```Advanced Macroeconomics - prof. Andrzej Cieślik
Problem 1
Solve the following problem:
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max U = Ct − aCt2 + β Ct+1 − aCt+1
Ct ,Ct+1
subject to:
(i) Ct + bt+1 + Kt+1 = Yt
(ii) Ct+1 = Yt+1 + (1 + r) bt+1 + (1 − δ) Kt+1
(iii) Yt+1 = A ln (1 + Kt+1 ),
where A > r + δ.
(a) Find the optimal values of Kt+1 , Yt+1 , Ct , Ct+1 , and bt+1 ;
(b) Given the optimal value of Kt+1 , find ∂Kt+1 /∂r. Does this make sense? Why?
(c) For Yt = 1, β (1 + r) = 1, and A = 2 (r + δ), find ∂bt+1 /∂r. Does the income or the substitution
effect dominate?
Problem 2
Solve the following optimization problem for a three-period lived individual:
max
Ct ,Ct+1 ,Ct+2
U = ln Ct + β ln Ct+1 + β 2 ln Ct+2
subject to:
(i) Ct + bt+1 = Yt
(ii) Ct+1 + bt+2 = Yt+1 + (1 + r) bt+1
(iii) Ct+2 = Yt+2 + (1 + r) bt+2
∗
∗
(a) Find the optimal values of Ct∗ , Ct+1
and Ct+2
;
∗
∗
(b) Find the value function V = U (Ct∗ , Ct+1
, Ct+2
), i.e. evaluate the total utility function at the optimal
values. What form does V have compared to U ?
Problem 3
Consider the neoclassical firm with internal adjustment costs. The production function takes the standard
Cobb-Douglas form Y = AK α L1−α , where A is a fixed level of technology. Imagine that the unit
adjustment cost function takes a specific form ϕ(I/K) = (b/2)(I/K), where b is an exogenous parameter.
The firm maximizes its value subject to the usual accumulation constraint. Assume no depreciation for
simplicity.
(a) Find and interpret the first order conditions.
˙
(b) Find the demarcation lines for q(t)
˙ and K(t),
and display a phase diagram of q (the shadow price of
capital) and K (the capital stock).
(c) Using your phase diagram discuss the effect of an unanticipated increase in A. What happens to the
steady-state values of q and K? Compare your result with the result for the neoclassical case with
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(d) Using your phase diagram discuss the effect of an unanticipated increase in b. What happens to the
steady-state values of q and K? Compare your result with the result for the neoclassical case with
(e) Using your phase diagram discuss the effect of an unanticipated increase in r. What happens to the
steady-state values of q and K? Compare your result with the result for the neoclassical case with
(f) Describe the time paths of K, I/K, and q when there is a permanent and anticipated increase in
the parameter b.
Problem 4
In a two period model suppose the government provides good G to a household that must be paid for
with taxes T on this household. Assume that the household receives this government good only in the
first period of their lives. The government must collect taxes in the first period to finance the provision of
the government good. The household chooses the optimal amount of consumption in each period taking
as given the government good. The household solves thus the following utility maximization problem:
max {U = ln Ct + β ln Ct+1 + a ln Gt }
Ct ,Ct+1
subject to the following constraint:
Ct +
Ct+1
= Yt − Tt
1+r
where a, β > 0.
(a) Find the optimal values of Ct , Ct+1 .
(b) Substitute the optimal values you obtained in a) back into the utility function to obtain an expression
for utility that depends on G and T .
(c) Now have the government choose the optimal level of T that maximizes the utility expression in b)
subject to the constraint that Gt = Tt (balanced budget).
(d) Given the solution to T obtained in c) substitute it back to the expressions for Ct and Ct+1 in a).
∗
(e) If β = 1 and r = 0 is Ct+1
> Ct∗ ? Why or why not?
Problem 5
Consider a consumer with the utility function in
Z ∞
c1−θ − 1
U (0) =
e−(ρ−n)t
dt
1−θ
0
who has an access to asset market which yields a return r (t). He receives a wage w from the firm in
which he works. His budget constraint is therefore:
a˙ (t) = r (t) a (t) + w (t) − c (t) − n · a (t)
The firm in the economy faces a neoclassical production function of the form F (K (t) , L (t) , A (t)),
where A (t) is the level of technology which increases at the exogenously given rate g.
(a) Assuming that output is sold at price P = 1, write down the firm profit function in terms of per
units of efficiency labor (i.e. the product of A and L). Find the FOCs for the firm.
(b) Assuming that economy is closed and using FOCs for the firm derived in (a), find an equation relating
the growth rate of consumption and capital per effective units of labor to consumption and capital
in effective units of labor.
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(c) What are the steady-state growth rates of the variables in effective units? What are their corresponding steady-state levels? Do they depend on parameter θ? Interpret.
(d) Using a phase diagram describe the transition of c and k if the initial value of the capital stock is
below the steady state one (i.e. k (0) < k ∗ ). How does this transition depend on the value of θ?
Problem 6
In class we have assumed that government purchases G in the Ramsey-Cass-Koopmans model do not
affect utility from private consumption. The opposite extreme is that government purchases and private
consumption are perfect substitutes. Specifically, suppose that all taxes are lump-sum taxes and the
lifetime utility function of consumers can be modified to be:
Z
U (0) =
1−θ
∞
e−(ρ−n)t
0
[c (t) + G (t)]
1−θ
−1
dt
(a) Find the first order conditions characterizing the optimal choice of the consumer.
(b) Find the first order conditions characterizing the optimal behavior of the firm producing output with
a standard Cobb-Douglas function assuming that there is a constant rate of capital depreciation δ.
