DXE_MAKR Macroeconomics
MUNI, Spring 2011
Assignment 2
Eva Hromádková
Obligatory problems:Endog - problem #1, RBC - problem #1
Deadline: April 14, 2011 by midnight (per email)
Endogenous growth model
Problem 1 - Utility Enhancing Government Expenditure.
Consider the growth model where firms have the linear production function y = Ak
where A > 0 is capital per capita and y is output per capita (for simplicity there is no
population growth and the size of the population is equal to one). Assume further that
the government expenditure g, which is a fixed fraction of output τ, i.e. g = τy = τAk,
contribute to the welfare of hte private agent. The government expenditure is financed
by lump-sum tax T. Thus infinitely lived household maximizes the following utility
∞
0
(ctgα
t )1−θ
− 1
1 − θ
e−ρt
dt
with α > 0.
1. Assuming perfect competition framework in production factor markets derive the
expression for the interest rate and wage rate. Assume for simplicity that the rate
of capital depreciation is equal to zero.
2. Set up the representative household’s optimisation problem. What are the state
and control variables? What is the current-value Hamiltonian?
3. Derive the F.O.C’s.
4. Derive the Euler equation. What is the growth rate of consumption at the steady
state? What is the steady state growth rate of capital?
5. Set up a social planner problem. What are the control variables? Discuss possible
sources of inefficiency of the decentralized equilibrium in this model.
Problem 2 - Effects of changes of parameters in Ramsey model.
Consider economy with knowledge spillovers where the representative household maximizes
its lifetime welfare subject to its flow budget constraint and No-Ponzi-Game
condition. Assume that the utility function is of CRRA type and that the size of the
household grows at the rate n > 0. There is a continuum of identical perfectly competitive
firms of mass one. The firm i has the production function Yi = AKα
i L1−α
i Kα
where
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0 < α < 1 and K =
1
0
Kidi is the aggregate capital. Assume further that there is a government
which takes household consumption and labor income at constant rates τc and
τw, respectively, and that the firms pay taxes from renting capital at constant rate τr.
Let the government consume the amount of the tax revenues from taxing consumption
and the rest is returned back to the households in the form of lump-sum transfers.
• Write down the household’s problem, firm;s problem and government budget con-
straints
• Derive the F.O.C’s.
• Solve for the growth rates of consumption, capital and output along the balanced
growth path.
• Which taxes can affect the long run growth rate and which cannot? Explain.
Problem 3 - Intermediate Inputs as Durables.
Assume the same setup as in the R&D model presented on the lecture. Suppose now
that the intermediate inputs, Xij, are infinite-lived durable goods. New units of these
durables can be formed from one unit of final output. The inventor of the jth type of
intermediate good charges the rental price Rj and the competitive producers of final
goods treat Rj as given.
1. How is Rj determined?
2. In the steady-state, what is the quantity Xj of each type of intermediate good?
3. What is the steady-state growth rate of the economy? How does this answer differ
from the case we had in class in which the intermediate inputs were perishable
goods?
4. If the intermediate goods are durables, then what kinds of dynamic effects arise
in the transition to the steady-state?
Problem 4 - Production function of new ideas discovery.
Consider the Romer (1990) version of the model with a variety of producer products, as
we had in class with the production function for firm i given by
Yi = Lβ
Y,i
A
j=1
X1−β
ij
where 0 < β < 1, Yi is output, LY,i is labor output and Xij is the employment of the
jth type of specialized intermediate (nondurable) output.
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• Derive the final good’s sector demand for labor and intermediate goods as a function
of the prices w and Pj.
• We assume that the inventor of good j retains a perpetual monopoly right over the
production and sale of the good Xj. As in class, one unit of final good combined
with a design can be transformed into one unit of intermediate good. Solve the
profit maximization problem for an intermediate-goods firm that owns a patent and
show that all intermediate goods sell for the same price. What is the instantaneous
profit π earned by an I-firm as a function of labor used in the final-good sector?
