ECON 210 - MICROECONOMIC THEORY
HOMEWORK 6
PROFIT
PROFESSOR JOSEPH GUSE
(1) The SWS makes sweaters out of labor and capital. The following production
function represents their technology
xl
f (xl , xk ) = 80 log min{ , xk } + 1
2
(a) How many sweaters are made with no inputs? ANSWER: Let’s verify
that using no inputs results in exactly zero output.
0
f (0, 0) = 80 log min{ , 0} + 1
2
= 80 log(1)
=0
(b) Using isoquants, draw a picture of this technology in labor × capital
space.
1
2
PROFESSOR JOSEPH GUSE
xk
Kapital
Perfect Complements
Output Expansion Path
e2 − 1
y=160
‘z’ = e2 − 1
y=80
e−1
‘z’=e − 1
2(e − 1)
2(e2 − 1)
Labor
xL
The isoquants (bold lines) are ‘L’ shaped reflecting a
perfect complement production function. Note that
doubling output (from 80 to 160 in the figure) requires more than double the inputs. The log function
makes the production function have decreasing returns
to scale.
(c) Derive the factor demands for labor and capital.ANSWER Note from
the figure that since we have perfect complements, the cost-minimizing
solution for any output level will always employ an input bundle with
twice as much labor as capital. Therefore we can search for the profit
maxizing input bundle along the one-dimensional output expansion path
illustrated in the figure. Imagine then a ‘unit’ bundle with exactly one
unit of capital and 2 units of labor. Let z be the number of such bundles
the firm chooses to employ. The profit maximization problem reduces to
max p80 log(z + 1) − z(2wL + wK )
z
where p is the price of output. The first set of terms is revenue and
the second set of terms is cost. Note that each unit bundle cost twice
ECON 210 - MICROECONOMIC THEORY
HOMEWORK 6
PROFIT
3
the labor wage plus the rental rate on capital. Now we just take the
derivative and set it equal to zero. This gives us
80p
− (2wL + wK ) = 0
+1
80p
⇒ z∗ =
−1
2wL + wK
To get the factor demands for labor and capital we need to just unwrap
the ‘z’ bundles. Remember in each on there in one unit of capital and
two units of labor. Hence
z∗
160p
−2
2wL = wK
80p
xK (wL , wK ) =
−1
2wL = wK
(2) Arlo Corp makes toy wagons out of labor, l, and machines, k. Their output
is given by the following production function
xL (wL , wK ) =
f (xl , xk ) = xal xbk
where xl is the quantity of labor used per day; xk is the number of machines
they own.
(a) Write down the first order conditions for profit maximization - one for
labor and one for capital. ANSWER For each input the marginal
revenue product w.r.t. that input should be equal to the wage paid to
that input. Hence
paxLa−1 xbK = wL
pbxaL xb−1
K = wK
(b) Assume a + b < 1. Solve for Arlo’s factor demands, given the price of
output p and factor prices wL and wK . ANSWER The two frist order
conditions give us two equations for our two ‘unknowns’ (xL , xK ). Using
the first question, let’s solve for xb−1
K in terms of xL
xb−1
K
=
wL
paxLa−1
b−1
b
Now let’s substitute this expression for xb−1
into the second equation.
K
This will give a single equation which only contains xL terms.
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PROFESSOR JOSEPH GUSE
pbxaL
wL
paxLa−1
b−1
b
= wK
1−b
a−1 b
pax
L
= wK
pbxaL
wL
1−b
(a−1)(1−b)
pa b
a+
b
= wK
pbxL
wL
1−b
(a−1)(1−b)
wL b
w
a+
K
b
⇒ xL
=
pb pa
1−b
a+b−1
wK wL b
b
⇒ xL
=
pb pa
1−b
b
1−a−b
1−a−b
pa
pb
⇒ xL =
wK
wL
By symmetry the factor demand for capital is given by
⇒ xK =
pa
wL
a
1−a−b
pb
wK
1−a
1−a−b
(c) Now assume a + b = 1. Now RTS is constant. Assume that instead of a
constant wage rate, Arlo faces an upward sloping local labor supply curve
given by wL (xL ) = αxL . Solve for factor demands.ANSWER Typically
with constant returns to scale, constant factor and output prices, there
is no well-defined solution to a firm profit maximization problem: If the
firm can make a profit with some input bundle in that case, then the
firm can double profit by doubling all inputs. In this problem we assume
constant RTS but we remove the assumption of constant factor prices.
With that, we now have good reason to believe that there is well-defined
solution to the profit maximization problem. Let’s write out profit as a
function of input choice.
π(xL , xK ) = pxaL xbK − wK xK − αx2L
ECON 210 - MICROECONOMIC THEORY
HOMEWORK 6
PROFIT
5
In this expression pxaL xbK is, as usual revenue - output price times output
- and wK xK is the cost of capital. Cost of labor is given by αx2L since,
now the wage paid to labor is itself αxL . We can get two first order
conditions by differentiating this expression with respect to xL and xK
and setting each derivative equal to zero.
