Econ 310 Microeconomic Theory II
Solutions to Midterm Exam Winter 2018
Solution 1 (a) The utility function is of the Cobb-Douglas form. It follows immediately
that the Marshallian demand is:
x1 =
y
y
; x2 =
:
2p1
2p2
Therefore, the indirect utility function is
v=
p
a
bx1 x2 =
s
a
b
y2
:
4p1 p2
(b) Let e be the expenditure. Then according to the following duality:
v(p; e(p; u)) = u;
we have the expenditure function:
u=
s
a
e2
b
) e=
4p1 p2
r
4p1 p2 ua
:
b
(c) We use the following duality:
1
xh1 (p; u) = x1 (p; e(p; u)) =
2p1
Similarly,
xh2 (p; u)
=
r
r
4p1 p2 ua
=
b
r
p2 u a
:
p1 b
p1 ua
:
p2 b
Solution 2 (a) The pro…t function is de…ned as:
(p; w)
max
x 0; y 0
py
w x
s:t: f (x)
y;
where p is the output price.
(b) Consider two arbitrary price vectors, (p; w) and (p0 ; w0 ). De…ne
p00 = t p + (1
w00 = t w + (1
t) p0
t) w0 ;
1
8 t 2 [0; 1] :
Let y
y (p; w), x x (p; w), y 0
y (p0 ; w0 ), x0
x (p0 ; w0 ), y 00
x (p00 ; w00 ) be the corresponding optimal solutions. Then we have
(p; w) = p y
)
t
p y 00
w x
t [p y 00
(p; w)
y (p00 ; w00 ) and x00
w x00
w x00 ]
(1)
and
(p0 ; w0 ) = p0 y 0
)
(1
w0
(p0 ; w0 )
t)
x0
(1
p0 y 00
w0
t) [p0 y 00
x00
w0
x00 ] :
(2)
Adding up (1) and (2), we have
t
(p; w) + (1
t [p y 00
w x00 ] + (1
= t p y 00 + (1
= [t p + (1
= p00 y 00
(p0 ; w0 )
t)
w00
t) p0 y 00
t) p0 ] y 00
x00
t) [p0 y 00
w0
x00 ]
t w x00
(1
t) w0
[t w + (1
t) w0 ]
x00
x00
(p00 ; w00 ) :
Solution 3 (a) The short-run cost minimization problem is:
min K + L
L
s:t:
p
KL
y:
The constraint must hold at equality:
p
KL = y:
Thus we can solve for
L(y; K) =
y2
:
K
The short-run cost function is given by
c(y; K) = K + L(y; K) = K +
y2
:
K
(b) To derive the long-run cost function, the …rm’s solves the following cost-minimization
problem:
y2
min c(y; K) = K +
K
K
2
The …rst-order condition is
y2
= 0:
K2
1
2
Therefore, in the long run K = y, c(y) = K + yK = 2y.
(c) In the long run, …rms earn zero pro…t. Therefore,
0=
= py
c(y) = py
2y = (p
2) y
Therefore, the long-run equilibrium price is p = 2.
Solution 4 (a) We …rst derive the individual …rm’s output supply. A …rm solves
max pq
3
q
q2
The …rst-order condition is:
p
2q = 0:
Therefore, the market-clearing condition is
Qd = Qs
p
50 p
= 24
)
2
2
)p = 2
(b) In the long run, an individual …rm’s pro…t must be equal to the entry cost, which is
6. Also, …rms solve the same pro…t maximization problem in the long run as in the short
run. Therefore, as we have derived in part (a), the …rst-order condition for a …rm has
p = 2q:
Thus, a …rm’s pro…t
= 6 is such that
= pq
3
q 2 = 2q 2
3
q2 = q2
3 = 6:
The above solves for q = 3 because production quantity can only be positive. Then, p =
2q = 6 in the long run. Let n be the number of …rms in the long run. Market-clearing
3
requires that
50
p
2
)
50
6
2
= nq
= 3n
)n =
4
44
6
ECON 310
Microeconomic Theory II
Midterm Exam
Winter 2018, Queen’s University
[Total number of pages: 1]
1. [30 pts] A consumer maximizes utility by consuming two goods. The prices of goods
are p1 and p2 . The income is y and the utility function is
p
U (x1 ; x2 ) = a bx1 x2 , where a > 1; b > 0:
(a) Derive Marshallian demand functions and the indirect utility function.
