Economic Model with Capital Accumulation and Human Capital, Lecture notes of Macroeconomics

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Macroeconomic Theory:

A Dynamic General Equilibrium Approach

Mike Wickens

University of York

Princeton University Press

Exercises and Solutions

Completed January 2010

Exercises

Chapter

2.1. We have assumed that the economy discounts s periods ahead using the geometric (or exponential) discount factor βs^ = (1 + θ)−s^ for {s = 0, 1 , 2 , ...}. Suppose instead that the economy uses the sequence of hyperbolic discount factors βs = { 1 , ϕβ, ϕβ^2 , ϕβ^3 , ...} where 0 < ϕ < 1. (a) Compare the implications for discounting of using geometric and hyperbolic discount fac- tors. (b) For the centrally planned model yt = ct + it ∆kt+1 = it − δkt

where yt is output, ct is consumption, it is investment, kt is the capital stock and the objective is to maximize Vt =

X^ ∞

s= βsU (ct+s)

derive the optimal solution under hyperbolic discounting and comment on any differences with the solution based on geometric discounting.

2.2. Assuming hyperbolic discounting, the utility function U (ct) = ln ct and the production function yt = Akt, (a) derive the optimal long-run solution. (b) Analyse the short-run solution. 2.3. Consider the CES production function yt = A[αk^1 t −^ γ^1 + (1 − α)n^1 t− 1 γ]^1 −^1 1 γ (a) Show that the CES function becomes the Cobb-Douglas function as γ → 1. (b) Verify that the CES function is homogeneous of degree one and hence satisfies F (kt, nt) = Fn,tnt + Fk,tkt.

(c) Compare these solutions with the discrete-time solution of Chapter 2.

Chapter 3

3.1. Re-work the optimal growth solution in terms of the original variables, i.e. without first taking deviations about trend growth. (a) Derive the Euler equation (b) Discuss the steady-state optimal growth paths for consumption, capital and output. 3.2. Consider the Solow-Swan model of growth for the constant returns to scale production function Yt = F [eμtKt, eνtNt] where μ and ν are the rates of capital and labor augmenting technical progress. (a) Show that the model has constant steady-state growth when technical progress is labor augmenting. (b) What is the effect of the presence of non-labor augmenting technical progress?

3.3. Consider the Solow-Swan model of growth for the production function Yt = A(eμtKt)α(eνtNt)β where μ is the rate of capital augmenting technical progress and ν is the rate of labor augmenting technical progress. Consider whether a steady-state growth solution exists under (a) increasing returns to scale, and (b) constant returns to scale. (c) Hence comment on the effect of the degree of returns to scale on the rate of economic growth, and the necessity of having either capital or labor augmenting technical progress in order to achieve economic growth.

3.4. Consider the following two-sector endogenous growth model of the economy due to Rebelo (1991) which has two types of capital, physical kt and human ht. Both types of capital are required

to produce goods output yt and new human capital iht. The model is

yt = ct + ikt ∆kt+1 = ikt − δkt ∆ht+1 = iht − δht yt = A(φkt)α(μht)^1 −α iht = A[(1 − φ)kt]ε[(1 − μ)ht]^1 −ε

where ikt is investment in physical capital, φ and μ are the shares of physical and human capital used in producing goods and α > ε. The economy maximizes Vt = Σ∞ s=0βs c 11 t−+−sσσ. (a) Assuming that each type of capital receives the same rate of return in both activities, find the steady-state ratio of the two capital stocks (b) Derive the optimal steady-state rate of growth. (c) Examine the special case of ε = 0.

Chapter 4

4.1. The household budget constraint may be expressed in different ways from equation (4.2) where the increase in assets from the start of the current to the next period equals total income less consumption. Derive the Euler equation for consumption and compare this with the solution based on equation (4.2) for each of the following ways of writing the budget constraint: (a) at+1 = (1 + r)(at + xt − ct), i.e. current assets and income assets that are not consumed are invested. (b) ∆at + ct = xt + rat− 1 , where the dating convention is that at denotes the end of period stock of assets and ct and xt are consumption and income during period t. (c) Wt = Σ∞ s=0 (1+^ ct+rs)s = Σ∞ s=0 (1+^ xt+rs)s + (1 + r)at, where Wt is household wealth. 4.2. The representative household is assumed to choose {ct, ct+1,...} to maximise Vt = P∞ s=0 βsU (ct+s), 0 < β = (^) 1+^1 θ < 1 subject to the budget constraint ∆at+1 + ct = xt + rtat where ct is consumption,

(b) If the household’s problem is to maximize expected discounted utility Vt = EtΣ∞ s=0βs^ ln ct+s (i) show that the solution is (^) c^1 t = Et[ (^) ct^1 +1 ] (ii) Using a second-order Taylor series expansion about ct show that the solution can be written as Et[ ∆c ctt+1 ] = Et[( ∆c ctt+1 )^2 ] (iii) Hence, comment on the differences between the non-stochastic and the stochastic solutions.

