DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
Economics 21: Microeconomics (Summer 2002)
Midterm Exam 1 - answers
Professor Andreas Bentz
instructions
You can obtain a total of 100 points on this exam.
Read each question carefully before answering it. Do not use any books or notes. The use of
non-programmable calculators, or the use of only the non-programming and non-graphing
functions of a programmable calculator, is permitted. You have 65 minutes to answer all
questions. Please use the space provided to answer questions; if you need more space, use the
back of the page.
There are 7 questions on this exam. These 7 questions are independent of each other.
Good luck.
Name: ……………………………………………………..…………
Section (circle as appropriate): 9L (section 1)
10 (section 2)
questions 1 - 7
1. Constance has the following utility function (sometimes referred to as a “constant elasticity of
substitution”, or “CES” utility function), defined over quantities of good 1 (x1) and good 2 (x2):
u(x1, x2) = (x10.5 + x20.5)2
a. (5 points) What is the marginal rate of substitution when Constance consumes 4 units of
good 1 and 25 units of good 2?
answer: MRS = -(MU1/MU2). MU1 = 2 (x10.5 + x20.5) (0.5) x1-0.5, and MU2 = 2 (x10.5 + x20.5)
(0.5) x2-0.5. Dividing one by the other we get MRS = - x20.5/x10.5. At the bundle (4, 25), that
is MRS = -5/2 = - 2.5.
b. (5 points) Good 1 costs p1 and good 2 costs p2, and Constance has wealth m. Write up
the Lagrangean function, and derive the first-order conditions without solving for the
optimal x1 and x2.
answer: L = (x10.5 + x20.5)2 - λ (p1x1 + p2x2 - m) with the first-order conditions:
(i)
2 (x10.5 + x20.5) (0.5) x1-0.5 = λ p1
(ii)
2 (x10.5 + x20.5) (0.5) x2-0.5 = λ p2
(iii)
p1x1 + p2x2 = m
c.
(5 points) If you were to solve the Lagrangean in part (b) of this problem, you would get
the following answer for the optimal consumption of good 1 (which of course depends on
p1, p2, and m):
x1(p1, p2, m) =
p2
m
p1(p1 + p2 )
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
Is this good a normal or an inferior good? Why? Show by calculating the appropriate
derivative.
answer: The partial derivative of x1 with respect to m is p2/[p1(p1+p2)] which is greater
than zero, so that good 1 is a normal good.
2. Douglas has the following utility function over quantities of “timed” consumption in
period 1 (c1) and in period 2 (c2):
u(c1, c2) = c14c26
Douglas has income m1 in period 1, and income m2 in period 2. The interest rate is r. There is
inflation, at a rate π.
a. (5 points) What is the marginal rate of substitution when Douglas consumes 20 units in
period 1 and 30 units in period 2? (In the intertemporal choice model sometimes the
marginal rate of substitution is referred to as the “rate of time preference”.)
answer: You can use the positive monotonic transformation ln(.) to make the utility
function easier to work with. Therefore use the utility function 4 ln(c1) + 6 ln(c2), which
makes taking derivative easier. Now, MRS = -(MU1/MU2). MU1 = 4/c1, and MU2 = 6/c2.
Therefore MRS = -(4c2)/(6c1). At the bundle (20, 30), this is -1.
b. (15 points) Write up, and solve (for c1 and c2), the Lagrangean for Douglas’s constrained
maximization problem.
answer: First, the budget constraint is c2 = m2 + [(1+r)/(1+π)] (m1 - c1). But it makes the
math look less cluttered if you use (1+ρ) = [(1+r)/(1+π)], so that the constraint looks as
follows: c2 = m2 + (1+ρ) (m1 - c1). For the utility function, again, use the (natural log)
positive monotonic transformation.
L = 4 ln(c1) + 6 ln(c2) - λ (m2 + (1+ρ) (m1 - c1) - c2)
(i)
4/c1 = - λ (1+ρ)
(ii)
6/c2 = - λ
(iii)
c2 = m2 + (1+ρ) (m1 - c1)
Using (ii), we know that λ = - 6/c2. Putting this into (i) we get 4/c1 = (6/c2) (1+ρ), or
c2 = (6/4) c1 (1+ρ). Putting this into the budget constraint (iii), we get (6/4) c1 (1+ρ) =
m2 + (1+ρ) (m1 - c1), or (dividing both sides by (1+ρ)): (6/4) c1 = m2/(1+ρ) + (m1 - c1),
Solving for c1, we get (10/4) c1 = m2/(1+ρ) + m1, or c1 = (4/10) [m2/(1+ρ) + m1]. From this
follows that c2 = (6/10) [m2 + (1+ρ) m1].
c.
