ProblemSet 2

Econ W3213

Intermediate Macroeconomics

Spring 2013

Due at the beginning of class Thursday February 21nd

(or in mailbox 20 in IAB 1022 before that point)

1. Fiscal Stimulus in a Neoclassical Model. The economic model that we have been developing

up until now in the course has been “Neoclassical” in the sense that all markets have been

assumed to be competitive and we have abstracted from all “market failures.” In this question,

we explore the implications of fiscal stimulus – i.e., increases in government purchases – on

output and consumption in this Neoclassical model.1 Later in the class, we will study fiscal

stimulus in a Keynesian model.

Consider a Robinson Crusoe economy (i.e., an economy populated by a large number of

identical households). In such an economy, all households will do the same thing since everyone

is identical. We can therefore represent the whole household side of the economy by one

“representative consumer,” which we refer to as Robinson Crusoe.

One somewhat tricky aspect of writing the model this way is that we will be using the same

symbols to represent both individual variables and the corresponding aggregate variables. For

example, we will use the symbol to represent individual consumption but also to represent

aggregate consumption.2 The same will be true for hours worked , government spending and

taxes . It is important to keep in mind when solving the problem that from the individual’s point

of view certain variables are exogenous – i.e., taken as given. For example, each household

doesn’t take into account how a change in consumption and hours worked will affect taxes and

government spending because from its perspective any change in its behavior has a trivial effect

on the aggregate and thus a trivial effect on taxes and government spending.

Suppose Robinson Crusoe’s preferences can be represented by the following utility function:

log log 1 log .

Here denotes consumption and denotes hours worked as in the models we have seen earlier

in the class. The new element is , which represents government purchases. We entertain the

possibility that Robinson Crusoe may value the things that the government purchases. This is

why shows up in Robinson Crusoe’s utility function. The degree to which Robinson Crusoe

1 Notice, that here I am careful to talk about government purchases as opposed to government spending.

Government spending includes transfers. Here we are focusing on purchases of goods and services by the

government, not transfers. Economists and others are not always careful to make this distinction. Sometimes they

will use the term government spending when they in fact mean to say government purchases. This distinction is

important since fiscal stimulus in the form of transfers can have very different effects on output and consumption

than purchases of goods and services by the government.

2 Formally, the idea here is that there are a continuum of identical households of length one (i.e., one household on

each point on the interval from zero to one). Each one makes the same consumption choice (since they are identical)

and so the aggregate is equal to the individual choice times the “number of households” which is represented by

the length of the interval and for simplicity is equal to 1 (as opposed to in the last problem set).

values government purchases is governed by the parameter . Suppose Robinson Crusoe’s

budget constraint is , where denotes the wage rate and denotes lump sum taxes

paid by Robinson Crusoe. Assume for simplicity that the government runs a balanced budget,

i.e., that . Notice, also, that the resource constraint in this economy implies that

, i.e., consumption plus government purchases cannot exceed the amount of output produced .

In this model, we will treat , , and as endogenous variables. All other variables – including

and – are considered exogenous.

a) (10 points) Derive Robinson Crusoe’s labor supply curve. (Hint: Since is exogenous,

Robinson Crusoe treats it as a constant.)

b) (20 points) Suppose the production function in the economy is and the wage is thus

given by as in Problem Set 1. Use this equation, Robinson Crusoe’s labor supply curve,

the economy’s resource constraint and/or the balanced budget equation to solve for output in

terms of only , , and .

c) (10 points) The government purchases multiplier is defined as the number of dollars that

output rises by when government purchases rise by one dollar. (You can assume for simplicity

that all the endogenous variables are denoted in dollars.) What is the government purchases

multiplier in this economy? If 0, what is the range of values that the government purchases

multiplier can take?

d) (10 points) Solve for consumption in terms of only , , and . Briefly comment on how an

increase in government purchases affects consumption.

e) (10 points) In one paragraph, discuss whether an increase in government purchases makes

Robinson Crusoe better or worse off. In particular, comment on whether Robinson Crusoe is

made better off in the case where he does not value the things the government purchases, i.e., if

0.

2. Ricardian Equivalence. In this question, we explore the consequences of different ways of

financing government purchases. In particular, we study the difference between tax financing

and debt financing of government purchases. The question we ask is, Is the stimulative effect of

the increase in government purchases greater or less when the government finances the stimulus

by an increase in taxes today as oppose to by debt? An important qualifier is that we study this

question under the simplifying assumption that the government can impose lump sum taxes. The

answer to this question is different if the government needs to impose distortionary taxes (i.e.,

labor income taxes or consumption taxes).

Suppose that Robinson Crusoe lives for two periods and that his utility from consumption is

log log .

Here we abstract for simplicity from endogenous labor supply and from the utility that Robinson

Crusoe receives from government spending. This turns out not t0 matter for the analysis at hand.

Robinson Crusoe’s budget constraints are

in period 1 and

1

in period 2. Here and denote Robinson Crusoe’s exogenous income in periods 1 and 2,

respectively, and denote lump sum taxes imposed by the government in periods 1 and 2,

respectively, and denotes the amount of bonds purchased by Robinson Crusoe in period 1.

denotes the exogenous interest rate on bonds. Suppose that Robinson Crusoe can borrow from

“abroad,” i.e., that there is a world bond market in which Robinson Crusoe (and the government)

can borrow and lend at the interest rate . This assumption implies that the amount of bonds that

Robinson Crusoe purchases need not equal the amount of bonds that the government issues.

a) (10 points) Derive Robinson Crusoe’s consumption Euler equation. I.e., derive a condition

that describes the optimal trade-off between consuming in period 1 versus saving and consuming

in period 2.

b) (10 points) Use the consumption Euler equation and the budget constraints to solve for

consumption in periods 1 and 2 in terms of only exogenous variables and parameters (i.e., , ,

, , , and ).

c) (10 points) Now suppose there is a war in period 1 and the government decides to spend to

purchase goods and services. The government needs to finance this spending. We consider two

polar extreme options. The first option is to raise in taxes in period 1 and thus pay for the

entire spending contemporaneously. The second option is to issue debt in period 1 and raise

enough taxes in period 2 to pay back the debt. If the government chooses the second option, how

high do taxes need to be in period 2?

d) (10 points) Use your results from parts (b) and (c) to answer the question, How does the

choice of the government between financing the government purchases by contemporaneous

taxes or debt affect consumption in periods 1 and 2. Comment briefly on the intuition for your

result.

e) For extra credit: Show that this same conclusion holds even if we allow for endogenous labor

supply with disutility of labor of the form used in question 1. (Hint: When maximizing with

respect to many variables, you take the partial derivative of the objective function with respect to

each and set these to zero. I.e., you maximize with respect to each endogenous variable

separately holding the other endogenous variables fixed.)