DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
workout set 1 - answers
1. For the Cobb-Douglas utility function u(x1, x2) = x1ax21-a, find the MRS for the bundle (2, 2),
the bundle (4, 4), and the bundle (6, 6). What is the MRS for the bundles (3, 1) and (6, 2)?
Try to draw (approximately - you don't have a value for a - but mark out what is important, i.e.
what you have just found out about the MRS in the first part of this question) the indifference
curves for this utility function.
Good 1 costs p1 per unit and good 2 cost p2 per unit, and the consumer has wealth m.
Find the optimal quantity of good 1 that this consumer will consume, and the optimal quantity
of good 2.
Find the consumer's demand function for goods 1 and 2. [Do this both ways: try to solve the
Lagrangean for the utility function x1ax21-a and for the positive monotonic transformation
ln(x1ax21-a). You will hopefully find that it is a lot easier if you first take the log.] How much
money does this consumer spend on good 1? How much on good 2? Which fraction of her
income does this consumer spend on good 1? On good 2?
Now find the consumer's Engel curve for good 1. What do you notice about this Engel curve
(i.e. does it have a special form)?
Is good 1 a normal or an inferior good? Is it a Giffen good?
(The reason you may be interested in questions about normal/inferior goods and Engel
curves is the following: suppose that econometric estimates tell us what a consumer's utility
function looks like [ECON20 teaches you how to make these estimates]. Then suppose a
government wants to know the impact of different ways in which health benefits could be
given: you could either subsidize healthcare [so that everyone pays only a fraction of what
healthcare they consume: this is a change in the price of healthcare], or you could give
people a certain amount of healthcare for free [which is like a change in income]. Of course
many other scenarios can be answered using the same tools.)
answer:
MRS = - MU1 / MU2 = (x21-a a x1a-1) / (x1a (1-a) x2-a) = - a/(1-a)·(x2/x1)
So at bundle (2, 2), the MRS is - a/(1-a)·(2/2) = - a/(1-a). Obviously, the MRS is the same at
bundles (4, 4) and (6, 6).
For the bundle (3, 1) the MRS is - a/(1-a)·(1/3). Obviously, the MRS is the same at bundle
(6, 2).
Utility functions with this property (the property that for bundles that are multiples of each
other, the slope of the indifference curve (MRS) is the same) are called homothetic utility
functions: along each ray from the origin, the indifference curves have the same slope.
This already implies the answer to the second part of the question (about the Engel curve): If
the slope of the indifference curves is the same for all bundles that are multiples of each
other, then that means that if we expand a consumer's income (i.e. we shift the budget
constraint out in a parallel fashion), then all the tangencies between indifference curves and
budget constraint (the optimal choices) are going to be picking out bundles that are multiples
of each other: if I increase income, the amount of good 1 (and good 2) the consumer wants to
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
consume increases proportionally with income. That is, the Engel curve is linear, as in the
picture below:
This is the math (using the positive monotonic transformation ln(·) of the utility function):
Lagrangean: L(x1, x1, λ) = a ln(x1) + (1-a) ln(x2) - λ (p1x1 + p2x2 - m).
First-order conditions:
(i)
(ii)
(iii)
a/x1 = λ p1, or a = λ p1x1
(1-a)/x2 = λ p2, or (1-a) = λ p2x2
p1x1 + p2x2 = m
Adding (i) and (ii), we get: 1 = λ (p1x1 + p2x2), which rewrites as 1 = λm, or λ = 1/m. Putting
this into (i) we get x1 = am/p1, and putting λ = 1/m into (ii) we get x2 = (1-a)m/p2.
So the demand function for good 1 is x1(p1) = am/p1, and that for good 2 is x2(p2) = (1-a)m/p2.
The consumer's expenditure on good 1 is p1x1, and we have just worked out that x1 = am/p1,
so the consumer's expenditure on good 1 is p1 am/p1, which is just: am (the p1's cancel). So
the consumer spends the fraction a of her income m on good 1. You can do the same for
good 2: you find that the consumer spends the fraction (1-a) of her income on good 2. That is
the interpretation of the exponents in a Cobb-Douglas utility function: they tell you the “budget
share” or which fraction (share) of your wealth (budget) you spend on each good.
The Engel curve for good 1 is x1(m) = am/p1, which is a linear function with slope a/p1. This is
just what we expected.
The Engel curve for good 1 is x1(m) = am/p1. That is, demand for good 1 increases as income
rises, that is dx1(m)/dm = a/p1 which is positive. This good therefore is a normal good. So it
can't be a Giffen good, which the following confirms: demand for good 1 decreases as the
price increases: dx1(p1)/dp1 = - am/p12, which is negative.
