Consider the optimal growth problem for a single agent, as seen in class. Suppose that the
government considers at time zero a tax on consumption. Taxes are anticipated with perfect
foresight. The consumer ignores what the government does with the proceeds of the taxes.
(a) The rate of the tax âœ“ is defined so that for every unit of consumption, the individual have to
pay 1 + âœ“. Show through a simple argument that when the tax is implemented the Euler equation
can be written as
"Ë™(t) = "(t)(â‡¢ ! r(t)),
[c(t)] = (1 + âœ“)"(t).
(b) The tax on consumption is announced at time zero, to be introduced from time T, and mantained
at the same constant rate forever after. Analyze the e↵ect of the tax on consumption and
output in the short and the long term.
(c) What if it is announced that the tax will be e↵ective from time T to time T + ", only?
2. Consider a firm whose level of cash-flow at time t is equal to
â‡¡(t) = F[K(t), L(t)] ! wL(t) ! I(t)
where w is the wage rate which is independent of time, and I(t) is the level of investment at time t.
The function F has diminishing returns to scale. You may assume (if you feel more comfortable)
that F[K, L] = K↵L", with ↵ + & < 1.
The variation of capital is given by
KË™ (t) = I(t).
There is no depreciation of capital. The level of capital at time zero K(0) is given. The firm chooses
I(t) and L(t), at any instant t. The value of the firm is
The firm behaves as a price taker and maximizes its value. The discount rate r is given.
(a) Show that the level of capital tends in the long-run to some value K?. What is the value of
F1[K(t), L(t)] in the long-run?
(b) Analyze the dynamics of investment for two values of K(0), one smaller than K?, the other
greater than K?. Use optimal control and call the shadow price of capital q.
(c) A tax on profits at the rate âœ“ is introduced at time 0. Taxable profits are defined as
p = F(K, L) ! wL.
What are the dynamics of investment after time zero?
(d) Assume now that the level of labor is fixed. There is a tax on profits at the rate âœ“. An
investment credit at the rate âŒ§ is put in place. For every dollar that is spent for investment, the
firm receives a tax credit of âŒ§ . It is assumed that the tax credits do not exhaust the tax liabilities
of the firm.
(1) The tax credit is put in place in time zero, and for only two years. What is the impact of the
policy on investment at time zero?
(2) The tax credit is put in place in a year from time zero, and for only two years. What is the
impact of the policy on investment at time zero?
3. Let c1, c2, c3 be arbitrary constants and show that
(a) xt = c1 + c22t solves xt+2 ! 3xt+1 + 2xt = 0.
(b) xt = c1 + c2(!1)t solves xt+2 = xt.
(c) xt = c1 + c22t ! t solves xt+2 ! 3xt+1 + 2xt = 1.
4. Samuelson (1939) proposed a model of economic fluctuations that can be described by the
Ct = bYt#1,
It = k(Ct ! Ct#1),
Yt = Ct + It + Gt,
where C, I, Y , and G represent consumption, investment, output, and government expenditures,
respectively, b and k are parameters, and G is constant over time.
(a) Determine the stationary solution of the level of output Yt.
(b) Show that depending on the values of the parameters b and k the level of output may exhibit
monotone convergence to the stationary value, dampened oscillations and convergence to the
stationary value, monotone explotion and exploding oscillations.
5. Calculate the steady state values of consumption and capital in the basic optimal growth model
that we have seen in class with the production function f(k) = log(1+k), the rate of time preference
0.05, and the depreciation rate 0.15.
6. Transform the equation
(f(kt) ! kt+1) = &u0
(f(kt+1 ! kt+2)f0
into an equivalent system of two first-order equations. Then linearize the system around the steady
state and discuss its stability properties. Then construct the corresponding phase diagram. Then
compare this with the optimal growth problem as seen in class. Use the obvious assumptions.
7. Solutions of the optimal growth model with variable labor supply are described by three equations
in the state variables (kt, ct, lt), where lt is labor supply at date t by the representative agent.
These equations are
ct = lt[f(kt) + (1 ! ()kt] ! lt+1kt+1,
(ct+1) = u0
1 ! ( + f0
(ct)w(kt) = h0
where w(k) âŒ˜ f(k)!kf0
(k) and h : R+ ! R is a smooth, increasing, convex function that measures
the disutility of work. Recall that the utility of consumption u : R+ ! R is a smooth, increasing,
The first two equations in this dynamical system are the same as in our basic optimal growth
model with fixed labor supply. The third equates the wage rate to the marginal rate of substitution
between leisure and consumption.
(a) Find a way to reduce this system to two dimensions by eliminating one of the state variables.
(b) Suggest how to proceed with the analysis in two dimensions.