ECON 5402

Public Economics: Taxation

Problem Set 2

Carleton University, 2015 Fall

Till Gross

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

th

.

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

?

`

1-?

.

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)

ah

1-a

,

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

k

). Who gains and loses from this tax? What could explain this?

1

(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

z

.

(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?

2

If the utility function is: Consumption Function:

U(c, `) = c ? ` 1-? . Y = (l + k) ah 1-a ,

Labor Supply:

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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