(b) Use the indirect utility function to derive the expenditure function.
(c) Use the Marshallian demand and expenditure functions to derive the Hicksian
demand function.
2. Consider a competitive …rm with a production function that is continuous and strictly
increasing.
(a) [5 pts] De…ne the …rm’s pro…t function.
(b) [10 pts] Prove that the pro…t function is convex in input and output prices.
3. Consider a competitive industry. Each …rm has the following production function::
p
f (K; L) = KL:
Let p be the output price and the input prices be wK = wL = 1. Let y denote the
output level.
(a) [10 pts] In the short run, capital is …xed at K. Derive the short-run cost function.
(b) [10 pts] Given the short-run cost function, derive the long-run cost function.
(c) [5 pts] There is no entry cost to this industry. Derive the equilibrium price p.
4. Consider an economy with 24 …rms in the short run. The market demand function
is given by p = 50 2Q. A …rm’s cost function is C(q) = 3 + q 2 .
(a) [10 pts] Derive the equilibrium price in the short run.
(b) [20 pts] Firms face a cost of 6 to enter this industry. Calculate the equilibrium
price and number of …rms in the long run.
1
Econ 310: Microeconomic Theory II
Assignment 2
Due: 11:59pm, Friday, February 18, 2022 (Kingston time)
[Note: A student must work individually and independently on all the
assignments.]
1. Let input prices be (w1 ; w2 ) and output price be p. Take as given the production
p
p
function f (x1 ; x2 ) = min 3 x1 ; x2 .
(a) [20 points] Derive the cost function.
(b) [10 points] Directly use the cost function to derive the output supply function.
2. Consider a competitive …rm which uses three inputs K (capital), L (labor), and A
(land) to produce output y. The input prices are (wK = 2; wL = 1; wA = 4) and the
output price is p = 1. The …rm has the following production function:
f (K; L; A) = A
p
p
L+ K :
In the short-run, the …rm’s land level is …xed at A, but can choose labor and capital
as it wishes. In the long-run, the …rm can also vary its land level.
(a) [20 points] Derive the short-run cost function given A.
(b) [10 points] Use the short-run cost function to derive the long-run cost function. [Hint: Use an approach similar to the example in class, where the pro…tmaximization problem can be decomposed into two steps. Here try to decompose
the long-run cost minimization problem into two steps.]
3. Consider an economy with 2 consumption goods and N consumers, all with the same
utility function:
u(x1 ; x2 ) = x1 x21 ; where and 2 (0; 1) :
The goods prices are p1 = 2 and p2 . Among the consumers, half of them each have
income y1 and the rest have income y2 .
(a) [20 points] There are n …rms operating in the competitive market for good 2.
Each …rm has the cost function c(q) = q 2 . Solve for the equilibrium price p2 .
(b) [20 points] Now consider an alternative market structure while keeping the same
group of consumers. Suppose the industry of good 2 is monopolistic, with the
cost function c(q) = q 2 q. Solve for the equilibrium price p2 . Does this market
structure give an e¢ cient equilibrium outcome? Brie‡y explain the intuition.
1
4. [20 points] There are n …rms in a competitive industry. The market demand function
is given by p = 50 2Q. A …rm’s cost function is C (q) = q 2 . The total surplus
502 n
. Suppose the government imposes a per-unit tax t > 0 on
is given by S1 = 4(n+1)
consumers. That is, for each unit of goods purchased, the consumer pays p + t. As
a result, the market demand function becomes p + t = 50 2Q. The tax goes to the
government and …rms only receive p for each unit of output sold. So the pro…t of a
…rm is pq q 2 . Derive the total surplus, denoted by S2 . Is S2 > S1 ? Explain the
intuition of your answer.
2
Production Theory
Now that we have studied the optimal consumer behaviour extensively, it
is time to turn to the producers. That is, the production …rms. Consumer
theory and producer theory are two pillars of microeconomics that serve
the development of economic studies.
Just like the consumers, the producers may …nd themselves solving one
of two types of decision problems, i.e., pro…t maximization and cost
minimization.
