Final Exam
ECON 391, Equili
ium in Market Economies
Jean Guillaume Forand∗
Winter 2021, Waterloo
Out of 35. A co
ect answer must include a co
ect explanation. Make sure your diagrams
are labeled properly.
1. [5] Answer TRUE or FALSE, along with an explanation.
(a) [1] The levels of poverty and inequality observed in the real world contradict the
Welfare Theorems for competitive markets.
(b) [1] Markets are the only exchange mechanisms that yield Pareto-efficient allocations
of goods.
(c) [1] Taxes make markets less efficient.
(d) [1] Consider a production economy in which firm a is twice as productive as firm b,
that is, firm b has production function f(z) and firm a has production function 2f(z).
Then firm b must shut down in all competitive equili
ia.
(e) [1] Private health insurance is expensive because markets should not be used to delive
health care.
2. [6] Consider a two-consumer economy in which ωA = (1, 1), ωB = (3, 0), uA(xA1 , x
A
2 ) =
xA1 x
A
2 and u
B(xB1 , x
B
2 ) = (x
B
1 )
2xB2 .
(a) [2] Derive the Pareto set of this economy.
(b) [2] Derive fair allocations for this economy.
(c) [2] Derive a competitive equili
ium for this economy that generates the allocations
you found in (b).
3. [8] Consider a one-consumer one-firm economy in which u(x) = xz + xq, ω = 1 and
f(z) = α ln (z + 1), where 1 < α < 2 is a parameter.
∗Room 131, Department of Economics, University of Waterloo, Hagey Hall of Humanities, Waterloo, Ontario,
Canada N2L 3G1. Office phone: XXXXXXXXXXx XXXXXXXXXXEmail: XXXXXXXXXX. Website: http:
arts.
uwaterloo.ca/~jgforand
1
http:
arts.uwaterloo.ca/~jgforand
http:
arts.uwaterloo.ca/~jgforand
http:
arts.uwaterloo.ca/~jgforand
(a) [2] Illustrate this economy in an Edgeworth box.
(b) [2] Derive a competitive equili
ium for this economy.
(c) [2] Derive the Pareto set of this economy.
(d) [2] How do your answers from (b) and (c) depend on α? Interpret your answer.
4. [6] Consider a one-consumer one-firm economy in which u(x) = (xz)
3(xq)
2, ω = 2 and
f(z) = z.
(a) [2] Illustrate this economy in an Edgeworth box.
(b) [2] Derive the Pareto set of this economy.
(c) [2] Fix some Pareto-efficient allocations from (b) and derive a competitive equili
ium
for this economy that generates these allocations.
5. [10] Consider a two-consumer exchange economy with uncertainty. Consumers can buy
and sell claims on consumption xi in state i = 1, 2 at unit price pi, which is realised with
probability qi. There are 2 units of the consumption good in state 1, which are entirely
owned by consumer A, and 2 units of the consumption good in period 2, which are entirely
owned by consumer B. Consumer A is risk-averse: her utility over money consequences xA
is uA(xA) = lnxA. Consumer B is risk-neutral: her utility over money consequences xB is
uB(xB) = xB.
(a) [2] Illustrate this economy in an Edgeworth box.
(b) [2] Is there idiosyncratic risk in this economy?
(c) [2] Is there aggregate risk in this economy?
(d) [2] Derive a competitive equili
ium for this economy.
(e) [2] Describe the consumers’ equili
ium insurance plans in this economy. Is insurance
actuarially fair? Are the consumers ever fully insured? Interpret your answer.
2
Solutions: Problem Set for Final Exam
ECON 391, Equili
ium in Market Economies
Jean Guillaume Forand∗
Winter 2021, Waterloo
1. Consider a one-consumer one-firm economy in which u(x) = min{xz, xq}, ω = 2 and f(z) =
3z.
(a) Derive a competitive equili
ium for this economy.
