Consumer and Producer Theory, ITAM
Instructor: Xinyang Wang
Assignment 5
Due: in class, on September 20th 2022
1. Consider an age-parametrized consumer’s problem: there is only one commodity, and
the consumption of it is denoted by a number x
0. The age parameter a ranges from
[0, 2]. At age a, the income is a2 , and the happiness is determined by a utility function
u(x, a) = 1
(x
a)2 (that is, this agent is satisfied by consuming a units of the commodity).
We denote this agent’s optimal happiness level at age a by V (a) and his optimal consumption
level by µ(a). Formally,
V (a) = max2 u(x, a)
x2[0,a ]
µ(a) = argmax u(x, a)
x2[0,a2 ]
a. In one coordinate system, draw the graph of functions u(x, 0), u(x, 0.5), u(x, 1), u(x, 1.5),
u(x, 2).
b. highlight points (µ(a), V (µ(a))) for a = 0, 0.5, 1, 1.5, 2 in your graphs in a.
c. Using a and b, draw the “upper envelope” of this class of optimization problems. That is,
draw the curve (µ(a), V (µ(a))) for a 2 [0, 2].
d. Compute V 0 (1) and give an interpretation of this number.1
e. Applying the maximum theorem to prove that µ is a continuous function. Please verify
carefully that the assumptions of the maximum theorem holds.
2. Fix a pair (p, w) where p >> 0 and w > 0 and assume u is concave and di↵erentiable. Prove if the pair (x⇤ ,
⇤
) satisfies the KKT condition for the consumer’s problem
maxx2RN+ :p·xw u(x), then, we have x⇤ solves the
⇤
max u(y) +
y2RN
+
(w
p · y)
Then, explain this link intuitively using the meaning of
1
⇤ 2
.
compare to the interpretation of as the shadow value of money
This is a trick that many theorists have used: for an individual consumer’s problem, at the optimum,
the utility function could be approximated by some quasi-linear utility function in money.
2
1
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ir
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(
constvuintl
it
as
wri
.
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