convex. There are M firms, each of them produces consumption goods with a constant return to scale technology out of labor and capital: Y = f (kf , lf ). Markets are competitive. Assume all prices are denoted in terms of the consumption good c (numeraire). Denote r as the real rental rate of capital (in terms of consumption goods) and w as the real wage. not in physical units 1. Write down the maximization problem of a household. Derive first order conditions, and the in- v 0 (lH ) tratemporal marginal rate of substitution u0 (c) = w. Solution: The households in this economy solve: max u(c) − v(lH ) subject to c ≤ k̄r + lH w + π c,lH π are the profits per household in terms of consumption goods (households are assumed to own the firms and get any excess profits). k̄ is the fixed per-household capital supply. We can write down the Lagrangian: \m L = u(c) − v(lH ) + λ(k̄r + lH w + π − c) The first-order conditions with respect to c, lH and λ are given by: u0 (c) = λ (15) λw = v 0 (lH ) (16) c = k̄r + lH w + π (17) Where (17) is simply the budget constraint of the household, holding with equality. From (15) and 4/6 Macroeconomics II Problem Set I (16) we find the Intratemporal marginal rate of substitution v 0 (lH ) =w (18) u0 (c) 2. Write down the maximization problem of a firm. Derive first order conditions for labor and capital demand. Solution: The firms in this economy solve: plata capitalului (obt din imprumut) plata fortei munca max π = f (kf , lf ) − kf r − lf w kf ,lf cifra de afaceri - cost which yields the usual first-order-conditions: no restrictions, optim liber cand pune indicele jos, intelegem ca derivata in raport cu indicele flf (kf , lf ) = w (19) fkf (kf , lf ) = r (20) From the properties of constant returns to scale and a competitive economy we know that π = 0. 3. Define the competitive equilibrium in this economy. Solution: A competitive equilibrium is defined by a set of prices such that all agents behave optimally and markets clear. Optimality implies: households choose c and lH optimally according to (18) and(17) firms choose lf and kf according to (19) and (20) market clearing implies: capital market: N k̄ = M kf |{z} | {z } supply demand labor market: N lH = M lf |{z} |{z} supply demand consumption good: M f (kf , lf ) = |{z} Nc | {z } supply demand 4. Use u(c) = ln(c), v(lh ) = lh and f (kf , lf ) = kfα lf1−α. Derive the equilibrium labor supply, real wage and rental rate of capital. Solution: Using u(c) = ln(c), v(lh ) = lh and f (kf , lf ) = kfα lf1−α (18) and (17) yield c=w (21) r lH = 1 − k̄ (22) w 5/6 Macroeconomics II Problem Set I where the last line follows from the properties of the firms production function, which implies zero profits due to the constant returns to scale assumption. From the firms optimality conditions in (19) and (20) we find α r= (23) k 1−α w = (1 − α)k α (24) kf r where k = lf. Thus the ratio w is: r α 1 α 1 = 1−α = w k (1 − α)k α 1−αk From capital market clearing we know: kf = N/M k̄ = nk̄ where we define n = N/M. Then we can nk̄ k̄ express the labor supply of firms through market clearing as lF = nlH. Thus k = nlH = lH and we can solve for lH with (22) r lH = 1 − k̄ w α lH =1− k̄ 1 − α k̄ Then: lH = 1 − α (25) Labor supply, real wage, and rental rate of capital are, respectively: ∗ lH =1−α  α ∗ k̄ w = (1 − α) 1−α α r∗ =  1−α k̄ 1−α 6/6