4.5. Suppose that households seek to maximize the inter-temporal quadratic objective function Vt = − 12 Et

X^ ∞

s= βs[(ct+s − γ)^2 + φ(at+s+1 − at+s)^2 ], β = (^) 1 +^1 r

subject to the budget constraint

ct + at+1 = (1 + r)at + xt

where ct is consumption, at is the stock of assets and xt is exogenous. (a) Comment on the objective function. (b) Derive expressions for the optimal dynamic behaviors of consumption and the asset stock. (c) What is the effect on consumption and assets of a permanent shock to xt of ∆x? Comment on the implications for the specification of the utility function. (d) What is the effect on consumption and assets of a temporary shock to xt that is unantici- pated prior to period t? (e) What is the effect on consumption and assets of a temporary shock to xt+1 that is antici- pated in period t?

4.6. Households live for periods t and t + 1. The discount factor for period t + 1 is β = 1. They receive exogenous income xt and xt+1, where the conditional distribution of income in period t + 1 is N(xt, σ^2 ), but they have no assets. (a) Find the level of ct that maximises Vt = U (ct) + EtU(ct+1) if the utility function is quadratic: U (ct) = − 12 c^2 t + αct, (α > 0).

(b) Calculate the conditional variance of this level of ct and hence comment on what this implies about consumption smoothing.

4.7. An alternative way of treating uncertainty is through the use of contingent states st, which denotes the state of the economy up to and including time t, where st^ = (st, st− 1 ,...) and there are S different possible states with probabilities p(st). The aim of the household can then be expressed as maximizing over time and over all possible states of nature the expected discounted sum of current and future utility Σt,sβtp(st)U[c(st)]

subject to the budget constraint in state st

c(st) + a(st) = [1 + r(st)]a(st−^1 ) + x(st)

where c(st) is consumption, a(st) are assets and x(st) is exogenous income in state st. Derive the optimal solutions for consumption and assets.

4.8. Suppose that firms face additional (quadratic) costs associated with the accumulation of capital and labor so that firm profits are

Πt = Akαt n^1 t −α− wtnt − it − 12 μ(∆kt+1)^2 − 12 ν(∆nt+1)^2

where μ, ν > 0 , the real wage wt is exogenous and ∆kt+1 = it −δkt. If firms maximize the expected present value of the firm Et[Σ∞ s=0(1 + r)−sΠt+s], (a) derive the demand functions for capital and labor in the long run and the short run. (b) What would be the response of capital and labor demand to (i) a temporary increase in the real wage in period t, and (ii) a permanent increase in the real wage from period t?

Chapter 5

where output yt and government expenditures gt are exogenous, and the aim is to maximize Σ∞ s=0βsU(ct+s) for β = (^) 1+^1 r , (a) find the optimal solution for taxes. (b) What is the household budget constraint? (c) Analyse the effects on taxes, debt and consumption of (i) a temporary increase government expenditures in period t (ii) an increase in output. 5.4. Assuming that output growth is zero, inflation and the rate of growth of the money supply are π, that government expenditures on goods and services plus transfers less total taxes equals z and the real interest rate is r > 0 , (a) what is the minimum rate of inflation consistent with the sustainability of the fiscal stance in an economy that has government debt? (b) How do larger government expenditures affect this? (c) What are the implications for reducing inflation? 5.5. Consider an economy without capital that has proportional taxes on consumption and labor and is described by the following equations

yt = Anαt = ct + gt gt + rbt = τ ct ct + τ wt wtnt + ∆bt+ U (ct, lt) = ln ct + γ ln lt 1 = nt + lt (a) State the household budget constraint. (b) If the economy seeks to maximize Σ∞ s=0βsU (ct+s, lt+s), where β = (^) 1+^1 r , derive the optimal steady-state levels of consumption and employment for given gt, bt and tax rates.