– Pro…t-max: The goal is to maximize the pro…t from selling output
subject to a production resource constraint.
– Cost-min: The goal is to minimize the cost of using production inputs
subject to a production resource constraint.
In fact, we will see next that the cost minimization problem solved by a
producer is exactly the same as that solved by a consumer. Let us …rst
introduce some relevant terminology.
Terminology
Production function: f : Rn
+ ! R+ ;
y = f (x) ;
where x
0 is the vector of n inputs and y
0 is the output.
Assumption 2 (Properties of the production function): Assume that
the production function f : Rn
+ ! R+ is continuous, strictly increasing and
f (0) = 0.
Marginal product:
M Pi
@f (x)
= fi (x) ;
@xi
which is the change of output as a result of an additional unit of input i.
Average product:
APi
f (x)
:
xi
Marginal rate of technical substitution (MRTS):
M RT Sij (x)
M Pi
f (x)
= i
;
M Pj
fj (x)
which is the rate at which one input can be substituted for another without
changing the amount of output produced.
The cost function
For a …rm, the cost function for all input prices w
0 and all output
levels y > 0 is obtained from solving the cost-minimization problem:
c (w; y )
”
minn w x
x2R+
s:t: f (x)
#
y :
If x (w; y ) solves the above cost-minimization problem, then
c (w; y ) = w x (w; y ) ;
where x (w; y ) is called the …rm’s conditional input demand.
Note that x (w; y ) depends on y , i.e., it is conditional on y . Thus x (w; y )
may or may not be pro…t maximizing.
Solving the cost-minimization problem
Because the objective function is strictly increasing, the constraint must
hold with equality:
f (x) = y:
(1)
Lagrangean:
L=
n
X
wixi
[f (x1; x2;
; xn )
i=1
FOCs:
@L
= wi
@xi
@f
= 0;
@xi
8i
y] :
)
wi
f (x)
= M RT Sij :
= i
wj
fj (x)
(2)
That is, the marginal rate of technical substitution between two inputs is
equal to the ratio of their prices. One can use (2) and (1) to further solve
for the conditional input demands.
Compare the cost function with the expenditure function
Expenditure function
2
6
e (p; u) = 4
Cost function
3
min p x
x
s.t. U (x)
u
7
5
2
6
c (w; y ) = 4
3
min w x
x
s.t. f (x)
y
7
5
Mathematically speaking, the two functions are identical. Consequently,
for every theorem we proved about the expenditure function, there is an
equivalent theorem for the cost function.
Properties of the cost function:
Theorem 10 If f is continuous and strictly increasing, then c (w; y ) is
(1) continuous; (2) strictly increasing in y for all w
0;
(3) increasing in w; (4) homogeneous of degree 1 in w:
c (tw; y ) = t c (w; y ) ;
8 t > 0;
(5) concave in w; (6) Shephard’s lemma:
xi (w; y ) =
@c (w; y )
@wi
[Proof same as proof to Theorem 3.]
8 i:
Properties of conditional input demands
Theorem 11 If f is continuous and strictly increasing and the associated cost
function c (w; y ) is twice continuously di¤erentiable. Then:
1. x (w; y ) is homogeneous of degree 0 in w:
x (tw; y ) = x (w; y )
8 t > 0;
2.
@xi (w; y )
@wi
0;
8 i:
Proof of Theorem 11
For part 1, given any t > 0, we have
”
minn (tw) x
x2R+
s.t. f (x)
#
y =t
”
minn w x
x2R+
s.t. f (x)
y
#
where the equality is because t > 0 is a constant. Therefore, the optimal
solutions to the former must be the same as those to the latter, which means
x (tw; y ) = x (w; y )
8 t > 0:
For part 2, recall Shephard’s lemma and yield
@xi (w; y )
@
=
@wi
@wi
@c (w; y )
@wi
!
@ 2 c (w ; y )
=
@wi2
0;
where the inequality is because c (w; y ) is concave in w by Theorem 10. QED
The short-run cost function
The cost function and input demand functions we have derived so far are
based on an implicit assumption that the decision-making …rm is able to
adjust all of the quantities of inputs. However, in reality sometimes …rms
are not able to do that for all of its inputs within a (relatively) short time.
