Prof. Stefania Albanesi The Ohio State University 8723 Macroeconomic Theory IIA Spring 2016 Assignment 1, due January 25, 2016.

Question 1 The purpose of this question is to shed light on one possible factor, lack of substitutability between capital and labor, that may help account for comovement in labor inputs over the business cycle. Consider a model where the preferences of the representative agent are given by: h i1−σ ψ ∞ c (1 − n) X , β ∈ (0, 1) , E0 β t u (ct , nt ) , u (c, n) = 1−σ t=0 where ψ, σ > 0, and satisfy the other restrictions needed for strict concavity (see the handout on adding variable labor to the canonical model). Suppose the resource constraint is given by: ct + kt+1 − (1 − δ) kt

≤ f (kt , nt , zt ) , ν h ν−1 ν−1 i ν−1 f (kt , nt , zt ) = αkt ν + (1 − α) nt ν exp (zt ) ,

where δ, 1 − δ, ν > 0 and zt has the following statistical representation: zt = ρzt−1 + εt , |ρ| < 1, and εt is a zero mean random variable which is independently distributed over time. Note that these assumptions imply that the mean of zt is 0. In addition, k0 > 0 is given and the following restrictions must hold: nt , kt+1 , 1 − nt , ct ≥ 0, for t = 0, 1, 2, ... a) Set up the planning problem associated with this economy. Show that the solution of the planning problem can be represented as a couple of policy rules n = h (k, z) and k 0 = g (k, z) . Explain why these also characterize the aggregate employment and capital decisions in the corresponding sequence of markets equilibrium. Using the strategy in the handout on introducing variable hours in the the canonical model, show that g is increasing in its first argument. b) We can interpret this model as a reduced form for a two sector model with the following specification of technology: ct ≤ f (kct , nct , zt ) , i ≤ f (kit , nit , zt ) , 1

and the following additional restrictions: nit , nct , kct , kit ≥ 0, kct + kit = kt , nit + nct = nt . Derive the policy rule for nc and show that it is given by:   ν 1−ν 1−α c (k, z) ν nc (k, z) = (1 − h (k, z)) , ψ exp (z) where c (k, z) is the policy rule for consumption. What is a necessary condition for comovement of employment in the different sectors? (Hint: If nc is increasing in n, what condition must be verified?) Can you derive a sufficient condition for comovement? How would you go about it? c) The share of income paid to labor, st , is defined as: st =

W t nt , yt

where yt = f (kt , nt , zt ) and Wt is the wage rate in the underlying market economy. Show that in a non-stochastic steady state, 1−ν  1 +δ−1 . 1 − s = αν β OBSERVATION: When ν 6= 1, the labor share formula in the non-stochastic steady state involves a combination of variables denominated in units of time (β and δ) and variables that do not have a time dimension (s, α, ν). E.g. the appropriate value for β and δ depends on whether we are using quarterly or annual data. This is important when we assign parameter values to match long run averages in the data. d) If for an annual model, β
= 1/1.03 and δ = 0.08, for a quarterly model 0.25 it must be that: β = 1/ 1.030.25 and δ = 1 − (1 − 0.08) ˜0.08/4. Assuming we want to match s = 0.64, in other words, we calibrate our model so that the statistic ”labor share” in the model is matched to the average value of the same statistic in the US data. What is the implied value for α for the annual model? What is the implied value for α in the quarterly model?

Question 2 This question is based on the following important evidence for the US economy: 1. Studies such as the Michigan Time Use Survey indicate that a typical married couple allocates 25% of its discretionary time to work in household production activities, including cooking, cleaning, childcare and so on. The typical married couple spends about 33% of its discretionary time working for paid compensation. 2

2. The post-war national income and product accounts indicate that investment in household capital, defined as purchases of consumer durables and residential structures, actually exceeds investment in market capital, defined as purchases of producer durables and non-residential structures, by about 15%. 3. Attempts to measure the value of the output from home production deliver estimates ranging from 20 to 50 percent of the value of measured market GNP. The purpose of this question is to explore implications of introducing home production in the canonical model with variable hours. Consider an economy populated by a continuum of unit measure of identical households, with preferences: U=

