CONCORDIA UNIVERSITY

Department of Economics

ECON 301 INTERMEDIATE MICROECONOMIC THEORY I

Instructor: Uma Kaplan

Summer 2015

MIDTERM EXAMINATION 1

May 28, 2015, 11:45AM-13:15PM Name: I.D:

Points Total: points

MULTIPLE CHOICE QUESTIONS (1 POINT EACH - 5 POINTS TOTAL)

1. Which of the following utility function may induce a consumer to choose an optimal

bundle that is a corner solution?

(a) u(x, y) = max(3x, 5y)

(b) u(x, y) = x1/4

y1/2

(c) u(x, y) = x5y

2 + 4

(d) u(x, y) = ln(x) + ln(y)

2. If Marie has a positively sloped Engel Curve with good X, it means that:

(a) Marie’s income and substitution effects for good X are always in the same

direction

(b) Good Y must be a normal good to Mary (who consumes only goods X and Y)

(c) All of the above

(d) None of the above

3. The Slutsky equation:

I. breaks down the total effect of a change in Px on the quantity demanded for X into

income and substitution effects

II. holds only when good X is a normal good

III. relies on the assumption that the consumer spends his entire income

(a) I

(b) I and II

(c) I and III

(d) I and II and III

4. When the price of x rises, Alex responds by changing his demand for x. The

substitution effect is the part of this change that represents his change in demand

a. holding the prices of substitutes constant.

b. if he is allowed to substitute as much x for y as he wishes.

c. if his money income is held constant when the price of x changes.

d. None of the above

5. Madonna buys only two goods. Her utility function is Cobb-Douglas. Her demand

functions have which of the following properties?

a. Her demand for one of the two goods does not depend on income.

b. Her demand for neither good depends on income.

c. Her demand for each of the goods depends on income and on the

prices of both goods.

d. Her demand for each of the two goods depends only on her income

and on the price of that good itself.

e. One of the goods is an inferior good and the other is a normal good.

LONG ANSWER QUESTIONS ( 30 POINTS TOTAL)

1. ( POINTS) A person has a utility function 1 1

( , ) ln( ) ln( )2 2

u x y x y . This utility

function gives “well-behaved” indifference curves.

(a) At any point (x, y), the marginal utility of goods X and Y are :

Answer:

MUx(x, y) = ( , ) 1

2

dU x y

dx x

MUy(x, y) =( , ) 1

2

U x y

y y

(b) This person faces positive prices Px > 0 and Py > 0, and has positive income M >0.

Derive his demand functions, x* = x(Px, Py, M) and y* = y(Px, Py, M).

Answer:

Because the indifference curves are well behaved, the optimal solution occurs at

the point of tangency between an indifference curve and the budget constraint.

This we can use the first order condition (FOC) to solve for the optimal

bundle:

Budget constraint must hold at optimal bundle:

Substitute y = (Px/ Py)x into budget constraint and get:

2. (12 POINTS) Consider a consumer who consumes two goods: CDs (good 1) and

pizzas (good 2). Her utility function can be represented by u(x1, x2) = x1x2. Her income

is m = $200. The prices are pi= $20 and p2 = $10.

(3 points) a) The utility function v(x1, x2) = In x1 + ln x2 also represents her

preferences. Can you explain why?

For the rest of the problem, you may pick either u or v as her utility function.

Answer:

a) Observe that v(x1, x2) = ln u(x1, x2). From 2(a), f(u) = ln u is a monotonic

transformation. Monotonic transformations of a utility function represent the same

preferences.

(4 points) b) Derive her optimal consumption bundle. Show the complete calculation,

and illustrate your answer graphically (draw the indifference curves and the budget

constraint).

Answer

b) Use the optimum condition

MRS = slope of budget line, and the budget constraint to derive the optimal

consumption bundle. In this case,

1 2

2 1

MU xMRS

MU x

The slope of the budget line is

1

2

202

10

p

p

Therefore, the optimal consumption bundle (x1, x2) satisfies

2

1

2x

x => x2 = 2x1

Substituting this into the budget constraint:

20x1 + 10x2 = 200,

we get x1 = 5. Since x2 = 2x1, x2 = 10.

So, her optimal consumption bundle is (x1, x2) = (5, 10).

(3 points) c)Now suppose the government levies a 25% tax on each CD the consumer

purchases. Derive her new consumption bundle and illustrate it on the same graph as

in part a.

Answer:

Now the price the consumer has to pay for CDs is

1p = 20(1 + 25%) = 25.

The slope of the budget line becomes

1

2

252

10

p

p

Therefore, the new optimal choice 1 2( , )x x satisfies

2

1

5

2

x

x

Using this in the new budget line

25x1 + 10x2 = 200,

We get 1x = 4, and 2x = 10. Therefore, the consumer’s new optimal consumption

bundle is (4, 10).

(2 points) d) Is the consumer better off or worse off? Explain. Compare your answer

with that to part c).

Answer: The consumer is worse off. Her utility is 50 before the tax and 40 after the tax.

3. (6 POINTS)

Miriam has quasilinear preferences over consumption of coffee (x1) and muffins (x2)

given by U(x1, x2) =5ln(x1) + x2. Her income is equal to m =$10, and p2 =$1, p1 =$5.

a) Find MRS as a function of consumptions of two commodities. Write down the

budget constraint and the expression necessary for finding the optimal consumption.

Answer:

The MRS for these preferences is MRS(x1, x2) = −5/ x1 . (Notice that this does not

depend on x2, over which utility is linear.) Thus we have , for an interior solution

(with x1 > 0 and x2 > 0) are:

(1) p1x1 + p2x2 = m

(2) MRS(x1, x2) = − p1/p2

⇒ − 5/x1= −p1/p2

b) Find optimal consumption of x1 and x2 (assume interior solution).

Answer:

Solving for the two above equations for x1 and x2 we get demand functions

x1 =5p2 /p1 and x2 = (m − 5p2 )/p2

as long as m ≥ 5p2 (if m < 5p2 we have a corner solution with x1 = mp1 and x2 = 0.)

c) Suppose price of coffee drops from p1 =5 to p'1 =1. Find the substitution effect.

Answer:

Given that m = 10 and p2 = 1, at price p1 = 5, our Lola’s demand for coffee x1 is x1(p1,

p2, m) = 5*1/5 = 1. (Also, x2(p1, p2, m) = (10− 5·1)/1 = 5.)

With a price of 1 1p , her demand is 1 1 2( , , ) 5*1/1 5x p p m

The total effect on demand for x1 resulting from the price change is

1 1 1 2 1 1 2( , , ) ( , , ) 5 1 4x x p p m x p p m .

To find the substitution effect, we first find 1 1 2 2m p x p x , the hypothetical income

that would be necessary for her to buy her initial bundle with the price change, which

is 1*1 1*5 6m .

Next we need to find the bundle Lola would actually choose if her income were m'= 6.

(We know she can afford the initial bundle, but that’s not necessarily what she buys

given the new relative prices.) At 1p= 1, p2 = 1 and m= 6, we have that

1 1 2( , , ) 5*1/1 5x p p m .

The substitution effect is then 1 1 1 2 1 1 2( , , ) ( , , ) 5 1 4sx x p p m x p p m .

d) Find the income effect.

Answer:

The income effect on the demand for x1 is 1 1 1 2 1 1 2( , , ) ( , , ) 5 5 0nx x p p m x p p m .

With quasilinear preferences, the good in which utility is linear (here, x2) “absorbs all

income effects.”