M1-TSE. Macro I. 2010-2011. Homework 1 Toulouse School of Economics Ernesto Pasten ([email protected] ) Frank Portier ([email protected] )

Macroeconomics I

Homework 1 October 20, 2010

1. Suppose production has the following CES (constant-elasticity of substitution) form: 1

Yt = {αKtσ + (1 − α) (At Lt )σ } σ where σ ≤ 1 and 0 < α < 1. (a) Does this production function have diminishing returns to each input? (b) Does this production function have constant returns to scale? 2. Consider a profit maximizing firm that chooses inputs Kt and Lt to maximize Πt = Yt − Rt Kt − Wt Lt where Yt satisfies the CES production function described in problem 1. (a) Write down the first order conditions that describe the profit maximizing choices of inputs. (b) Using these first order conditions show that the capital-labor ratio is a function of the relative input price Rt /Wt . (c) Compute the elasticity of the capital-labor ratio with respect to the relative input price. (The d(Kt /Lt ) elasticity is defined as − d(R ). Is this a constant? Explain. t /Wt ) 3. Let A(t), L(t), K(t), Y (t) denote the level of technology, labor, capital and output produced at time t. Assume that technology and labor evolve according to: dA = gA (t) , dt dL = nL (t) dt where g and n are the exponential rates of growth. Assume the production function is: Y (t) = F (K (t) , A (t) L (t)) where F (·) satisfies the assumptions given in class (including CRS), K(t) denotes capital and A(t)L(t) denotes effective units of labor. Capital accumulation satisfies dK = sY (t) − gK (t) dt 1

M1-TSE. Macro I. 2010-2011. Homework 1

where δ denotes the instantaneous rate of depreciation and s denotes the savings rate. (a) Derive the equation that describes the evolution of capital per effective unit of labor: k (t) =

K (t) A (t) L (t)

(b) Plot savings (investment per effective unit of labor) versus break-even investment as a function of k(t) (the necessary investment to keep k (t) constant). Show the effect of an increase in the savings rate on the steady-state k(t). (c) Plot the time paths of k(t) and output per worker in response to such an increase, starting from the original steady-state. 4. Within the context of the Solow growth model described in problem 3, assume that the production function has the Cobb-Douglas form Y (t) = K (t)α (A (t) L (t))1−α (a) Algebraically solve for the steady levels of k(t) and output per effective unit of labor y(t) = Y (t)/(A(t)L(t)). (b) What is the effect of an increase in (i) savings and (ii) the capital share, α. Provide economic intution for your results. 5. Consumption and the golden-rule: In the Solow growth model described in problem 3, consumption satisfies: C (t) = Y (t) − I (t) where gross investment I (t) = sY (t). (a) On the same graph, make a plot of output per effective unit of labor y(t), the break-even investment line (n + g + δ)k(t) and savings per effective unit of labor sy(t). (b) At the steady-state k ∗ , indicate on the graph the amount of output per effective unit of labor and consumption per effective unit of labor c(t) = C(t)/(A(t)L(t)). (c) Suppose the economy starts out at an initial steady-state k ∗ . Describe the time path of consumption per effective unit of labor in response to a one time increase in the savings rate. Does c(t) necessarily increase in the long-run? Explain. (d) Varying the savings rate will change the long-run level of consumption per effective unit of labor c(t). Can you provide a condition which would determine the optimal savings rate in the Solow growth model?

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