Solution. Letting m = 2pz+Π be the value of the consumer’s endowment and assum-
ing that pz, pq > 0, the consumer’s demand functions are x(p,m) = (m/pz+pq,m/pz+pq).
The firm’s supply functions are
φ(p) =
(0, 0) if pz/pq > 3,
any φ = (z, q) such that q = 3z if pz/pq = 3,
does not exist if pz/pq < 3,
and the firm’s co
esponding profits are
Π =
0 if pz/pq ≥ 3,does not exist if pz/pq < 3.
To find a competitive equili
ium, normalise p∗z = 1. In any equili
ium, the firm’s
optimal production plan must exist, so that from above we know that 1/p∗q ≥ 3. First,
can we have an equili
ium in which 1/p∗q > 3? In this case, the firm’s optimal output
level is q(p∗) = 0. However, the consumer’s demand for output is xq(p
∗) = 2/1+p∗q > 0
(where I have used the fact that Π∗ = 0 at these prices). Therefore, the market
clearing condition for output is not satisfied. In words, at such prices the firm shuts
down but because the consumer still derives income from her endowment of the input
∗Room 131, Department of Economics, University of Waterloo, Hagey Hall of Humanities, Waterloo, Ontario,
Canada N2L 3G1. Office phone: XXXXXXXXXXx XXXXXXXXXXEmail: XXXXXXXXXX. Website: http:
arts.
uwaterloo.ca/~jgforand
1
http:
arts.uwaterloo.ca/~jgforand
http:
arts.uwaterloo.ca/~jgforand
http:
arts.uwaterloo.ca/~jgforand
Figure 1: Competitive equili
ium for Question 1
good, she has a positive demand for output. Second, can we have an equili
ium in
which 1/p∗q = 3? In this case, any level of output q
∗ = 3z∗ is optimal for the firm, and
its profits are 0. The consumer’s demand for output is xq(p
∗) = 3/2. Therefore, prices
(1, 1/3), allocations x∗ = (3/2, 3/2) and φ∗ = (1/2, 3/2), along with profits Π∗ = 0 form a
competitive equili
ium. This is illustrated in Figure 1.
(b) Derive the Pareto set of this economy.
Solution. Pareto-efficient allocations are solutions to the problem
max
0≤xz≤2,xq≥0
min{xz, xq} subject to 3[2− xz] = xq.
This utility-maximisation problem is illustrated in Figure 2. We know that the optimal
consumption bundle will be such that x∗z = x
∗
q , so that along with x
∗
q = 3[2− x∗z], we
have that x∗z = x
∗
q = 3/2. Therefore, allocations x
∗ = (3/2, 3/2) and φ∗ = (1/2, 3/2) are
the unique Pareto-efficient allocations for this economy.
2. Consider a one-consumer one-firm economy in which u(x) = xz +
√
xq and f(z) =
√
z.
(a) Derive the Pareto set of this economy.
2
Figure 2: Pareto set for Question 1
Solution. Pareto-efficient allocations are solutions to the problem
max
0≤xz≤ω,xq≥0
xz +
√
xq subject to
√
ω − xz = xq,
which is illustrated in Figure 3. Any solution to this problem in which x∗z, x
∗
q > 0 must
solve the first-order condition
∂
∂xz
u(x∗)
∂
∂xq
u(x∗)
= f ′(ω − x∗z), or 2
√
x∗q =
1
2
√
ω − x∗z
.
By substituting x∗q =
√
ω − x∗z, we have that x∗q = (1/2)
4/3 and x∗z = ω − (1/2)
8/3.
Any solution to the Pareto problem with x∗z = 0 would have to be such that
∂
∂xz
u(x∗)
∂
∂xq
u(x∗)
∣∣∣∣
x∗z=0
≤ f ′(ω − x∗z)
∣∣∣
x∗z=0
,
which, using that x∗q =
√
w in this case, reduces to ω ≤ (1/2)8/3. In words, if the
consumer’s endowment of input is small, then the Pareto-efficient allocations have
the firm use up all of the input to produce output, leaving the consumer with no
consumption of input. This is illustrated in Figure 4.
