Econ 302 Exam 1 McLeod

NAME

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PSU ID #

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PLEASE ANSWER ALL QUESTIONS IN THE

SPACE PROVIDED.
1. (20 total points) Suppose a consumer’s utility function is given by U(X,Y) = X*Y. Also, the consumer
has $144 to spend, and the price of X, PX = 9, and the price of Y, PY = 9.

a) (4 points) How much X and Y should the consumer purchase in order to maximize her utility?

X* = 8; Y* = 8

b) (4 points) How much total utility does the consumer receive?

U = 64 (Okay if consistent with a))

c) (4 points) Now suppose PX decreases to 1. What is the new bundle of X and Y that the consumer will
demand?

X** = 72; Y** = 8

d) (4 points) How much money would the consumer need in order to have the same utility level after the
price change as before the price change?

M = 48

e) (4 points) Of the total change in the quantity demanded of X, how much is due to the substitution effect
and how much is due to the income effect?

XC = 24, so:

SE = XC – X* = 24 – 8 = 16
IE = X** - XC = 72 – 24 = 48
2. (24 total points) Suppose there are two consumers, A and B, and two goods, X and Y. Consumer A is
given an initial endowment of 9 units of good X and 14 units of good Y. Consumer B is given an initial
endowment of 7 units of good X and 2 units of good Y. Consumer A’s utility function is given by:

UA(X,Y) = X3*Y,

And consumer B’s utility function is given by

UB(X,Y) = X*Y.

Therefore, consumer A’s marginal utilities for each good are given by:

MUX = 3X2Y
MUY = X3

Also, consumer B’s marginal utilities for each good are given by:

MUX = Y
MUY = X

a) (8 points) Suppose the price of good Y is equal to one. Calculate the price of good X that will lead to
a competitive equilibrium.

PX = 2
b) (4 points) How much of each good does each consumer demand in equilibrium?

Consumer A’s Demand for X: 12

Consumer A’s Demand for Y: 8

Consumer B’s demand for X: 4

Consumer B’s demand for Y: 8

c) (2 points) What is the marginal rate of substitution for consumer A at the competitive equilibrium?

MRSA = -2
d) (10 points) Illustrate the situation in an Edgeworth Box. Be sure to label your box carefully and
accurately. Identify the initial endowment and label it W. Identify the competitive equilibrium and label
it D. Draw the budget constraint that each consumer faces and identify the values where it intercepts the
perimeter of the Edgeworth Box (there are two different intercepts to identify).

The Edgeworth Box should have the following dimensions:

X-dimension = 16; Y-dimension = 16. I will illustrate this in class. The Budget constraint
should intercept B’s X-axis at 8 . It should also intercept A’s X-axis at 16 (at the lower
right corner of the Edgeworth Box).

Dimensions: 2 points
W: 2 points
D: 2 points
Budget constraint intercepts: 2 points (1 point each)
Labeling: 2 points (1 point for labeling the x and y axes; 1 point for labeling the graph with
A and B)
3. (8 total points) Suppose there are two consumers, A and B. There are two goods, X and Y. There is a
TOTAL of 8 units of X and a TOTAL of 8 units of Y. The consumers’ utility functions are given by:

UA(X,Y) = X + 2Y
UB(X,Y) = X*Y

For each of the following allocations, answer True if the allocation is Pareto Efficient, and FALSE if the
allocation is not Pareto Efficient.

i) Consumer A gets 2 units of X and 4 units of Y, and Consumer B gets 6 units of X and 4 units of Y.

FALSE

ii) Consumer A gets 4 units of X and 6 units of Y, and Consumer B gets 4 units of X and 2 units of Y.

TRUE

iii) Consumer A gets 2 units of X and 0 units of Y, and Consumer B gets 6 units of X and 8 units of Y.

FALSE

iv) Consumer A gets 0 units of X and 6 units of Y, and Consumer B gets 8 units of X and 2 units of Y.

FALSE
4. (12 total points) Suppose a consumer’s utility function is given by U(X,Y) = MIN(2X,Y). The Price
of X is PX = 3 and the price of Y is PY = 1.

a) (8 points) Draw the Income Consumption Curve for the following values of M: M = 20; M = 40; M =
60. Be sure to label your graph carefully and accurately. You do not need to draw the indifference
curves that run through each of the consumer’s optimal bundles, but your graph should include accurately
drawn budget constraints associated with each level of income, and show the consumer’s optimal bundle
for each budget constraint.