∞ X

β t u (cmt , cht , nmt , nht ) ,

t=0

where β ∈ (0, 1) is the discount factor. The instantaneous utility function is defined over consumption of a market good cm , consumption of a home good ch , hours worked for paid compensation nm and hours devoted to home production, nh . Leisure is given by: lt = 1 − nmt − nht , where 1 is the time endowment. Assume that u is continuously differentiable and concave with u1 , u2 > 0 and u3 , u4 < 0, where ui denotes the partial derivative of u with respect to the i−th argument, for i = 1, 2, 3, 4. In each period, the individual is subject to a market budget constraint, that allocates income to three uses, the purchase of market consumption, the purchase of business capital, km , and the purchase of household capital, kh . Business capital is used in market production while household capital is used in home production, and their respective depreciation rates are given by δm and δh , respectively. If wt and rt are the wage rate and the rental rate on business capital, respectively, the budget constraint is given by: cmt + kmt+1 + kht+1 ≤ wt nmt + rt kmt + (1 − δm ) kmt + (1 − δh ) kht .

(1)

The rate of transformation between business and household capital is one, so that their relative price will also be equal to 1. Households are also subject to the home production constraint at each date: cht = g (nht , kht , zht ) ,

(2)

where g is the home production function and zht is a home production technology shock. Assume that g is increasing and concave in labor and capital. Note that there are implicitly no uses for the home production good except for current consumption. 3

a) Define the following problem: V (cm , nm , kh , zh ) = max u (cmt , g (nht , kht , zht ) , nmt , nht ) . nh

(3)

Show that V is continuous, increasing in cm and kh , decreasing in hm , and concave in its first three arguments. This implies that the following problems: max

∞ X

β t u (cmt , cht , nmt , nht ) , P roblem1

(4)

t=0

subject to (1) and (2), and: max

∞ X

β t V (cmt , nmt , kht , zht ) , P roblem2

(5)

t=0

subject to (1) have the same solution. b) Suppose that: u = b ln (c) + (1 − b) ln (l) ,

(6)

where c = C (cm , ch ) is a consumption composite and l = 1 − nh − nm is leisure. Derive the analytical expression for the function V defined in (3) in the following three cases: i) c = cm + ch and g = a0 kh + a1 nh and g = a0 kh + a1 nh ii) c = cam c1−a h and g = khη n1−η iii) c = cam c1−a h h Does home production matter for case iii)? c) Consider case i). The specification for u implies that there is a negative wealth effect on the total supply of labor. Is there a wealth effect on the supply of labor to the market, nm , in the model with home production corresponding to i)? What are the resulting implications for the elasticity of the aggregate (market) labor supply curve corresponding to this specification? d) Suppose that u satisfies (6), that C (cm , ch ) = cm +ch and that g (nht , kht , zht ) = 1−η η kht (zht nht ) . The economy is also populated by identical firms operating the following constant returns to scale technology for the production of the market good: 1−θ θ f (kmt , nmt , zmt ) = kmt (zmt nmt ) . The parameters η and θ are the capital share in home and market production and zm and zh are labor augmenting technology shocks that hit market and home production. We assume that: zmt

= λt zm0 ,

zht

= λt zh0 .

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The firms maximize: Πt = f (kmt , nmt , zmt ) − rt kmt − wt nmt , taking as given rt , wt and zmt . The resource constraint is given by: cmt + xt = yt , where yt is market output and xt is investment. Capital evolves according to the law of motion: kt+1 = (1 − δm ) kmt + (1 − δh ) kht + xt , and kt = kht + kmt . Since capital in each sector may depreciate at different rates, also define: xmt

=

kmt+1 − (1 − δm ) kmt ,

xht

=

kht+1 − (1 − δh ) kht .

Define a sequence of markets equilibrium for this economy. e) Derive the first order necessary condition for the household and firm problem. Show that there exists a balanced growth path in which cm , ch , y, kh and km grow at the same rate, λ. Display the expression for xm /km and xh /kh along the balanced growth path. Show that on the balanced growth path nmt = nm and nht = nh .

Question 3 Exercises 2.8, 2.9, 2.10 in Stokey and Lucas with Prescott.

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