(5 points) Suppose the nominal interest rate is 20% and the inflation rate is 10%. Using
the exact formula, what is the real interest rate?
answer: The real rate of interest, ρ, is defined by: (1+ρ) = [(1+r)/(1+π)]. That is, ρ =
[(1+r)/(1+π)] - 1. For the given numbers, ρ = [(1.2)/(1.1)] - 1 = 9.1%.
3. Annie’s demand for good 1 (x1), which is a function of the price of good 1 (p1), the price of
good 2 (p2), her income (m), and parameters of Annie’s utility function (a and b) looks as
follows:
a
m
 + b ⋅ p2

 p1 
x1(p1, p2, m) = 
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
a. By calculating the appropriate derivative, answer the following three questions (i - iii): for
which values of the parameters a and b (if it doesn’t matter which value a parameter
takes, write “this parameter can be anything”) is good 1:
i.
(5 points) … a normal good?
a −1
m
1
answer: We need dx1/dm > 0. The derivative (w.r.t. income) is a ⋅   ⋅ . We
p1
 p1 
know that m > 0, p1 > 0 (income and prices are always positive), so the derivative
will be > 0 if also a > 0. b can be anything.
ii.
(5 points) … a Giffen good?
answer: The question here is when dx1/dp1 > 0. The derivative (w.r.t. own price)
a −1
m


 p1 
is − a ⋅ 
 m 
⋅  2  . Again, we know that m > 0, p1 > 0, so the derivative will be
p 
 1 
> 0 if a < 0. b can be anything.
iii.
(5 points) … a (gross) substitute for good 2?
answer: We need dx1/dp2 > 0. The derivative (w.r.t. cross price) is b. That is
positive if b > 0. a can by anything.
b. (10 points) Suppose a = 2, b = 0, p2 = $10, and m = $50. What is the price elasticity of
demand when Annie consumes 100 units of good 1?
2
 50 
answer: The values give us x1(p1, p2, m) =   . This of course is a constant elasticity
 p1 
demand function, so the elasticity is constant at -2. To work out this result, note that if
Annie consumes 100 units good 1 (x1 = 100), then p1 must be $5. The elasticity is defined
as (dx1/dp1)(p1/x1) = [- 2(50/p1)(50/p12)][p1/x1] = -2.
c.
(5 points) Suppose a = 2, b = 0, p2 = $10, and m = $50. Annie buys good 1 from a
monopolist, and the monopolist wonders about the rate at which its revenue would
increase if it were to sell just a little more of good 1 to Annie. You are the consultant for
this monopolist and your answer for the rate at which revenue increases with output is:
answer: You know that MR = p(1 + 1/elasticity). Since we worked out in part (b) that
elasticity is -2, MR = p(1 - 1/2) = (1/2)p.
4. (8 points) The market demand curve for a good is as follows:
x(p) = a - b·p
where p is the price of the good, x is its quantity, and a and b (with a, b > 0) are parameters in
this demand curve (this is just the linear demand curve familiar from ECON01). Calculate
marginal revenue.
answer: First, calculate revenue. We need everything to be a function of x (so we can then
take the derivative to find marginal revenue), so first you need to rewrite the demand function
as an inverse demand function (i.e. solve for p in terms of x). That is, p = (a/b) - (1/b)x. Then
calculate revenue, which is price times quantity, i.e. [(a/b) - (1/b)x] x = (a/b) x - (1/b) x2. The
derivative w.r.t. x (that is, marginal revenue) is: (a/b) - 2(1/b) x. Note that this marginal
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
revenue curve has the same intercept as the inverse demand curve (a/b), but its slope is
twice as steep. This is the mathematical justification for why in ECON01 you drew the
marginal revenue curve as you did: twice as steep as the (inverse) demand curve.
5. (5 points) Following the announcement of an impending interest rate increase, the price of a
perpetuity that pays $5 every year (from next year onwards) falls to $100. That means that
the market expects the interest rate to be cut to what rate?
answer: The present value of a perpetuity with yearly payments of $5 is $5/r. If the price of
this perpetuity is $100, that means that $100 = $5/r, or r = 0.05. That is, the market expects
an interest rate of 5%.
6. (5 points) Today’s date is 07/12/2002. You are considering buying a bond with face value
$200, and a coupon of $10, with a maturity date of 07/12/2005. If you buy the bond you will
get the first coupon payment today. The interest rate is 5%. How much should you just be
willing to pay for this bond?
answer: The present value of this bond is $10 + $10/(1.05) + $10/(1.05)2 + $210/(1.05)3 =
$210.
7. (12 points) Evaluate (i.e. true or false?) the following statement by drawing a diagram that
shows both the Hicks decomposition and the Slutsky decomposition: “For a price increase,
and a normal good, the Hicks substitution effect is always larger (in absolute value) than the
Slutsky substitution effect.”
answer: True. See diagram below
x2
BS
BH
C
A
x1
Substitution effect (Slutsky): A → BS
Substitution effect (Hicks): A → BH
The Hicks substitution effect is larger than the Slutsky substitution effect.
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