• Assume that the production function in R&D sector for discovering new ideas is
more general than we had in class: the quantity of labor necessary to invent a new
product is ˆη = η
Lλ−1
R A
capturing the externality effects of the stock of ideas A and
the total labor searching for new ideas LR. Assume that < 1 and 0 < λ < 1.
Assume further that population L grows at rate n =
˙L
L
.
1. Explain the meaning of ˙A = LR
η
and interpret the meaning of ˆη. Using the
above general production function specify the rate of technological change
˙A
A
as a function of the labor used in R&D sector and of the level of technology?
Derive the expression for the rate of technological change along a balanced
growth path (BGP). Is this model a true endogenous growth model? Why or
why not? [Hint: Use the fact that along a BGP
˙LR
LR
∗
= n.]
2. Write down the formula for the present discounted value of a blueprint for
an I-good V (t). Show that it implies the arbitrage condition r = π(t)
V (t)
+ V˙(t)
V (t)
.
Explain the meaning of the arbitrage condition. Prove that the profit grows
at rate ˙π(t)
π(t)
= n. Using this and the arbitrage condition along a BGP derive
the expression for the present value V ∗
(t) along the BGP.
3. Explain the meaning of free entry into the R&D sector. Write down the
expression which captures free entry into the R&D sector for this economy.
4. Assume that the household divides his unit time endowment between the
working time in the R&D - lR and the working time in the final goods sector
- lY . So lR + lY = 1 and the aggregate labor in the R&D and the final good
sector are LR = lRL and LY = lY L respectively. Write down the household
problem for this economy. What are the state and control variables for this
problem?
5. Explain in words the nature of all market failures (inefficiencies) that emerge
from the decentralized model.
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RBC
Problem 1 - RBC model, optimisation under uncertainty
Consider the following simple RBC model. Preferences are given by
E0
∞
t=0
βt
[bc1−η
t + (1 − b)l1−η
t ]
1
1−η 0 < β < 1
where l = 1 − n is leisure and production technology is given by
yt = ezt
kα
t n1−α
t
and the resource constraint
ct + kt+1 − (1 − δ)kt = yt
1. Set up the social planner’s problem for the economy and derive the first order
conditions.
2. Instead of a social planner, assume there are households that maximize utility and
firms that maximize profit. Households and firms interact in competitive markets.
Write down the decision problem of the representative households and firms. Derive
the first order conditions and show that they imply the same equilibrium as
derived from the social planner approach.
3. For a given value of consumption and wage, how does the increase in b affect the
labor supply curves?
Problem 2 - RBC model with endogenous labor and capital utilisation
YOU CAN ALSO TRY TO SOLVE IT USING HAMILTONIAN, I JUST PUT IT
HERE IF YOU WOULD LIKE TO TRY SOME RECURSIVE.
There is a centralized economy with one good that can be consumed and invested.
The social planner maximizes expected value of the discounted lifetime utility,
E0
∞
t=0
βt C1−σ
t
1 − σ
−
N1+χ
t
1 + χ
, σ > 0, χ > 0
where C and N are consumption and working hours. The budget constraint and technology
are given by:
Kt+1 = Yt + (1 − δ(ut))Kt − Ct
Yt = At(Ktut)α
N1−α
t
At+1 = Aρ
t vt, K(0), A(0) given.
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The depreciation rate is endogenous and given by δ(ut) = µuθ
t , where µ > 0 and
θ > 1 are parameters and ut is capital utilization rate chosen by social planner. At is
autocorrelated productivity shock, and vt is the i.i.d random variable such that E[ln vt] =
0.
1. (3 points) Write down Bellman equation for the transformed problem. Clearly
denote state and control variable(s).
2. (3 points) Derive First Order Condition(s) for the problem.
3. (1 point) Write down Envelope Theorem condition(s)
4. (2 points) Plug in ET into FOC and derive the Euler equation for this problem.
5. (5 points) Log-linearize equations of the model: capital law of motion, FOC’s,
Euler equation, productivity shock law of motion
6. (4 points) Assume At = ¯A for all times t. Derive marginal product of capital and
the real interest rate in the steady state. Interpret your answer.
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