2. A consumer has the following utility function: u(x1, x2) = ln(x1) + x2. What is the MRS at the
bundles (2, 1), (2, 2) and (2, 3)? Try to draw (again, approximately) the indifference curves for
this utility function.
Her wealth is m, and she faces prices p1 and p2 for goods 1 and 2, respectively.
Find the optimal quantity of good 1 that this consumer will consume, and the optimal quantity
of good 2.
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Derive the consumer's demand curve for good 1, and her demand curve for good 2. Then
derive the Engel curve for good 1. What do you notice about this Engel curve?
Just as an example, suppose that the price of good 1 is $1 and the price of good 2 is $1, and
that the consumer’s wealth is 50 cents. What are the optimal quantities of good 1 and good
2? What’s going on?
answer:
A utility function of this form is called a quasilinear utility function. This is because utility
changes in a linear fashion as one of the variables is changed: if we increase x2 by 1, then
utility goes up by 1. If we increase x2 by 2, then utility goes up by 2, etc. The indifference
curves for a utility function like this look as follows:
The Lagrangean for this problem is L(x1, x1, λ) = ln(x1) + x2 - λ (p1x1 + p2x2 - m).
The necessary first order conditions are:
(i)
(ii)
(iii)
1/x1 = λ p1
1 = λ p2
p1x1 + p2x2 = m
From (ii) we obtain λ = 1/p2. Replacing λ in (i) with this expression, we obtain 1/x1 = (p1/p2) or
x1 = p2/p1. Put this into (iii) to get p2 + p2x2 = m, which rewrites as x2 = m/p2 - 1.
So the demand function for good 1 is: x1(p1) = p2/p1.
The demand function for good 2 is: x2(p2) = m/p2 - 1. (The result that x2 does not depend on
p1 is specific to this particular form of the utility function [especially, the fact that I have
chosen ln(x1), and is not a general feature of all quasilinear utility functions.)
The Engel curve for good 1 is just: x1(m) = p2/p1. This Engel curve does not depend on wealth
at all, that is it must be vertical. The next diagram illustrates:
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
Actually, here we really need to start worrying about corner solutions. As an example,
suppose that the price of good 1 is $1 and the price of good 2 is $1, so that x1 = 1, and x2 = m
- 1. What does the math predict for your consumption of good 2 when your income is 50
cents? It says that you will consume -1/2 unit of good 2 and 1 unit of good 1. This is clearly
stupid: you cannot consume negative quantities of goods. (You can see what goes on if you
draw a budget constraint for very low income into your diagram: the tangency is below the
horizontal axis. And this tangency is all the math can describe.) So when you get a clearly
"stupid" result of negative consumption, this alerts you to the fact that not all is well with the
math, and you have to look at what the diagram tells you - viz. that for low incomes you do
reduce your consumption of good 1. In this case, you’d just draw a diagram and that would
tell you the right answer: you consume zero of good 2, and you spend all your remaining
money (50 cents) on good 1, that is you buy 1/2 unit of good 1 (corner solution). This is the
reason we ruled out corner solutions - they make our life difficult!
3. A consumer has the following utility function: u(x1, x2) = x1 + x2. What is the MRS at the
bundle (2, 4) and at the bundle (3, 3) [these bundles are, of course, on the same indifference
curve]. Which preferences does this utility function therefore represent?
answer: Again, the MRS is (minus) the ratio of the marginal utilities. Here it is: - 1/1. The
slope of the indifference curve is -1 everywhere (regardless of what x1 and x2 are). So the
indifference curves are straight lines with a slope of -1 … they represent perfect substitutes in
the ratio 1:1.
4. I have the following (quasilinear) preferences over consumption (c) and leisure (l):
u(l, c) = √l + c.
I have no wealth, but instead have to work in order to earn money which I can then spend on
consumption. Each day, I have 24 hours that I can allocate to work and leisure (and nothing
else), and I want you to tell me what I will do. My wage rate is w per hour, and each unit of
the consumption good costs p. For this labor-leisure choice problem you should know from
ECON01 that the budget constraint is (24 - l)·w = p·c. (If you don’t remember, make sure you
know where it comes from. Remember that each budget constraint is of the form: money out
= money in. Here, "money out" is the money you spend on consumption. "Money in" is the
money that you earn from work: the number of hours you work times your hourly wage.)
Now make your prediction about how much time I will spend in leisure activities based on the
assumption that I am rational.
