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MIDTERM 1: Feb. 7, 2022 ECO 3153

Question 1 (1 point)

If we know that an individual's preferences are monotonic. We then know that

The utility function is unique up to any positive monotonic transformation
All answers are correct
More is preferred to less
Two answers are correct
Indifference curves are positively sloped
No answer is correct
Individuals will have a preferred consumption bundle that is independent of income
Doubling the consumption bundle will provide double utility
Three answers are correct

Question 2 (1 point)
Suppose that the indifference curves (when graphed with x1 on the horizontal axis and x2 on the vertical
axis) are straight lines with a slope of-2. From this we know that

These two goods are perfect substitutes, and the individual is willing to give up 2 units of good 2 for 1
unit of good 1

These two goods are perfect substitutes and the individual is willing to give up 1/2 unit of good 2 for 1
unit of good 1

No answer is correct, more information is required

These two goods are perfect complements the individual is willing to give up 2 units of good 2 for 1 unit
of good 1

These two goods are perfect complements the individual is willing to give up 1/2 unit of good 2 for 1
unit of good 1

Question 3 (2 points)
If we know that the indifference curves are strictly convex, then we also know that

The utility function is strictly convex
The utility function is strictly quasi concave
Two answers are correct
Individuals prefer a balanced consumption bundle to consuming mostly one good
No answer is correct
Three answers are correct
The marginal utility of a good falls with consumption
Four answers are correct
The marginal rate of substitution of good 2 for good 1 falls as good 1 is increased


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Question 4 (2 points)
Let the utility function be homogeneous, then we know that

The utility function may reflect quasi-linear preferences but not necessarily
Four answers are correct
Two answers are correct
Three answers are correct
The goods cannot be perfect complements
The utility function is homothetic
All answers are correct
The marginal rate of substitution varies along a ray from the origin
Doubling the consumption function will lead to a doubling of utility
No answer is correct

Question 5 (2 points)
The Lagrange multiplier in the utility maximization problem tells us

By how much utility increases with a one unit increase in the consumption of good xi
Whether or not an interior solution exists
No answer is correct
Three answers are correct
Two answers are correct
By how much expenditures must increase for a one unit increase in utility
By how much utility increases with a one unit increase in income


Question 9 (1 point)
Consider the utility function U(x1, x2) = x12x22. We know that

Three answers are correct
MRS is increasing
Two answers are correct
Preferences are well behaved
No answer is correct
MRS is decreasing
The utility function is strictly convex

Question 10 (1 point)
Consider the utility function U(x1, x2) = x12x22. We know that

U(x1, x2) = x1x2 describes the same preferences
Marginal utility is diminishing
Two answers are correct
Marginal utility is increasing
Three answers are correct
Individuals are irrational


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Question 11 (2 points)
Consider the utility function U(x1, x2) = x12x22. We know that

For any given prices, individuals will consume these two goods in the same ratio independent of income
All answers are correct
Two answers are correct
Doubling of prices and doubling of income will not affect the optimal bundle of goods
The demand for x1 is independent of the price of x2
Four answers are correct
Individuals will spend half of their income on good 1 and half on good 2
Preferences are monotonic
Three answers are correct

Question 12 (2 points)
If two goods are perfect complements, then

two answers are true
no answer is true
the MRS is constant as the consumption of good 1 increases
the MRS decreases as the consumption of good 1 increases
marginal utility of good 1 is decreasing
marginal utility of good 1 is increasing
the assumption of monotonicity holds
marginal utility of good 1 is constant
the MRS increases as the consumption of good 1 increases
three answers are true

Question 13 (2 points)
Suppose that the first order condition from the utility maximizing problem holds. We then know that

preferences are strictly convex
expenditures are minimized
all answers are correct
four answers are correct
utility is maximized
the utility function is strictly quasi concave
no answer is correct
preferences are well behaved

Question 14 (3 points)
Let utility be: U(x,y) = 20x – x2 + 18y -3y2. The consumer has no budget constraint, but x + y cannot
exceed 15. What is the utility maximization bundle of x and y (fractions can be consumed)?

(3, 10)
no answer is correct
(3.5, 11.5)
(5, 10)
(1, 14)
4

(4, 11)
(4.5, 10.5)
(2, 10)

WRITTEN ANSWER QUESTIONS

Question 6 (12 points)
Let's suppose that an individual's utility function is U(x1, x2) = 2x1 + x21/2

a. On a large, well labelled, diagram, sketch in two indifference curves (let x1 be on the horizontal axis
and x2 on the vertical axis) one for U=10 and another for U=20. Suppose that p1=4 and p2 = 2 and
income = $10, draw in the budget constraint. (3 marks)

b. Determine the equilibrium bundle of x1 and x2, and indicate this equilibrium on your graph. (4 marks)

c. Prove that the demand for x2 is independent of income. (3 marks)

d. What is the minimum amount of income necessary for an interior solution? (2 marks)

Question 7 (8 points)
The individual's indirect utility function is: I/(2p1+p2)

a. Calculate the demand for x1 and x2 (3 marks)

b. Calculate the compensated demand for x1 and x2 (2 marks)

c. Write out a Utility function that represents these preferences and describe the optimal bundle of x1
and x2. (3 marks)

Question 8 (11 points)
Let's suppose that the individual's preferences are described by the utility function U(x1, x2) =
x11/2x21/2.

a. Write out the Lagrangian for this problem. Solve the Lagrange problem. What does the solution give
you? (4 marks)

b. Let p1=p2=1 and income = $20. What is the equilibrium bundle? (2 marks)

c. Then suppose that the price of good 1 changes to $2. Calculate the new equilibrium bundle. (1 mark)

d. You now have two points on the Marshallian demand curve for x1 and two points on the Hicksian
demand curve for x1. Using this information, sketch these two demands on the same well-labelled graph
and comment on their relative slopes. (4 marks)