5.6 (a) What is the Ramsey problem of optimal taxation?

(b) For Exercise 5 find the optimal rates of consumption and labor taxes by solving the asso- ciated Ramsey problem.

Chapter 6

6.1. (a) Consider the following two-period OLG model. People consume in both periods but work only in period two. The inter-temporal utility of the representative individual in the first period is U = ln c 1 + β[ln c 2 + α ln(1 − n 2 ) + γ ln g 2 ]

where c 1 and c 2 are consumption and k 1 (which is given) and k 2 are the stocks of capital in periods one and two, n 2 is work and g 2 is government expenditure in period two which is funded by a lump-sum tax in period two. Production in periods one and two are

y 1 = Rk 1 = c 1 + k 2 y 2 = Rk 2 + φn 2 = c 2 + g 2

Find the optimal centrally-planned solution for c 1. (b) Find the private sector solutions for c 1 and c 2 , taking government expenditures as given. (c) Compare the two solutions. 6.2 Suppose that in Exercise 6.1 the government finances its expenditures with taxes both on labor and capital in period two so that the government budget constraint is

g 2 = τ 2 φn 2 + (R − R 2 )k 2

where R 2 is the after-tax return to capital and τ 2 is the rate of tax of labor in period two. Derive the centrally-planned solutions for c 1 and c 2.

6.3. (a) Continuing to assume that the government budget constraint is as defined in Exercise 6.2, find the private sector solutions for c 1 and c 2 when government expenditures and tax rates are pre-announced.

Chapter 7

7.1. An open economy has the balance of payments identity xt − Qxmt + r∗ft = ∆ft+

where xt is exports, xmt is imports, ft is the net holding of foreign assets, Q is the terms of trade and r∗^ is the world rate of interest. Total output yt is either consumed at home cht or is exported, thus yt = cht + xt. Total domestic consumption is ct; yt and xt are exogenous. (a) Derive the Euler equation that maximises P∞ s=0 βs^ ln ct+s with respect to {ct, ct+1,...; ft+1, ft+2,...} where β = (^) 1+^1 θ. (b) Explain how and why the relative magnitudes of r∗^ and θ affect the steady-state solutions of ct and ft. (c) Explain how this solution differs from that of the corresponding closed-economy. (d) Comment on whether there are any benefits to being an open economy in this model. (e) Obtain the solution for the current account. (f) What are the effects on the current account and the net asset position of a permanent increase in xt?

7.2. Consider two countries which consume home and foreign goods cH,t and cF,t. Each period the home country maximizes Ut =^ h c (^) H,tσ−^ σ^1 + c (^) F,tσ−^ σ^1 i^ σ−σ 1

and has an endowment of yt units of the home produced good. The foreign country is identical and its variables are denoted with an asterisk. Every unit of a good that is transported abroad has a real resource cost equal to τ so that, in effect, only a proportion 1 − τ arrives at its destination. PH,t is the home price of the home good and P (^) H,t∗ is the foreign price of the home good. The

corresponding prices of the foreign good are PF,t and P (^) F,t∗. All prices are measured in terms of a common unit of world currency. (a) If goods markets are competitive what is the relation between the four prices and how are the terms of trade in each country related? (b) Derive the relative demands for home and foreign goods in each country. (c) Hence comment on the implications of the presence of transport costs. Note: This Exercise and the next, Exercise 7.3, is based on Obstfeld and Rogoff (2000). 7.3. Suppose the model in Exercise 7.2 is modified so that there are two periods and inter- temporal utility is Vt = U(ct) + βU (ct+1)

where ct =^ h c (^) H,tσ−^ σ^1 + c (^) F,tσ−^ σ^1 i^ σσ− 1

. Endowments in the two periods are yt and yt+1. Foreign prices P (^) H,t∗ and P (^) F,t∗ and the world interest rate are assumed given. The first and second period budget constraints are

PH,tyt + B = PH,tcH,t + PF,tcF,t = Ptct PH,t+1yt+1 − (1 + r∗)B = PH,t+1cH,t+1 + PF,t+1cF,t+1 = Pt+1ct+1,

where Pt is the general price level, B is borrowing from abroad in world currency units in period t and r∗^ is the foreign real interest rate. It is assumed that there is zero foreign inflation. (a) Derive the optimal solution for the home economy, including the domestic price level Pt. (b) What is the domestic real interest rate r? Does real interest parity exist? (c) How is r related to τ? 7.4. Suppose the "world" is compromised of two similar countries where one is a net debtor. Each country consumes home and foreign goods and maximizes

Vt =

X^ ∞

s=0^ β

s (cαH,t+s 1 c −^1 F,t− σα+s)^1 −σ

(c) Comment on any differences with the solutions in Exercise 7.4.