Now make your prediction about how much time I will spend in leisure activities based on the
assumption that I am rational. The time that I do not spend in leisure activities I spend
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
working (i.e. supplying labor). Find out my labor supply function. Will I work more as my wage
increases? Why or why not?
answer:
Review: The budget constraint is of the form money out = money in.
money out is money spent on consumption, i.e. pc [buying c units of consumption at a price
of p each]
money in is money earned through work, i.e. (24 - l)w [I have 24 hours - if I spend l hours in
leisure activities, I am spending 24 - l hours working. Each hour of work gives me a wage of
w.]
So the budget constraint is: pc = (24 - l)w
The Lagrangean for this problem is: L(l, c, λ) = √l + c - λ (pc - (24 - l)w)
The necessary first order conditions are:
(i)
(ii)
(iii)
1/(2√l) = λw
1 = λp
pc = (24 - l)w
From (ii) we get λ = 1/p. Putting this into (i) we have 1/(2√l) = w/p, or rewritten: l = (p/2w)2.
Finally, putting this into (iii) we get pc = (24 - (p/2w)2)w, or c = 24w/p - (p/4w).
Since I spend l(w) = (p/2w)2 hrs in leisure activities, I will work n(w) = 24 - l(w) = 24 - (p/2w)2
hours.
Will I work more as my wage increases? We just need to find dn(w)/dw = 2(p/2w)(p/2w2),
which is positive, so I will work more as my wage increases.
You have just worked out the labor supply model that should be familiar from ECON01.
5. There are 100 identical individuals who are consumers in the market for education. Each
consumer has the following utility function over education (e) and other goods (y): u(e, y) =
e1/3 + y. Education costs p per unit, and "other goods" cost $1 per unit. Each individual has
the same wealth, m. Derive an individual's demand curve for education. Then find the market
demand curve.
Sketch the market demand curve.
What is the elasticity of market demand when price is 0.33 and the total amount of education
sold is 100? What is the elasticity of market demand when price is 3 and the total amount of
education sold is 3.70?
You are currently selling 100 unit at a price of 0.33 each to this market. What is your marginal
revenue, i.e. what would happen if you expanded output slightly (by slightly lowering price)?
answer:
As before, write up the Lagrangean, to solve for the individual demand curve. Here, the
Lagrangean is L(e, y, λ) = e1/3 + y - λ(pe + y - m). The necessary conditions are:
(i)
(1/3) e-2/3 = λp
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
(ii)
(iii)
1=λ
pe + y = m
using (ii) and plugging it into (i), we get e = (3p)-3/2. (And then we could get from (ii) what the
demand for other goods is, viz. y = m - 3-3/2p-1/2.)
So (3p)-3/2 is one individual's demand curve. But this is the same for all 100 individuals, so we
just need to add up the 100 individuals' demand curve to get the market demand E(p) =
100(3p)-3/2.
The elasticity is (dE(p)/dp)(p/E(p)), or 100(3)-3/2(-3/2)p-5/2(p/E(p)). If you put in the numbers for
p and E(p), you will see that the elasticity is always -3/2. (More directly, you know that E(p) =
100(3p)-3/2. Just put that in the expression for the elasticity to get an elasticity of -3/2, for any
point on the demand curve.
Demand curves with this property (elasticity is the same everywhere) are called constant
elasticity of demand curves. (When properly they should be called constant elasticity of
demand demand curves.) Below is what it should look like (quantity is measured in 100's).
Marginal revenue of course is therefore p(E) (1+1/(-3/2)), or (1/3) p(E), so it is always
positive: whenever you increase quantity (lower price), your revenue will increase.
6. A consumer's preferences over two goods (c1 and c2) can be represented by the following
utility function:
u(c1, c2) = c1c2.
You may want to call c1 "general consumption this period", and c2 "general consumption next
period". The way this consumer can trade off consumption this period against consumption
next period (her budget constraint) is as follows: c1 + c2/(1+r) = m1 + m2/(1+r). (Here, m1 is
what she earns in period 1 and m2 is what she earns in period 2; r is the interest rate.)
How much does this consumer consume this period (what is c1), and how much does she
consume next period (what is c2)? The difference between how much you earn in period 1
and how much you consume in period 1 is how much you save; if it's a negative number it is
how much you borrow. . Suppose that the interest rate is 10% (r = 0.1). This consumer's
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
income in this period (m1) is 40 and her income (m2) next period is 22. How much does she
save (or borrow)?
answer:
The Lagrangean is: L(c1, c2, λ) = c1c2 - λ[c1 + c2/(1+r) - m1 - m2/(1+r)]. Taking the first-order
conditions and solving, you obtain:
1


c 1 = 0.5 m1 +
m2  .
1
r
+


For the given figures, your consumption in period 1 is:
1


c 1 = 0.5 40 +
22 = 30.
1 .1 

Since this consumer consumes less this period than your endowment, she is saving. She
saves (lends) 10.
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