For example, a …rm signs a contract to rent an o¢ ce space. Then the
rent is …xed for the duration of the contract. As another example, certain
employees are under contracts with the …rm to receive a …xed level of
salaries for the duration of the contract (say, 3 years). Therefore, these
salaries are not to be changed as long as the contracts are still in place.
Therefore, we need a way to analyze the …rm’s optimal decisions in such
“short-run”circumstances.
Notations:
– x: vector of variable inputs
– x: vector of …xed inputs
– w: price vector of variable inputs
– w: price vector of …xed inputs
Given the knowledge of the cost-minimization problem, it is fairly straightforward to set up the short-run cost-min problem. The key is to acknowledge that some of the inputs are held …xed and cannot be chosen. Thus,
the short-run cost function is de…ned as:
sc (w; w;y ; x)
min w x + w x
x
s:t: f (x; x)
y:
If x (w; w;y ; x) solves the above problem, then
sc (w; w;y ; x) =
|
w x (w; w;y; x) +
{z
}
total variable cost
w x
| {z }
total …xed cost
Solving the above short-run problem involves the exact same method as
solving a general cost-min problem.
Note that sc (w; w;y ; x)
c (w; w;y ). This is because in solving the
former problem, some of the inputs are …xed and thus cannot be optimally
adjusted. The latter problem, however, does not su¤er this problem. Here
the …rm gets to choose all of the inputs optimally. Therefore, the latter
must achieve a lower (although may not be strictly lower) cost than the
former. To put it in plain terms, the …rm’s “hands are tied” in the former
situation, compared to the latter, and thus will not get to obtain a strictly
better outcome.
Pro…t maximization
The pro…t function is de…ned as:
(p; w)
max p y
x; y
w x
s:t: f (x)
y;
where p is the output price.
Because the objective function is strictly decreasing in x, the constraint
must hold with equality: f (x) = y . Using the constraint to eliminate y
in the objective function, the above problem becomes
max
x 0
p f (x)
w x:
FOCs:
@f (x)
p
@x
| {z i }
MR
)
w
| {zi }
MC
= 0;
8i
f (x )
wi
= i
= M RT Sij :
wj
fj (x )
x
x (p; w): input demand function
y
y (p; w) = f (x ): output supply function
(1)
Properties of the pro…t function
Theorem 12 If f is continuous and strictly increasing, then for p
w 0, the pro…t function (p; w) is continuous and
1. increasing in p ;
2. decreasing in w;
3. homogeneous of degree 1 in (p; w):
(tp; tw) = t
4. convex in (p; w);
(p; w) ;
8 t > 0;
0 and
5. Hotelling’s lemma:
@ (p; w)
= y (p; w)
@p
@ (p; w)
= xi (p; w) ;
@wi
8 i:
Theorem 13 (Properties of output supply and input demand functions)
Let (p; w) be twice continuously di¤erentiable. Then the output supply and
input demand functions have the following properties:
1. Homogeneous of degree 0 :
y (tp; tw) = y (p; w) ;
xi (tp; tw) = xi (p; w) ;
8t>0
8 t > 0; 8 i:
2. Own-price e¤ects:
@y (p; w)
@p
0;
@xi (p; w)
@wi
0;
8 i:
Proof of Theorem 13
Part (1):
(tp; tw) =
max tp y
tw x
s:t: f (x)
y
= t max p y
w x
s:t: f (x)
y = t (p; w) :
x; y
x; y
This proves that the pro…t function is homogeneous of degree 1 in (p; w) and
that the optimal solutions to the …rst problem must be the same as those to
the second one. Thus, the output supply and input demand functions must be
homogeneous of degree 0 in (p; w).
For part (2), recall Hotelling’s lemma:
y (p; w) =
@ (p; w)
:
@p
Di¤erentiate the above w.r.t. p:
@y (p; w)
@ @ (p; w)
=
@p
@p
@p
where the inequality is because
@
@xi (p; w)
=
@wi
@wi
QED
!
@ 2 (p; w)
=
@p2
0
(p; w) is convex in p. Similarly,
@ (p; w)
@wi
!
