Fall 2023: Practice Problem set 1
Micro 1
September 29, 2023
Total: 225 points
1. For the following two good scenarios draw the budget set and check whether it is
convex.
(a) (10 points) Suppose good 1 always has the same price, while good 2 has a
bulk discount. After consuming xh units of good 2, extra units of the good
are available for purchase at half the previous price of good 2. Consider the
case when p1 = 2, p2 = 2, m = 8, and xh = 2.
(b) (10 points) Suppose that the government is worried about over-consumption
of good 2, so they make purchases of good 2 over x̄ illegal–units purchased
above this amount come at an extremely high fine, which effectively prevents
anyone from purchasing more than that many units. Consider the case when
p1 = p2 = 1, m = 5, and x̄ = 3.
(c) (10 points) In this case assume that the per unit prices are fixed as p1 , p2 > 0
and income at m > 0. However, there are two wrinkles. First, there is a
fixed cost k > 0 involved in receiving any strictly positive amount of good
1. In addition there is rationing of good 1, so that no more than xH > 0 is
available at any price. Consider the budget set for the special case in which
p1 = p2 = 1.
2. Suppose $100 is to be divided up between two decision makers DM1 and DM2.
Suppose that DM1 only gets enjoyment from whoever has the highest amount of
money, regardless of who this individual is. However, DM2 only gets enjoyment
from whoever has the lowest amount of money, regardless of who this individual
is. Define x1 as the money going to DM1 and x2 as the money going to DM2.
(a) (10 points) Illustrate possible indifference curves for DM1 with x1 on the hori-
zontal axis and x2 on the vertical axis. Write possible equations for indifference
curves of this form.
(b) (10 points) Illustrate possible indifference curves for DM2 with x1 on the hori-
zontal axis and x2 on the vertical axis. Write possible equations for indifference
curves of this form.
3. (10 points) Draw the IC for the following utility function:
u(x1 , x2 ) = min {x1 + 3x2 , 3x1 + 2x2 }
1
4. (20 points) Show whether each of the following are (i) complete and (ii) transi-
tive. For all parts assume x̃, ỹ are two dimensional real vectors
(a) x̃ % ỹ if and only if x1 > y1 and x2 > y2
(b) x̃ % ỹ if and only if x1 > y1 or x1 = y1 and x2 ≥ y2
5. For the following ICs check whether they satisfy strict monotonicity and convexity.
If yes, show how, and if not, give a counter-example.
(a) (5 points) Vertical ICs, the direction of improvement: East
(b) (5 points) upward sloping IC, the direction of improvement: NE
6. For the following ICs check whether they satisfy strict convexity. If yes, show how,
and if not, give a counter-example.
(a) (5 points) perfect substitutes
(b) (5 points) perfect complements
(c) (5 points) horizontal ICs, the direction of improvement: North
7. Consider the Cobb-Douglas utility function,
1 2
u(x) = ln x1 + ln x2 .
3 3
(a) (10 points) Fix p1 = p2 = 1 and write the formula for the corresponding Engel
curve and income offer curve. Illustrate these graphically. Find the income
elasticity.
(b) (10 points) Fix price p2 = 1 and m = 1 and write the formula for the cor-
responding demand curve and price offer curve. Illustrate these graphically.
Find the price elasticity.
8. Consider the utility function,
u(x) = 3x51 x22 .
(a) (10 points) Fix p1 = p2 = 1 and write the formula for the corresponding Engel
curve and income offer curve. Illustrate these graphically. Find the income
elasticity.
(b) (10 points) Fix price p2 = 1 and m = 4 and write the formula for the cor-
responding demand curve and price offer curve. Illustrate these graphically.
Find the price elasticity.
9. Consider the following utility function
√
u(x1 , x2 ) = x 1 + x2
(a) (15 points) Show that u(x1 , x2 ) is strictly concave function. [Use any method
you want]
(b) (10 points) Solve for the optimal choice when p1 = 41 , p2 = 1 and m = 10.
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(c) (5 points) Find the indirect utility function at these parameter values.
10. Quasi-linear Utility Function Suppose the utility function for Linda is given
by,
u (x1 , x2 ) = ln x1 + x2
(a) (2 points) What would be her demand for x1 and x2 when p1 = 4, p2 = 5 and
m = 10.
(b) (3 points) What would be her demand for x1 and x2 when p2 becomes 12.
(Assume that Linda’s demand cannot be negative for any good and p1 and m
do not change.)
(c) (10 points) Derive the demand function for Linda for both goods when prices
of x1 and x2 are respectively p1 and p2 and Linda’s income is m.
(d) (5 points) Fix p1 = p2 = 1, solve for the Engel curve and Income-Offer curve
(IOC). Carefully consider the case where the consumer chooses a boundary
solution.
11. Assume that demand for three goods has been evaluated at the following three
price vectors:
p1 = (1, 2, 2); p2 = (2, 1, 2); p3 = (2, 2, 1);
with income m = 12 in all cases. Consider two different data sets specifying
x1 ≡ x(p1 , 12), x2 ≡ x(p2 , 12), and x3 ≡ x(p3 , 12).
(a) (10 points) Suppose that x1 = (4, 2, 2), x2 = (2, 4, 2), and x3 = (2, 2, 4). Show
that there are no cycles in the affordability matrix. What does this imply for
the existence of a utility function rationalizing these choices?
(b) (10 points) Suppose instead that x1 = (2, 3, 2), x2 = (2, 2, 3), and x3 =
(3, 2, 2). Show that there is a cycle in the affordability matrix. What does
this imply for the existence of a utility function rationalizing these choices?
12. (10 points) Consider the following choice data:
p1 = (3, 1, 2); x1 = (0, 1, 3); m1 = 7
p2 = (2, 3, 1); x2 = (3, 0, 1); m2 = 7
p3 = (1, 2, 3); x3 = (1, 3, 0); m3 = 7.
Does this data satisfy SARP? Does it satisfy WARP?
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