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Exercises M&I week 2: Wednesday meeting
Problem 1
Alvin’s preferences for consumption goods X and Y are shown in the diagram below.
At any consumption bundle with the quantity of good X exceeding the quantity of good Y (that is, a bundle
located below the 45 degree line, like point A), Alvin’s marginal rate of substitution of good X for good Y
is:
A) diminishing.
B) positive.
C) constant and positive.
D) zero.
Problem 2
Sally consumes two goods, X and Y. Her utility function is given by the expression
U = 3 · XY2.
The current market price for X is $10, while the market price for Y is $5. Sally’s current income is $500.
a) Sketch a set of two indifference curves for Sally in her consumption of X and Y, one where U = 60
and one where U = 72.
b) Write the expression for Sally’s budget constraint. Graph the budget constraint and determine its
slope.
c) Determine the X,Y combination which maximizes Sally’s utility, given her budget constraint. Show
her optimum point on a graph. (Partial units for the quantities are possible.)
d) Calculate the impact on Sally’s optimum market basket of an increase in the price of X to $15. What
would happen to her utility as a result of the price increase?
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Problem 3
Janice Doe consumes two goods, X and Y.
Janice has a utility function given by the expression: U = 4X0.5 Y0.5.
The current prices of X and Y are 25 and 50, respectively.
Determine Janice’s optimal mix of X and Y.
a. Y = 2X
b. Y = ½ X
c. Y = 4/5 X
d. Y = 5/4 X
Problem 4* (Answer to part (a) is published in advance on BB)
A consumer has a utility function U(x, y) = ln (x + a) + y. The price of good x is px = 1, the price of good y is
py = 2, and his income is 10. (Remember: by the rules of differentiation, dln(x)/dx = 1/x).
(a) Let the parameter a in the consumer’s utility function be equal to 1. Derive the marginal rate of substitution
(MRS), and show that the law of diminishing MRS is met. Derive the optimal consumption bundle. Use the
Lagrange method (if desired, check the document called Appendix Chapter 4 in Blackboard).
(b) Let the parameter a in the consumer’s utility function instead equal 3. Derive again the consumer’s optimal
consumption bundle.
Problem 5
A firm’s production function is q = 2K0.5 + 5L0.5, where q is output per hour, K is the amount of capital used
per hour and L is the amount of labour used per hour.
Capital costs r = $20 per hour and labour costs w = $100 per hour.
a) Determine whether or not capital and labour show diminishing returns.
b) Determine the number of units of capital and of labour, if the firm wants to produce 45 units per hour.
Use the Lagrange method (if desired, check the document called Appendix Chapter 7 in Blackboard).
Problem 6
A firm has production function Q = 4L1/2 K1/2. Suppose that the price of labour is $5 and the price of capital
is $20. The firm desires to produce 200 units of output. How much labour will be hired to minimize the costs
of producing 200 units of output? (Hint: the optimal ratio of labour to capital is: L = 4K.)
a. 200
b. 25
c. 50
d. 100
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Exercises M&I week 2: Friday meeting
Problem 1
The next exercise (partly modified) can also be found at the bottom of Unit 3.6, just before the topic of
Income Effect (IE) and Substitution Effect (SE) starts. Indeed, part 3 of the question below forms already
sort of a stepping stone to IE and SE.
EXERCISE 3.7 YOUR PRODUCTION FUNCTION
1. Consider yourself as a student. What could bring about a technological improvement in your ‘production
function’ and those of your fellow students?
2. Draw a diagram to illustrate how this improvement would affect your feasible set of grades and study
hours. Start from Figure 3.9 in CORE Unit 3.4.
3. Analyse what might happen to your choice of study hours; during the lecture, compare your choice with
the ones your fellow students have made.
Problem 2
The next exercise is a MC question (no. 3.11) shown in Unit 3.9.
We will discuss it (briefly) anyhow, as it treats the basics of IE and SE nicely.
QUESTION 3.11 CHOOSE THE CORRECT ANSWER(S)
Figure 3.20 depicts a model of labour supply and consumption for the US in 1900 and 2013. The wage rate
is shown to have increased between the two years.
Which of the following is/are true?
The substitution effect corresponds to the steepening of the budget constraint. This is represented by
the move from point A to point D.
The income effect corresponds to the parallel shift in the budget constraint outwards due to the higher
income. This is represented by the move from point A to C.
As shown, the income effect dominates the substitution effect, leading to a reduction in the hours of
work.
If Americans had had different preferences, they might have responded to this wage rise by
reducing their free time.
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Problem 3
A consumer’s original utility maximizing market basket of consumption goods X and Y is shown in the
figure below as point A. Following a price change of good X, the consumer’s utility maximizing market
basket changes to point B.
The substitution effect of the price change in X on the quantity of X purchased is:
a. the change from F2 to F3.
b. the change from F3 to F2.
c. the change from F2 to F1.
d. the change from F1 to F2.
Problem 4
Consider the following two antipoverty programmes: (a) A payment of €10/day is to be given this year to each
person who was classified as poor last year; and (2) each person classified as poor will be given a benefit equal
to 25 percent of the wage income he earns each day this year. Assume that leisure is a normal good.
Also note that the graph starts with 0 hours and not with 8 hours, as the CORE textbook shows.
(a) Assuming poor people have the option of working at €4/hr, show in the graph how each programme
would affect the daily budget constraint of a representative poor worker during the current year.
(b) Which programme would be most likely to reduce the number of hours worked? Explain, using income
effects and substitution effects.
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Problem 5
Tony and Carmela are economists. In an attempt to limit their son Anthony Junior’s (known as AJ) use of the
family car, they charge him a user fee of 20 cents/mile. At that price he still uses the car more than they would
like, so they decide to raise the user fee to 40 cents/mile. In the diagram below, you find the budget line that
AJ faces with a 20 cents/mile fee, the budget line that he faces with a 40/cents mile fee, and the corresponding
optimal bundles.
On the diagram, indicate the total effect, the substitution effect, and the income effect of an increase in the fee
from 20 cents to 40 cents.
other goods
40
cents
20
cents
gasoline
Problem 6
Consider a student whose final grade increases with the number of hours spent studying. Her choice is
between more free time and higher grades, both of which are ‘goods’.
Which of the following are the same as her marginal rate of transformation between the two goods?
Note: More than one answer can be correct!
A.
The number of percentage points the student would gain by giving up another hour of free time.
B.
The opportunity cost of free time.
C.
The number of percentage points the student would lose by taking another hour of free time.
D.
The slope of the student’s feasible frontier.
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Problem 7
The following diagram shows a farmer’s choice between free time and grain before (A) and after (B) an
improvement in technology.
Based on this information, which of the following statements is/are correct?
Note: More than one answer can be correct!
1) The MRT of the new feasible frontier at B is larger than the MRT of the old feasible frontier at A.
This gives the farmer an incentive to take more free time after the technology improvement.
2) If the MRS of the indifference curve at B is larger than the MRS of the indifference curve at A, then
this gives the farmer an incentive to take more free time after the technology improvement.
3) The farmer will definitely choose a point to the right of B after the technology improvement.
4) The farmer may choose a point on either side of B after the technology improvement.