When M = $20, the budget constraint will have a vertical intercept (Y-axis) of 20 and a
horizontal intercept (X-axis) of 20/3. The bundle chosen is (X = 4, Y = 8).

When M = $40, the budget constraint will have a vertical intercept (Y-axis) of 40 and a
horizontal intercept (X-axis) of 40/3. The bundle chosen is (X = 8, Y = 16).

When M = $60, the budget constraint will have a vertical intercept (Y-axis) of 60 and a
horizontal intercept (X-axis) of 20. The bundle chosen is (X = 12, Y = 24).

M X Y
20 4 8
40 8 16
60 12 24

 Each Budget Constraint: 1 point (both intercepts must be correct) (3 total points)
 Each Bundle: 1 point (both X and Y values must be correct) (3 total points)
 Labeling Each Axis: 1 point each (2 total points)

b) (4 points) Sketch the graph of the consumer’s Engel curve for good X.

The Engel Curve is the graph of the first two columns above. M is on the vertical axis, and X is on
the horizontal axis. It is the graph of M = 5X.

 Each combination of M and X: 1 point (3 points total)

 Labeling the graph: 1 point
 NOTE: IF THE GRAPH IS NOT LABELED CORRECTLY, IT IS NOT AN ENGEL
CURVE, SO 0 POINTS.
5. (8 points) Suppose a consumer’s utility function is given by U(X,Y) = X +2Y. The consumer has $30
to spend (M = $30). The price of the goods are P X = 3 and PY = 2. Draw the consumer’s budget
constraint and identify the optimal bundle that the consumer should choose in order to maximize her
utility. Also, draw the indifference curve that runs through this bundle. Your graph should be labeled
carefully and accurately.

 The Budget Constraint has Y-Intercept = 15 and X-Intercept = 10.

 The Optimal Bundle is X* = 0; Y* = 15.
 The Indifference Curve has Y-Intercept = 15 and X-Intercept = 30

 Budget Constraint: 2 points (both intercepts must be correct).

 Optimal Bundle: 2 Points (both X and Y values must be correct).
 Indifference Curve: 2 points (both intercepts must be correct).
 Labeling Graph: 2 points (1 point for each axis).
6. (12 total points) Suppose there are two consumers, A and B, and 2 goods, X and Y. The utility
functions of each consumer are given by:

UA(X,Y) = MIN(X, 4Y)

UB(X,Y) = X + 4Y

The initial endowments are:

A: X = 4; Y = 2
B: X = 4; Y = 6

a) (8 points) Using an Edgeworth Box, illustrate the initial endowments. Also, for each consumer, draw
the indifference curve that runs through their bundle. Be sure to label your graph carefully and
accurately.

In Class

b) (4 points) Is the initial allocation Pareto Efficient?

No
7. (8 points) Carefully explain the difference between the Marginal Rate of Transformation and the
Marginal Rate of Substitution. Your answer should include an accurate definition of each of these
concepts as well as stating the value that each of them takes.

The Marginal Rate of Substitution is the rate at which a consumer is willing to trade one
good for another. It is equal to the slope of the consumer’s indifference curve. Put another
way, it is the amount of the good on the vertical axis that a consumer is willing to give up to
obtain an extra unit of the good on the horizontal axis. The value that it takes is equal to -
MUX/MUY.

The Marginal Rate of Transformation is the rate at which a consumer is able to trade one
good for another. It is equal to the slope of the consumer’s budget constraint. Put another
way, it is the amount of the good on the vertical axis that a consumer has to give up to
obtain an extra unit of the good on the horizontal axis. The value that it takes is equal to –
PX/PY.

 2 points for each correct explanation.

 2 points for each correct value.
8. (4 points) Suppose a consumer is spending all of their money on a bundle containing Good X and
Good Y. Also assume the consumer has a Cobb-Douglas utility function so their indifference curves are
strictly convex. If |MRS| > |MRT|, carefully explain what the consumer could do to increase their utility.
Be specific.

The consumer can increase utility by buying more X and less Y.

2 points if they only say buy more X (as this would put the consumer off of the budget constraint).
9. (4 points) Consider the following statement: “If an animal can fly, then it has wings.” Assume this
statement is true.

For each of the following statements, write “True” if it is true and “False” if it is false:

Having wings is a necessary condition for being able to fly. TRUE

Having wings is a sufficient condition for being able to fly. FALSE

Being able to fly is a sufficient condition for having wings. TRUE

Being able to fly is a necessary condition for having wings. FALSE