Exercises

Exercise 8.1. Consider the consumption allocation decision of an infinitely-lived household with (a continuum of) L (t) members at time t, with L (0) = 1. Suppose that the household 337

has total consumption C (t) to allocate at time t.

(1) Show that the problem of the household can be written as

subject to

and subject to the budget constraint of the household,

where i denotes a generic member of the household, A (t) is the total asset holding of the household, r (t) is the rate of return on assets and W (t) is total labor income.

(2) Show that as long as u (∙) is strictly concave, this problem becomes

subject to

Exercise 8.2. Derive (8.7) from (8.9).

Exercise 8.3. Suppose that the consumer problem is formulated without the no-Ponzi game condition. Construct a sequence of feasible consumption decisions that provides strictly greater utility than those characterized in the text.

Exercise 8.4. Consider a variant of the neoclassical model (with constant population growth at the rate n)in which preferences are given by

and there is population growth at the constant rate n. How does this affect the equilibrium? How does the transversality condition need to be modified? What is the relationship between the rate of population growth, n, and the steady-state capital labor ratio k*?

Exercise 8.5.

Exercise 8.6. Explain why the steady state capital-labor ratio k* does not depend on the form of the utility function without technological progress but depends on the intertemporal elasticity of substitution when there is positive technological progress.

Exercise 8.7. (1) Show that the steady-state saving rate s* defined in (8.23) is de­

creasing in ρ, so that lower discount rates lead to higher steady-state savings.

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(2) Show that in contrast to the Solow model, the saving rate s* can never be so high that a decline in savings (or an increase in ρ) can raise the steady-state level of consumption per capita.

Exercise 8.8. In the dynamics of the basic neoclassical growth model, depicted in Figure

8.1, prove that the c = 0 locus intersects the k = 0 locus always to the left of kg_θid. Based on this analysis, explain why the modified golden rule capital-labor ratio, k*, given by (8.21) differs from kg_θid.

Exercise 8.9. Prove that, as stated in Proposition 8.7, in the neoclassical model with labor­augmenting technological change and the standard assumptions, starting with k (0) > 0, there exists a unique equilibrium path where normalized consumption and capital-labor ratio monotonically converge to the balanced growth path. [Hint: use Figure 8.1].

Exercise 8.10. Consider a neoclassical economy, with a representative household with pref­

erences at time t = 0:

There is no population growth and labor is supplied inelastically. Assume that the aggregate production function is given by Y (t) = F [A (t) K(t),L (t)] where F satisfies the standard assumptions (constant returns to scale, differentiability, Inada conditions).

(1) Define a competitive equilibrium for this economy.

(2) Suppose that A (t) = A (0) and characterize the steady-state equilibrium. Explain why the steady-state capital-labor ratio is independent of θ.

(3) Now assume that A (t) = exp (gt) A (0), and show that a balanced growth path (with constant capital share in national income and constant and equal rates of growth of output, capital and consumption) exists only if F takes the Cobb-Douglas form,

(4) Characterize the balanced growth path in the Cobb-Douglas case. Derive the com­mon growth rate of output, capital and consumption. Explain why the (normalized) steady-state capital-labor ratio now depends on θ.

Exercise 8.11. Consider the baseline neoclassical model with no technological progress.

(1) Show that in the neighborhood of the steady state k*, the law of motion of k (¢) ? K (¢) /L (¢) can be written as

where ξ₁ and ξ₂ are the eigenvalues of the linearized system.

(2) Compute these eigenvalues show that one of them, say ξ₂, is positive.

(3) What does this imply about the value of

(4) How is the value of ηι determined?

(5) What determines the speed of adjustment of k (t) towards its steady-state value k*?

Exercise 8.12. Derive closed-form equations for the solution to the differential equations of transitional dynamics presented in Example 8.2 with log preferences.

Exercise 8.13. (1) Analyze the comparative dynamics of the basic model in response

to unanticipated increase in the rate of labor-augmenting technological progress will increase to g' > g. Does consumption increase or decrease upon impact?

(2) Analyze the comparative dynamics in response to the announcement at time T that at some future date T⁰ > T the discount rate will decline to ρ' < p. Does

consumption increase or decrease at time T.

Exercise 8.14. Consider the basic neoclassical growth model with technological change and CRRA preferences (8.30). Explain why θ > 1 ensures that the transversality condition is always satisfied.

Exercise 8.15. Consider a variant of the neoclassical economy with preferences given by

with 7 > 0. There is no population growth. Assume that the production function is given bywhich satisfies all the standard assumptions and A (t) =

exp (gt) A (0).

(1) Interpret the utility function.

(2) Define the competitive equilibrium for this economy.

(3) Characterize the equilibrium of this economy. Does a balanced growth path with positive growth in consumption exist? Why or why not?

(4) Derive a parameter restriction ensuring that the standard transversality condition is satisfied.

(5) Characterize the transitional dynamics of the economy.

Exercise 8.16. Consider a world consisting of a collection of closed neoclassical economies J. Each j ∈ J has access to the same neoclassical production technology and admits a rep­resentative household with preferencesCharacterize

the cross-country differences in income per capita in this economy. What is the effect of the 10% difference in discount factor (e.g., a difference between a discount rate of 0.02 versus 0.022) on steady-state per capita income differences? [Hint: use the fact that the capital share of income is about 1/3].

Exercise 8.17. Consider the standard neoclassical growth model augmented with labor supply decisions. In particular, there is a total population normalized to 1, and all individuals have utility function

where l (t) ∈ (0,1) is labor supply.

