Public Economics: Taxation
Problem Set 2
Carleton University, 2015 Fall
For this problem set, you can work with your classmates, but everybody should hand in their own
copy. Please indicate on your copy who you are working with. I recommend to start working on it
on your own and then compare results with your classmates. When you derive your answers, please
make clear what the answer path is supposed to be. This assignment is due on November 6
1 Welfare Losses and Tax Incidence
There is a small open economy with two types of agents, high-skilled and low-skilled. The utility
function is type-independent and of the form
U(c, `) = c
c is consumption and ` is leisure. Agents have one unit of time. Perfectly competitive firms produce
the consumption good according to the function
Y = (l + k)
where l and h are low- and high-skilled labor. k is the amount of capital in the economy, which is
supplied by foreigners such that the rate of return is equal to r
. To guarantee that the high-skilled
earn more and that capital is utilized, assume that r
* < a and r
* < aa(1 – a)
1-a. The price of
consumption is one and goods can be traded at no cost with the rest of the world.
(a) Find labor supply. What is consumption as a function of the wage? What are wages wl and
wh and capital supply k as a function of parameters only (i.e. as functions of ?, a, and r
(b) The government introduces a small tax on capital at rate t
k > 0, so that the price of capital
for producers is now r
(1 + t
). Who gains and loses from this tax? What could explain this?
(c) The government tries to maximize a utilitarian welfare function, the sum of the utilities of both
types of agent. It can use linear taxes on each type of labor and capital. What characterizes
the optimal tax system? [Hint: There is only one equation that needs to be satisfied. Justify
why.] What is the trade-off between efficiency and redistribution?
(d) What is your biggest concern as to what makes this model ill-suited for the analysis of tax
policy? What would you change to make it reasonable? Be specific about which functions have
to change in what way.
2 Ramsey Taxation with an Untaxable Factor
Consider a standard Ramsey taxation problem as shown in class, but assume the following production
function f(k, z, n) satisfying CRS and the Inada conditions. All cross-derivatives are positive,
so fzk > 0. z is another factor that is supplied by the owner (the representative household), let’s
say it is land. The owner incurs a loss to utility from providing land to the market, since they could
instead use it for their own enjoyment, so uz(c, `, z) < 0. The return to land is denoted by s and
taxes on it are t
(a) Derive the private sector’s first-order conditions.
(b) Derive the implementability constraint, set up the government Lagrangean using this, and
briefly explain why optimal steady-state capital taxes are still zero.
(c) Assume now that the government cannot tax the new factor z. What additional constraint
does this impose on the government?
(d) With this additional constraint, show that optimal steady-state capital taxes are positive. What
could explain it?
If the utility function is: Consumption Function:
U(c, `) = c ? ` 1-? . Y = (l + k) ah 1-a ,
Demand for a commodity x is D(q) with a decreases in q = p + t
Supply for commodity x is S(p) with an increases in p
Equilibrium is satisfied under the condition: Q = S(p) = D(p + t)
Begin from t = 0 and S(p) = D(p). So as to characterize dp/dt: the result of a tax increase on price, which regulates the load of tax:
Adjust dt to causes change in dp so that equilibrium holds:
S(p + dp) = D(p + dp + dt) ?
S(p) + S’(p)dp = D(p) + D’(p)(dp + dt) ?
S’(p)dp = D’(p)(dp + dt) ?
dp /dt = D’(p) /S’(p) – D’(p)
Therefore with the derivation above:
c ? ` 1-? = v(l + k) ah 1-a
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