Econ 305 Midterm 2
July 12, 2012
D. Andolfatto
Name (yours and TA)
Instructions. Limit your answers to the space provided below each question. The exam is worth
80 marks. Exam begins at 12:30AM and ends at 2:00PM.
[1] [35 marks] Consider an individual with preferences defined over current and future consumption
(1  2 ) = (1 )12 +  (2 )12  which implies
µ ¶ µ ¶12
2
1
 (1  2 ) =

1
The individual has an exogenous earnings profile (1  2 ) and can borrow or lend freely at the
(gross) real interest rate 
(a) Explain what the preference parameter  is measuring here (i.e., provide an interpretation).
The parameter  determines the weight that an individual places on future consumption in their
objective function. A larger  means that more weight is placed on future consumption (relative
to current consumption). As such,  can be thought of as measuring how “patient” an individual
is.
(b) Write down the lifetime budget constraint. Explain what this constraint means.
The lifetime budget constraint is 1 +2  ≤ 1 +2  This constraint implies that the present
value of lifetime spending cannot exceed wealth (the present value of lifetime earnings).
(c) State the choice problem in plain language. Now state the choice problem mathematically.
An individual’s choice problem is to choose the best feasible lifetime consumption plan (choose
a consumption plan that maximizes utility subject to their lifetime budget constraint). Mathematically, we can write this as
max (1  2 ) subject to 1 + 2  ≤ 1 + 2 
1 2
(d) Derive the “consumption function”—i.e., the demand for current consumption 1 ( 1  2 )
Demonstrate that desired consumption is proportional to wealth.
We know that the optimal consumption plan (the solution to the problem above) must satisfy
the “tangency condition:” (i)  (1  2 ) =  and (ii) 1 + 2  = 1 + 2 . Using   =
 −1 (2 1 )12  one can derive:
¡
¢−1 h
2 i
1 +
1 ( 1  2 ) = 1 +  2

1
(e) How does consumer demand 1 react to an increase in the real interest rate ? Explain (and
depict the experiment using a diagram with budget lines and indifference curves.)
We can rewrite the consumption function like this: 1 = () () where  () is wealth
(present value) and 0  ()  1 is the marginal propensity to consume out of wealth (note, it is
not necessary to do this for answering the question—it just makes it easier). Both () and  ()
are decreasing functions of  and so 1 is a decreasing function of 
Explanation: The real interest rate  is the price of current consumption (measured in units
of future consumption). In general, a change in the interest rate will have substitution and wealth
effects. The wealth effect will depend on whether the person is a lender or borrower. The substitution effect makes the person substitute out of 1 and into the cheaper commodity, 2  For these
preferences, it appears that the substitution effect always dominates the wealth effect. (Diagram
has new indifference curve on steeper budget line with lower 1 )
(f ) How does consumer demand 1 react to an increase in expected future earnings 2 ? Explain
how a financial market allows people to spend their future income.
Consumer demand is an increasing function of expected earnings. A financial market allows
people to borrow against their future income. Borrowing against future income is like spending
your future income today, because you will no longer be able to spend your future income in the
future (it will have to be used to pay back creditors).
(g) Suppose that some people are not permitted to spend their future income. That is, in addition
to the lifetime budget constraint, a person is subject to a “no borrowing constraint” 1 ≤ 1 
Assume that the borrowing constraint binds and depict on a diagram. Explain why consumer
demand in this situation responds to income rather than wealth.
Draw a “kinked” budget line as in lecture with corner solution; i.e., (1  2 ) = (1  2 )
Explanation: If a person cannot borrow, they cannot spend their future income today—they
can only spend their current income today. Consequently, consumption depends on income, rather
than wealth (the present value of one’s lifetime income stream).
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[2] [15 marks] Consider the economy described in question 1 above. In that question, we took 
as exogenous (i.e., we assumed a small open economy). Let us now assume that the economy
is closed. Since there is no capital investment or government sector in this model economy,
the income expenditure identity implies 1 = 1  This, in turn, implies 1 (∗  1  2 ) = 1 
(a) Use this latter condition (together with the 1 derived from question 1) to find an expression
for the equilibrium real rate of interest ∗ 
The condition is
Solving...
¡
¢−1 h
2 i
1 + ∗  2
1 + ∗ = 1

1
 =

∗
µ
2
1
¶12
(b) How is ∗ predicted to respond to an increase in the expected growth rate of the economy
(2 1 )? Explain.
The real interest rate is predicted to be an increasing function of the expected growth rate.
Explanation: Higher growth means that future output/income is more plentiful than current
output/income. The relative scarcity of current output drives up its market price—the real rate of
interest.
Explanation (alternate 1): Suppose 2 increases and 1 stays fixed. The expectation of higher
future income increases wealth. An increase in wealth implies a higher demand for current consumption. Since the current supply of output is fixed, it’s price  must rise to clear the market.
Explanation (alternate 2): Suppose 2 increases and 1 stays fixed. The expectation of higher
future income increases wealth. An increase in wealth implies a higher demand for current consumption. Since current income remains fixed, people must borrow to satisfy their increased consumption
demand. But as this is a closed economy, the excess demand for consumption cannot be met by
imports. Consequently, the interest rate must rise to a point where there is no longer an excess
demand for consumption.
(c) Some economists argue that when households become more uncertain of the future, they spend
less and save more. One way to model this “shock” here is to imagine an economy-wide
increase in the parameter  How is ∗ predicted to respond to an increase in ? Explain.
The model predicts that ∗ is a decreasing function of 
Explanation: As explained above,  can be interpreted as a measure of patience. A higher 
means that people suddenly attach a greater value to future consumption. The way to “send” more
income to the future is to save more. But if everyone wants to save more, the interest rate must
fall (they cannot save more, because this is a closed economy and there is no investment.)
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[3] [10 marks] Capital spending today  contributes to future output  () where  is an increasing
and concave function (  0 ()  0 and  00 ()  0 ). Suppose that people expect (either
rationally or irrationally) a future tax  on the return to their capital investments. Let 
denote the gross real rate of interest. Then the net present value of investing  today is given
by
(1 −  ) ()
 () = − +

(a) What level of  maximizes  ()? (Write down the mathematical condition and explain what
it means.)
A necessary condition is  0 ( ) = 0 or
1=
(1 −  ) 0 ( )

The left-hand-side measures the marginal cost of investment, and the right-hand-side measures the
marginal benefit of investment. Investing one unit of output today costs one unit of output. This
one unit of investment produces  0 () extra units of future output. The producer only gets to keep
the after-tax return of this investment, (1 −  ) 0 () And because this return occurs in the future,
it must be discounted by  so that it can be measured in the same units as the investment cost.
The optimal level of investment,   just balances the marginal cost and benefit of investment.
(b) What is predicted to happen to investment demand if the business sector anticipates a higher
future  ? Explain.
From the condition above, an increase in  means a decrease in   
Explanation: An increase in the tax rate on investment returns lowers the after-tax return on
capital spending. Entrepreneurs are motivated to scale back on their investment plans because
some projects that were worthwhile to undertake at a low tax rate are now no longer worthwhile
at a high tax rate.
[4] [20 marks] A government agency has planned a temporary increase in its purchases of nondurable goods and services from the private sector. There is a debate, among members of
this agency, concerning the best way to finance this expenditure. One side insists on a temporary increase in income and sales taxes (sufficient to keep the budget balanced). The other
side is against any tax increase, and instead favours deficit finance (issuing bonds).
(a) You are asked to serve as a consultant in this debate. What would you recommend and why?
(b) How would your answer to part (a) change if the increase in expenditure was permanent,
instead of temporary? Explain.
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