F OC[xL ] :
paxLa−1 xbK − 2αxL = 0
F OC[xK ] :
pbxaL xb−1
K − wK = 0
The rest is just algebra. Using F OC[xK ], we can solve for xbK in terms
of xL .
xbK =
wK
bpxaL
b
b−1
Now substitute this expression in for xbK in equation F OC[xL ].
b
b−1
w
K
− 2αxL = 0
paxLa−1
bpxaL
b
−ab
wK b−1
a−1 b−1
− 2αxL = 0
⇒paxL xL
bp
b
(a−1)(b−1)−ab
wK b−1
b−1
⇒paxL
− 2αxL = 0
bp
b
1−a−b
wK b−1
b−1
− 2αxL = 0
⇒paxL
bp
b
1−a−b−b+1
wK b−1
b−1
⇒paxL
− 2α = 0
bp
Note the last step in the sequence above was gotten by dividing the whole
equation thru by xL .1 Rearranging we end up with.
1Thanks
to Joey MacDonald for this step.
6
PROFESSOR JOSEPH GUSE
2−a−2b
b−1
xL
2α
=
pa
bp
wK
b
b−1
b
pa bp 1−b
=
⇒xL
2α wK
b
pa bp 1−b
⇒xL =
2α wK
2−a−2b
1−b
The last step is justified by the fact that 2 − a − 2b = 1 − b, since by
assumption a + b = 1. Substituting this back into our expression for xK
would yield the factor demand for capital.
(3) Suppose that the Rigel Corporation makes electricity (Y) using coal (C) and
labor (L) and land (D) according to the following production function
1
2
2
Y = 200C 5 L 5 D 5
(a) What are the returns to scale of Rigel’s electricity production? Show
your work. ANSWER:
2
1
2
f (αC, αL, αD) = 200(αC) 5 (αL) 5 (αD) 5
2
1
2
2
1
2
= 200(α 5 α 5 α 5 )C 5 L 5 D 5
= αf (C, L, D)
Hence scaling up each factor by any multiple α scales up output by α as
well. This is the definition of constant returns to scale.
(b) Suppose that Rigel’s land holdings are fixed in the medium-run. What
are the returns to scaling up just the coal and labor inputs? ANSWER:
1
2
2
f (αC, αL, D) = 200(αC) 5 (αL) 5 D 5
2
1
2
1
2
= 200(α 5 α 5 )C 5 L 5 D 5
3
= α 5 f (C, L, D)
Hence scaling up just the labor and coal inputs by a factor of α while
3
keeping land fixed results in just α 5 times the output.
ECON 210 - MICROECONOMIC THEORY
HOMEWORK 6
PROFIT
7
(c) What is the marginal product of labor? of coal? ANSWER:
200 2 −4 2
∂Y
=
C 5L 5 D5
∂L
5
∂Y
400 −3 1 2
=
C 5 L5 D5
∂C
5
(d) Show that the marginal rate of techinical substitution between labor and
land is decreasing. ANSWER:
∂Y
∂L
∂Y
∂D
=
2
−4
5
2
1
40C 5 L
2
D5
−3
80C 5 L 5 D 5
D
=
2L
This is the amount of land required to replace a unit of labor in order to
maintain a constant level of output. Notice that it is a function of both
land (D) and labor (L), but does not depend on the third input coal (C).
In particular it is increasing land and decreasing in labor. In words, the
more land you have, the more is required to replace lost labor; The more
labor you have the less land is required to replace lost labor. This means
that our technology obeys the principle of diminishing marginal rate of
techinical substitution.
(e) Write down an expression for Rigel’s profit as a function of prices and
input levels. Use pY for the price of electricity, wC for the price of coal,
wL for the price of labor, and wD for the price of land.
ANSWER:
2 1 2
π(pY , wc , wL , wD ) = 200py C L 5 D 5 − wC C − WL L − WD D
5
(f) Write down the first order conditions for profit maximization. ANSWER: There are three of them. One for each input.
400 −3 1 2
∂π
= 0 ⇒ pY
C 5 L 5 D 5 = wC
∂C
5
∂π
200 2 −4 2
= 0 ⇒ pY
C 5 L 5 D 5 = wL
∂L
5
∂π
400 2 1 −3
= 0 ⇒ pY
C 5 L 5 D 5 = wD
∂D
5
(g) Suppose that in the short run labor (L) and land (D) are fixed at levels, L̄
and D̄. Draw a picture in Y × C space showing the production function,
8
PROFESSOR JOSEPH GUSE
iso profit lines and the firm’s profit maximizing choice of coal. How does
the price of electricity affect the optimal choice of coal?ANSWER:
Short-Run Optimal Choice of Choice of Coal
Output (Y)
Isoprofit lines have slope
equal to wPYC
f (C, L̄, D̄)
Y∗
pY Y ∗ − w C C ∗ − F C
C∗
Coal
Since the slope of the iso-profit lines are pwYc , an increase in the price of electricity (PY ) would make them
flatter. Since the short-run production function is concave, this would cause the optimal choice of coal (C ∗ )
to increase.