Any solution to the Pareto problem with x∗q = 0 would have to be such that
∂
∂xz
u(x∗)
∂
∂xq
u(x∗)
∣∣∣∣
x∗q=0
≥ f ′(ω − x∗z)
∣∣∣
x∗q=0
,
3
Figure 3: Pareto problem for Question 2
which, using that x∗z = ω in this case, reduces to 0 ≥ 1/2
√
ω−x∗z|x∗z=0 ≈ ∞. This
condition never holds, so that x∗q = 0 is never optimal. In words, if the firm is
producing no output, then its marginal productivity is a
itrarily high, whereas if
the consumer is not consuming any output, then her marginal utility from output
is also a
itrarily high. Therefore, it would benefit the consumer and be feasible to
produce some output in this case, so that no-production is Pareto-dominated. This is
illustrated in Figure 5.
Therefore, the Pareto set of this economy contains only allocations x∗ = (ω−(1/2)8/3, (1/2)4/3)
and φ∗ = ((1/2)8/3, (1/2)4/3) if ω > (1/2)8/3, and it contains only allocations x∗ = (0,
√
ω)
and φ∗ = (ω,
√
ω) if ω ≤ (1/2)8/3.
(b) Fix some Pareto-efficient allocations from (a) and derive a competitive equili
ium fo
this economy that generates these allocations.
Solution. Suppose that ω > (1/2)8/3, so that the Pareto-efficient allocations from
(a) have x∗z, x
∗
q > 0. The slope of the consumer’s indifference curve through the
Pareto-efficient allocations is 2
√
x∗q = 2
1/3. Therefore, the prices for the equili
ium
that we are constructing must be p∗ = (1, (1/2)1/3). Given prices p∗, we know that
the firm’s optimal production plan is φ∗ = (1/4(p∗q/p∗z)
2, 1/2(p∗q/p∗z)) = ((1/2)
8/3, (1/2)4/3),
with co
esponding profits Π∗ = p∗q2/4p∗z = (1/2)
8/3. Therefore, prices p∗, allocations
x∗ = (ω − (1/2)8/3, (1/2)4/3) and φ∗ = ((1/2)8/3, along with profits Π∗ = (1/2)8/3 form a
4
Figure 4: Corner solution with x∗z = 0
Figure 5: No corner solution with x∗q = 0
5
Figure 6: Competitive equili
ium generating Pareto-efficient allocations
competitive equili
ium. This is illustrated in Figure 6.
3. Consider a one-consumer one-firm economy in which u(x) = αxz+xq, ω = 1 and f(z) = z
2.
(a) Derive a competitive equili
ium for this economy.
Solution. No competitive equili
ium exists for this economy. To see this, conside
any equili
ium prices p∗. Given any level of input z > 0, the firm’s profit from
production plan φ = (z, z2) is z[p∗qz − p∗z]. This grows without bound as z → ∞, to
that no optimal production plan exists for the firm (refe
ing back to ECON 290, notice
that the firm has increasing returns to scale). Intuitively, given any input and output
prices, the firm wants to “grow” as much as possible, so that no scale of production is
optimal because any larger scale would increase profits.
(b) Derive the Pareto set of this economy.
Solution. Pareto-efficient allocations are solutions to the problem
max
0≤xz≤1,xq≥0
αxz + xq subject to (1− xz)2 = xq,
which is illustrated in Figure 7. From the Figure, it follows that if α > 1, then
the unique Pareto-efficient allocations are x∗ = (1, 0) and φ∗ = (0, 0) (this case is
illustrated in Figure 7), if α < 1 then the unique Pareto-efficient allocations are x∗ =
(0, 1) and φ∗ = (1, 1), whereas