(1) Define a competitive equilibrium.

(2) Set up the current value Hamiltonian that each household solves taking wages and interest rates as given, and determine first-order conditions for the allocation of consumption over time and leisure-labor trade off.

(3) Set up the current value Hamiltonian for a planner maximizing the utility of the representative household, and derive the necessary conditions for an optimal solu­tion.

(4) Show that the two problems are equivalent given competitive markets.

(5) Show that unless the utility function is asymptotically equal to

for some h(∙) with h⁰ (∙) > 0, there will not exist a balanced growth path with constant and equal rates of consumption and output growth, and a constant level of labor supply.

EXERCISE 8.18. Consider the standard neoclassical growth model with a representative household with preferences

where G (t) is a public good financed by government spending. Assume that the production function is given by Y (t) = F [K(t),L (t)], which satisfies all the standard assumptions, and the budget set of the representative household is C (t) + I (t) ≤ Y (t), where I (t) is private investment. Assume that G (t) is financed by taxes on investment. In particular, the capital accumulation equation is

and the fraction τ (t) of the private investment I (t) is used to finance the public good, i.e., ^{G (t}) = ^{τ (t) i (t)}.

Take the sequence of tax rates

(1) Define a competitive equilibrium.

(2) Set up the individual maximization problem and characterize consumption and in­vestment behavior.

(3) Assuming thatcharacterize the steady state.

(4) What value of τ maximizes the level of utility at the steady state. Starting away from the state state, is this also the tax rate that would maximize the initial utility level? Why or why not?

EXERCISE 8.19. Consider the neoclassical growth model with a government that needs to finance a flow expenditure of G. Suppose that government spending does not affect utility and that the government can finance this expenditure by using lump-sum taxes (that is, some amount T (t) imposed on each household at time t irrespective of their income level and capital holdings) and debt, so that the government budget constraint takes the form

where b (t) denotes its debt level. The no-Ponzi-game condition for the government is

Prove the following Ricardian equivalence result: for any sequence of lump-sum taxes [T (t)]∞₀ that satisfy the government’s budget constraint (together with the no-Ponzi- game condition) leads of the same equilibrium sequence of capital-labor ratio and consump­tion. Interpret this result.

Exercise 8.20. Consider the baseline neoclassical growth model with no population growth and no technological change, and preferences given by the standard CRRA utility function (8.30). Assume, however, that the representative household can borrow and lend at the exogenously given international interest rate r*. Characterize the steady state equilibrium and transitional dynamics in this economy. Show that if the economy starts with less capital than its steady state level it will immediately jump to the steady state level by borrowing internationally. How will the economy repay this debt?

Exercise 8.21. Modify the neoclassical economy (without technological change) by intro­ducing cost of adjustment in investment as in the q-theory of investment studied in the previous chapter. Characterize the steady-state equilibrium and the transitional dynamics. What are the differences between the implications of this model and those of the baseline neoclassical model?

Exercise 8.22. * Consider a version of the neoclassical model that admits a representa­tive household with preferences given by (8.30), no population growth and no technological progress. The main difference from the standard model is that there are multiple capital goods. In particular, suppose that the production function of the economy is given by

where K_m denotes the m^th type of capital and L is labor. F is homogeneous of degree 1 in all of its variables. Capital in each sector accumulates in the standard fashion, with for m = 1,..., M. The resource constraint of the economy is for all t.

(1) Write budget constraint of the representative household in this economy. Show that this can be done in two alternative and equivalent ways; first, with M separate assets, and second with only a single asset that is a claim to all of the capital in the economy.

(2) Define an equilibrium.

(3) Characterize the equilibrium by specifying the profit-maximizing decision of firms in each sector and the dynamic optimization problem of consumers.

(4) Write down the optimal growth problem in the form of a multi-dimensional current­value Hamiltonian and show that the optimum growth problem coincides with the equilibrium growth problem. Interpret this result.

(5) Characterize the transitional dynamics in this economy. Define and discuss the appropriate notion of saddle-path stability and show that the equilibrium is always saddle-path stable and the equilibrium dynamics can be reduced to those in the one-sector neoclassical growth model.

(6) Characterize the transitional dynamics under the additional assumption that invest­ment is irreversible in each sector, i.e.,for all t and each m = 1,...,M.

EXERCISE 8.23. Contrast the effects of taxing capital income at the rate τ in the Solow growth model and the neoclassical growth model. Show that capital income taxes have no effect in the former, while they depress the effective capital-labor ratio in the latter. Explain why there is such a difference.

Exercise 8.24. Let us return to the discrete time version of the neoclassical growth model. Suppose that the economy admits a representative household with log preferences (i.e., θ = 1 in terms of (8.30)) and the production function is Cobb-Douglas. Assume also that δ = 1, so that there is full depreciation. Characterize the steady-state equilibrium and derive a difference equation that explicitly characterizes the behavior of capital stock away from the steady state.

Exercise 8.25. Again in the discrete time version of the neoclassical growth model, suppose that there is labor-augmenting technological progress at the rate g, i.e.,

For simplicity, suppose that there is no population growth.

(1) Prove that balanced growth requires preferences to take the CRRA form

(2) Assuming this form of preferences, prove that there exists a unique steady-state equilibrium in which effective capital-labor ratio remains constant.

(3) Prove that this steady-state equilibrium is globally stable and convergence to this steady-state starting from a non-steady-state level of effective capital-labor ratio is monotonic.