Chapter 8

8.1. Consider an economy in which money is the only financial asset, and suppose that house- holds hold money solely in order to smooth consumption expenditures. The nominal household budget constraint for this economy is

Ptct + ∆Mt+1 = Ptyt

where ct is consumption, yt is exogenous income, Pt is the price level and Mt is nominal money balances. (a) If households maximize Σ∞ s=0βsU(ct+s), derive the optimal solution for consumption. (b) Compare this solution with the special case where β = 1 and inflation is zero. (c) Suppose that in (b) yt is expected to remain constant except in period t + 1 when it is expected to increase temporarily. Examine the effect on money holdings and consumption. (d) Hence comment on the role of real balances in determining consumption in these circum- stances.

8.2. Suppose that the nominal household budget constraint is ∆Bt+1 + ∆Mt+1 + Ptct = Ptxt + RtBt

where ct is consumption, xt is exogenous income, Bt is nominal bond holding, Mt is nominal money balances, Pt is the general price level, mt = Mt/Pt and Rt is a nominal rate of return. (a) Derive the real budget constraint. (b) Comment on whether or not this implies that money is super-neutral in the whole economy. (c) If households maximize Vt = Σ∞ s=0βsU(ct+s, mt+s) where the utility function is

U (ct, mt) =

h (^) cαt m (^1) t−α αα(1−α)^1 −α

i 1 −σ 1 − σ

levels of consumption that result from these choices for the economy in Exercise 8.4 when there is a single consumption good ct.

8.6. Consider the following demand for money function which has been used to study hyper- inflation mt − pt = −α(Etpt+1 − pt), α > 0

where Mt= nominal money, mt = ln Mt, Pt= price level and pt = ln Pt. (a) Contrast this with a more conventional demand function for money, and comment on why it might be a suitable formulation for studying hyper-inflation? (b) Derive the equilibrium values of pt and the rate of inflation if the supply of money is given by ∆mt = μ + εt

where μ > 0 and Et[εt+1] = 0. (c) What will be the equilibrium values of pt if (i) the stock of money is expected to deviate temporarily in period t + 1 from this money supply rule and take the value m∗ t+1, (ii) the rate of growth of money is expected to deviate permanently from the rule and from period t + 1 grow at the rate υ.

Chapter 9

9.1. Consider an economy that produces a single good in which households maximize Vt =

X^ ∞

s= βs

ln ct+s − φ ln nt+s + γ ln M Ptt++ss

, β = (^) 1 +^1 r

subject to the nominal budget constraint

Ptct + ∆Bt+1 + ∆Mt+1 = Ptdt + Wtnt + RBt.

where c consumption, n is employment, W is the nominal wage rate, d is total real firm net revenues distributed as dividends, B is nominal bond holdings, R is the nominal interest rate, M is nominal money balances, P is the price level and r is the real interest rate. Firms maximize the present value of nominal net revenues

Πt =

X^ ∞

s=0^ (1 +^ r)

−sPt+sdt+s

where dt = yt − wtnt, the real wage is wt = Wt/Pt and the production function is yt = Atnαt. (a) Derive the optimal solution on the assumption that prices are perfectly flexible. (b) Assuming that inflation is zero, suppose that, following a shock, for example, to the money supply, firms are able to adjust their price with probability ρ, and otherwise price retains its previous value. Discuss the consequences for the expected price level following the shock. (c) Suppose that prices are fully flexible but the nominal wage adjusts to shocks with probability ρ. What are the consequences for the economy?

9.2. Consider an economy where prices are determined in each period under imperfect compe- tition in which households have the utility function

U [ct, nt(i)] = ln ct − η ln nt(i)

with i = 1, 2. Total household consumption ct is obtained from the two consumption goods ct(1) and ct(2) through the aggregator ct = c φt(1)φ(1φ −ct (2)φ) 11 −−φφ