(4) Julie makes cookies (Y) using labor (L), sugar (S) and corn-syrup (R) according to the following production function.
3
3
Y = 100L 7 (2S + R) 7
(a) What are Julie’s returns to scale?
(b) What is MRtS between sugar (S) and corn syrup (R)? Interpret your
answer in words. ANSWER:
3
3
100L 7 (2S + R) 7 2
∂Y /∂S
= 73
−4
3
∂Y /∂R
100L 7 (2S + R) 7
7
=2
−4
ECON 210 - MICROECONOMIC THEORY
HOMEWORK 6
PROFIT
9
In words, every cup of sugar taken out of production, must be replaced
with 2 cups of cornsyrup, if output is be maintained at the same level.
Note this rate of exchange is constant; it does not depend on which input
bundles it is evaluated at. Note also that the law of diminishing MRtS
only holds weakly. That is the MRtS between these two inputs neither
increases nor decreases. This is because they are perfect substitutes.
(c) What is the MRtS between labor (L) and sugar (S)? Interpret your
answer in words. ANSWER:
3
3
100L 7 (2S + R) 7
∂Y /∂L
= 37
−4
3
∂Y /∂S
100L 7 (2S + R) 7 2
7
(2S + R)
=
2L
−4
In words, when evaluated at input bundle (L, S, R), an hour of labor
cups of sugar.
taken out of production, must be replaced with (2S+R)
2L
Hence the more labor the process is using the less sweetener is required to
replace a lost unit of labor, which is precisely the meaning of diminishing
MRtS.
(d) (Optional) Suppose that the price of sugar, wS = 3, the price of labor
wL = 10 and the price of corn syrup wR = 1. How many cookies should
Julie make as a function of the price of cookies pY ?ANSWER: Note that
according the the MRtS between sugar and corn syrup, these two inputs
are perfect substitutes for each other. A cup of sugar peforms exactly
the same service as 2 cups of corn syrup in this production process.
Therefore, a profit maximizing firm will buy either corn syrup or sugar,
depending on their relative prices. Since the MRtS between is 2, the
firm should buy sugar if its price is less than twice the price corn syrup
and should buy only corn syrup if the opposite is true. By the prices
given in the problem, sugar is three times the price of syrup, so the firm
will buy only syrup and set S = 0. Hence we can write down a reduced
production function...
Y (L, S, R|
3
3
wS
> 2) = 100L 7 R 7
wR
The first order conditions (FOCs) for profit maximization must be satisfied at the optimal choices of Labor, L∗ and corn syrup R∗ .
10
PROFESSOR JOSEPH GUSE
−4
3
3
pY M PL = wL ⇒ pY 100L∗ 7 R∗ 7 = 10
7
−4
3
3
pY M PR = wR ⇒ pY 100L∗ 7 R∗ 7 = 1
7
Following the method laid out in the appendix to the profit maximization
chapter in Varian, we can re-write these in terms of output Y. To see
∗3
−4
how first, in the first equation above, re-write L∗ 7 as LL∗7 and similarly,
R∗
−4
7
3
as
R∗ 7
L∗
in the second equation to get
3
3
3 100L∗ 7 R∗ 7
= 10
pY M PL = wL ⇒ pY
7
L∗
3
3
3 100L∗ 7 R∗ 7
pY M PR = wR ⇒ pY
=1
7
R∗
Notice that the numerators on the left-hand side of both equations is the
output level evaluated at the optimal choice of input bundle, Y (L∗ , R∗ ),
which itself must be the profit-maximizing output level. Hence we have
Y (L∗ , R∗ )
3
⇒ pY
= 10
7
L∗
3
Y (L∗ , R∗ )
pY M PR = wR ⇒ pY
=1
7
R∗
This gives us expressions for the factor demands in terms of the profitmaximizing output level
pY M PL = wL
3
pY Y (L∗ , R∗ )
70
3
pY M PR = wR ⇒ R∗ = pY Y (L∗ , R∗ )
7
∗
∗
∗
Now let Y stand for Y (L , R ) and plug these demand functions back
into the production function
pY M PL = wL ⇒ L∗ =
3
Y = 100
pY Y ∗
70
∗
73 3
pY Y ∗
7
73
Factoring out the Y ∗ and moving it to the LHS and combining terms on
the RHS we get
ECON 210 - MICROECONOMIC THEORY
HOMEWORK 6
PROFIT
11
37 37
3
3
pY
pY
6 = 100
70
7
Y ∗7
6
11
3pY 7
∗ 71
⇒Y = 10 7
7
6
3pY
∗
11
⇒Y = 10
7
Y∗
Wow! if the price of cookies is $1, the firm would want to make about 619
million cookies to maximize profit! Note that cutting the price of cookies
in half makes this number drop considerably to 9.7 million cookies. In
general a doubling of price will increase this firm’s supply of cookies by
64 times (26 ). Try drawing a picture of the supply curve with output on
the horizontal axis and price on the vertical.