Intermediate Microeconomics UA10
Practice Questions for Mid-Term 1 Fall 2025
1. (a) Define what it means for a preference order ≿ on a choice set X to be transitive.
(b) Define what it means for a preference order ≿ on a choice set X to have a utility function representation u : X → R.
(c) Given utility function u(x1, x2) identify which of the four following transformations do and which do not represent the same underlying preference ordering and explain why.
i. v(x1, x2) = −10 + 15u(x1, x2)
ii. v(x1, x2) = 10 − 15u(x1, x2)
iii. v(x1, x2) = 50 + u(x1,x2)/1
iv. v(x1, x2) = (−1 + u(x1, x2))2
2. Suppose a consumer with standard preferences initially chooses from a budget set with m = 12, p1 = 6, and p2 = 3, and then from a new budget set where the price of good 1 has decreased to p
′
1 = 3, with p2 and m staying the same.
(a) Draw the two budget sets described above, and add indifference curves for a case in which good 1 is an ordinary good.
(b) On your graph from part (a), illustrate the Slutsky decomposition by drawing a third budget set and indifference curve associated with the substitution effect. Make sure to label the change in demand corresponding to the substitution effect and the change in demand corresponding to the income effect that in combination define the price effect.
3. Imagine a student receives income y1 today (from a part-time job) and expects income y2 in the future (after graduation). The interest rate on savings is r. The student chooses how much to save today.
(a) Write the budget equations for period 1 and period 2, and show how saving today determines consumption tomorrow.
(b) Suppose the interest rate r rises. Use the Slutsky decomposition to explain how this change affects the student’s saving choice. Identify the substitution effect and the income effect.
(c) How do the substitution and income effects influence the student’s decision to save more or less today? Discuss all cases.
4. Consider the Cobb-Douglas utility function,
u(x1, x2) = 9/4ln x1 + 9/5ln x2.
(a) Compute the marginal rate of substitution (MRS) of good 1 for good 2.
(b) Show how to use MRS condition, the price ratio, and the budget constraint together to solve for the demand functions ˆx1(p1, p2, m) and ˆx2(p1, p2, m).
(c) Confirm that your solution satisfies the expenditure shares property.
5. Assume that choice data for each of two goods has been observed at three distinct price vectors p
1
,p
2
, p
3
:
p1 = (3, 1);
p2 = (2, 3);
p3 = (2, 1);
with income m = 12. Suppose that the corresponding choice data are:
x1 = (3, 3);
x2 = (4, 2);
x3 = (6, 2/3);
Here, for instance, x1
is chosen at price vector p1.
(a) Derive the 3-by-3 affordability matrix, where the rows correspond to price p
1
, p2
, p3
respectively and the columns to the choice data x
1
, x2
, x3
.
(b) Are there any cycles in this matrix? If so, illustrate them on the affordability matrix.
(c) What does your answer above imply for existence of a strictly monotone utility function rationalizing these choices?
6. Consider an endowment economy with two individuals, A and B, with identical Cobb-Douglas utility functions
u
A,B(x1, x2) = 0.5 ln x1 + 0.5 ln x2,
and with endowments,
(ω1
A
, ω2
A
) = (1.5, 0.5);
(ω1
B
, ω2
B
) = (0.5, 1.5).
(a) Draw the Edgeworth Box for this economy and mark the set of all Pareto improvements over the endowment point (also known as “the cigar of gains from trade”).
(b) Confirm that p
∗
1 = p
∗
2 = 2
1 with each consuming (1, 1) is an equilibrium (you can take for granted the expenditure shares formula for Cobb-Douglas demand).
(c) Illustrate this equilibrium in the Edgeworth box by drawing the corresponding budget set and indifference curves.