=
@ 2 (p; w)
@wi2
0:
The short-run pro…t function
The short-run pro…t function is de…ned as:
(p; w; w; x)
max p y
y; x
w x
w x
s:t: f (x; x)
y:
The solutions y (p; w; w; x) and x (p; w; w; x) are called the short-run
output supply function and variable input demand function. They
satisfy:
(p; w; w; x) =
p f (x (p; w; w; x) ; x)
w x (p; w; w; x)
w x:
Example
Given the following production function:
f (x1; x2) = ln
1
2
x x
1 2
!
;
derive the conditional input demand functions and the cost function.
Solution
Let w = (w1; w2). Then the cost function is given by
c (w; y ) =
min w1x1 + w2x2
x1 ;x2
s.t.
ln
1
x2 x
1 2
!
y:
Note that the constraint must hold with equality because the objective function
is strictly increasing. Set up the Lagrangean:
L = w1x1 + w2x2
”
ln
1
x2 x
1 2
!
#
y :
The …rst-order conditions are:
1
2x
1 x1 2
=0
1
2
x12 x2
@L
= w1
@x1
1
x2
@L
= w2
@x2
1
1
x2 x
= 0:
1 2
Dividing the above two equations yields
1
2x
x1 2
x2
w1
=
=
:
1
w2
2
x
1
2×12
1
2
x x
Use the above to eliminate x2 in the constraint, ln
1
x2
w1
2
x1
1
w2
)
)
x1 =
!
1 2
!
= y:
= ey ;
w2ey
2w1
1
w1
x2 = 2 x1 = 2 3
w2
!2
3
w1
w2
!1
3
2y
e3 :
Then substitute the input demand functions in to the objective function and
solve for the cost function:
c (w; y ) = w1
w2ey
2w1
!2
3
+ w2
1
23
w1
w2
!1
3
2y
e3 :
[End of solution.]
Example
Given the following cost function:
c (w; y ) =
y 2w12
2
(w1 + w2)
;
derive the output supply function.
To solve this problem, let us …rst analyze the relation between the cost-min
problem and the pro…t-max problem.
Relation between cost-minimization and pro…t-maximization
Before we start to solve this problem, let’s revisit the pro…t function:
(p; w)
max
x 0; y 0
py
w x
s:t: f (x)
y:
We can break this problem into two sub-problems:
(i) Take y as given, solve the cost-minimization problem:
(
max
x 0
w x s:t: f (x)
y
)
,
(
min w x s:t: f (x)
x 0
We can derive c (w; y ) from solving the above.
y
)
c (w ; y ) :
(ii) Given the obtained c (w; y ), choose y to solve the pro…t-maximization
problem:
max p y
y 0
c (w; y ) :
Solving this problem yields the output supply function y (p; w).
To summarize, cost-minimization is a sub-problem of pro…t-maximization.
Solution to the previous example
Because the cost function c (w; y ) is given, we can use it to set up the pro…tmaximization problem:
max p y
y
c (w; y ) = max p y
y
y 2w12
2
(w1 + w2)
The …rst-order condition is:
p
2 y w12
= 0;
2
(w1 + w2)
which solves for
p
2
y (p; w) =
(
w
+
w
)
1
2 :
2w12
[End of solution.]
:
4. Consider an industry of n competitive firms. For each firm, production requires 3
inputs: capital (K), labor (L) and electricity (E). The production function is given
by:
f (K, L, E) = Kł (min (1#, ]).
(a) In the short run, capital input is fixed at K*. However, firms are free to choose
the levels of L and E. Let y denote the output level. Input prices are wk = 2
and w1 = we = 1. Derive the short-run cost function for an individual firm.
(b) In the long run, firms can also vary the capital level. Derive a firm’s optimal
choices of K, L and E in the long run. Derive its long-run cost function. Let
p be the price of output. Solve for the output supply function and the profit
function.
(c) There are 100 consumers in the economy. Among these consumers, 50 of them
have an income of 10 each and the rest have an income of 20 each. Let y denote
a consumer’s income. For each consumer, the indirect utility function is given
by
v (p, y) = 5 lny – 2 Inp.
i. Solve for the Marshallian demand function for an individual consumer.
ii. Calculate the equilibrium market price: p(n).
ii. There is an entry fee to this competitive market, o. Due to the pandemic,
a third of the firms have dropped out of the industry. Solve for the number
of firms currently remain operating
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