(c) Describe the general equilibrium in this economy using (a) and (b), and draw a phase diagram in
the (c, k) space.
(d) Assuming that this economy is in the steady-state describe the effects of a temporary increase in
government purchases on the paths of capital k, consumption c, output y and the interest rate r over
time.
Problem 7
Consider the RCK model we saw in class. Suppose that households produce output at home and choose
consumption and capital so as to maximize their utility:
Z ∞
U (0) =
e−(ρ−n)t ln c (t) dt
0
subject to their budget constraint:
k˙ (t) = (1 − ty ) Ak α − tL − c − (δ + n) k + v
where: c is consumption per capita and k (0) > 0 is given, where ty is the constant proportional
income tax, tL is a lump-sum tax and v are transfers from the government.
(a) Suppose that the government spends all the tax revenue to finance its consumption which does not
affect consumer’s utility neither firm’s productivity.
α
ty Ak (t) + tL = g, with v = 0.
Find FOCs for all private individuals taking tax rates as given.
(b) Use a phase diagram in c and k to explain the dynamic behavior of the economy when the government
decides to increase the income tax ty to finance the increased spending on government services g.
What happens to the steady-state capital stock?
(c) Repeat the experiment in (a) assuming this time that the government increases the lump sum tax
rate tL instead of ty . If the result is different, explain why.
(d) Suppose now that the government returns taxes to private individuals through lump sum transfers.
That is, repeat the experiments discussed in (a) and (b).
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Problem 8
Let H d be a demand function for housing. It is proportional to the amount of adult people in the
economy N and it is a decreasing function of the rental price of a housing unit R : H d = f (R)N , where
f is a monotonic function with f 0 (R) < 0. Let us assume that the housing market is efficient so there
are no arbitrage opportunities:
R(h) + P˙
=r
P
where r is the real interest rate on bonds, P is the price of a standardized unit of housing, h is the
stock of houses per adult person, h = H/N and R(h) is the rent function which can be derived from
H d = f (R)N . The producers of houses face a decreasing returns technology. Maximization of profits
yields the following function for the housing supply:
H˙ + δH = ψ(P )N
where ψ 0 (P ) > 0 and δ is the constant depreciation rate. Let n denote the rate of growth of adult
population n = (dN/dt)/N .
(a) Rewrite the dynamic model in terms of P and h (housing per adult person). Display a phase diagram
in P and h.
(b) Suppose we are in the 1960s and we know that we are having a baby boom, i.e. a temporary increase
in n. To be more specific, in 1960 we announce that between 1980 and 1989 the growth rate of
adult population will increase from n to n0 > n. Use the phase diagram to show what should be
the effects on housing prices, residential investment and stock of houses between 1960 and today?
Explain intuitively the behavior of all variables.
(c) Imagine that we are in the steady state with a constant price P ∗ and a constant stock of houses per
person h∗ , and that the government suddenly and permanently imposes rent controls. What are the
immediate effects of the rent controls on the price of houses? What are the long run effects on the
stock of houses? Are rent controls always good for the poor?
Problem 9
In the Solow model, there is a golden rule, that gives an optimal rate of savings sgold and associated with
that rate level of capital per worker kgold that maximize consumption per worker c in the steady state.
Without technological progress, the condition for finding the kgold is: f 0 (kgold ) = δ + n.
(a) Show in the Solow model that if k ∗ > kgold then the economy is dynamically inefficient, i.e. we could
increase both short-term and long-term (steady state) level of consumption per person.
(b) Compare the steady state level of capital per person in the RCK model without the technological
progress k ∗ with Solow model’s kgold . Determine, whether the RCK economy can be dynamically
inefficient. Explain, why (not)? Hint: use the assumption in the RCK model: ρ − n > 0.
(c) Introduce the technological progress. Derive kˆgold in the Solow model where g > 0.
(d) Prove that in the RCK model with the technological progress the steady state level of capital per
efficient worker is lower than kˆgold . Use the transversality condition:
Z t
lim at · exp −
(rv − n) dv
=0
t→0
0
Problem 10
Using the phase diagram of the Ramsey model, identify all of the possible steady states. Determine the
stability of each of them (stable, unstable, saddle-path stable). Why, despite the multitude of steady
state candidates in the RCK model, have we concluded that only one of those candidates fulfils all of
the model’s conditions? What role does the transversality condition play?
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Problem 11
Since Thomas Malthus published his An Essay on the Principle of Population, there is an ongoing
argument that because of the fixed land supply, the economic growth will come to an end and result
in mass famine. To determine the truth value of such claim, use the modified RCK model (or Solow
β
model) with the following production function: Yt = Ktα (At Lt ) T 1−α−β , where T denotes the fixed
land supply, L grows with the rate of n, and A grows with the rate of g.
Problem 12
Consider the investment model where neoclassical firms bear adjustment costs ϕ (It /Kt ) when changing
their capital stock. The government on the one hand introduces the tax on the production with tax rate
of t, but on the other it allows to deduce from the tax dues a part ψ of the investment expenses (ψIt ).
(a) Derive the firm’s FOCs, assuming that they buy the investment goods at price pIt = 1, hire labour
paying the wage wt , and sell their final product at price pt = 1.
(b) Using the FOCs, find the separating curves for q˙t = 0 and K˙ t = 0, and draw them in phase diagram
in a (K, q) plane.
(c) Using the phase diagram, analyse the impact of an unexpected and permanent increase in tax
deductible ψ on q, I and K w in both short and long term.
(d) What would happen, if this change was only temporary?
(e) What would happen if instead of a reduction in ψ the government would unexpectedly and permanently decrease the tax rate t?
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