962201 Intermediate Microeconomics
Khon Kaen University
Suggested Solution 1
Problem Set 5-7. Do the following functions exhibit increasing, constant, or decreasing returns to
scale? What happens to the marginal product of each individual factor as that factor is increased,
and the other factors held constant?
a. q = 3L + 2K
Solution:
This function exhibits constant returns to scale. For example, if L is 2 and K is 2 then q is 10. If L is
4 and K is 4 then q is 20. When the inputs are doubled, output will double. Each marginal product is
constant for this production function. When L increases by 1 q will increase by 3. When K increases
by 1 q will increase by 2.
b. q = ( 2L + 2K )
1
2
Solution:
This function exhibits decreasing returns to scale. For example, if L is 2 and K is 2 then q is 2.8. If L
is 4 and K is 4 then q is 4. When the inputs are doubled, output will be less than double. The
marginal product of each input is decreasing.
This can be determined using calculus by
differentiating the production function with respect to either input, while holding the other input
constant. For example, the marginal product of labor is
2
2 ( 2L + 2K ) 2
1
Since L is in the denominator, as L gets bigger, the marginal product gets smaller. If you do not
know calculus, then you can choose several values for L, find q (for some fixed value of K), and then
find the marginal product. For example, if L = 4 and K = 4 then q = 4. If L = 5 and K = 4 then
q = 4.24. If L = 6 and K = 4 then q = 4.47. Marginal product of labor falls from 0.24 to 0.23.
c. q = 3LK2
Solution:
This function exhibits increasing returns to scale. For example, if L is 2 and K is 2 then q is 24. If L
is 4 and K is 4 then q is 192. When the inputs are doubled, output will more than double. Notice
also that if we increase each input by the same factor λ then we get the following:
q ′ = 3 ( λL ) ( λK ) = λ 3 3LK 2 = λ 3q
2
Since λ is raised to a power greater than 1, we have increasing returns to scale.
Date: Jan 19th, 2009
Page 1
Instructor: A. N.
962201 Intermediate Microeconomics
Khon Kaen University
Suggested Solution 1
The marginal product of labor is constant and the marginal product of capital is increasing. For any
given value of K, when L is increased by 1 unit, q will go up by 3K2 units, which is a constant
number. Using calculus, the marginal product of capital is MPK = 6LK. As K increases, MPK will
increase. If you do not know calculus then you can fix the value of L, choose a starting value for K,
and find q. Now increase K by 1 unit and find the new q. Do this a few more times and you can
calculate marginal product. This was done in part b above, and is done in part d below.
1
2
d. q = L K
1
2
Solution:
This function exhibits constant returns to scale. For example, if L is 2 and K is 2 then q is 2. If L is
4 and K is 4 then q is 4. When the inputs are doubled, output will exactly double. Notice also that if
we increase each input by the same factor λ then we get the following:
1
1
1
1
q ′ = (λL)2 (λK )2 = λL2 K 2 = λq
Since λ is raised to the power 1, we have constant returns to scale. The marginal product of labor is
decreasing and the marginal product of capital is decreasing. Using calculus, the marginal product of
capital is
1
1 L 2
MPK = ⎛⎜ ⎞⎟
2⎝K ⎠
For any given value of L, as K increases, MPK will increase. If you do not know calculus then you
can fix the value of L, choose a starting value for K, and find q. Let L = 4 for example. If K is 4
then q is 4, if K is 5 then q is 4.47, and if K is 6 then q is 4.89. The marginal product of the 5th unit
of K is 4.47 – 4 = 0.47, and the marginal product of the 6th unit of K is 4.89 – 4.47 = 0.42.
we have diminishing marginal product of capital.
Hence
You can do the same thing for the marginal
product of labor.
1
2
e. q = 4L + 4K
Solution:
This function exhibits decreasing returns to scale. For example, if L is 2 and K is 2 then q is 13.66.
If L is 4 and K is 4 then q is 24. When the inputs are doubled, output will less than double. The
marginal product of labor is decreasing and the marginal product of capital is constant. For any
given value of L, when K is increased by 1 unit, q will go up by 4 units, which is a constant number.
To see that the marginal product of labor is decreasing, fix K=1 and choose values for L. If L = 1
Date: Jan 19th, 2009
Page 2
Instructor: A. N.
962201 Intermediate Microeconomics
Khon Kaen University
Suggested Solution 1
then q=8, if L=2 then q = 9.65, and if L = 3 then q = 10.93. The marginal product of the second unit
of labor is 9.65 – 8 = 1.65 and the marginal product of the third unit of labor is 10.93 – 9.65 =1.28.
Marginal product of labor is diminishing.
Problem Set 5-9. An Orcish Horde has two plants for producing Orc juggernauts, one in Flint and
one in Inkster. The Flint plant produces according to fF(x1, x2) = min{x1, 2x2} and the Inkster plant
produces according to fI(x1, x2) = min{2x1, x2}, where x1 and x2 are the inputs – peons and lumbers,
respectively.
a. On the graph below, use blue ink to draw the isoquant for 40 juggernauts at the Flint plant. Use
red ink to draw the isoquant for producing 40 juggernauts at the Inkster plant.
Solution:
x2
40
35
x1 = 20
x1 = 40
min{2x1, x2} = 40
x2 = 40
B
A
30
C
25
min{x1, 2x2} = 40
x2 = 20
20
20 25
30 35
x1
40
b. Suppose that the Horde wishes to produce 20 juggernauts at each plant. How much of each input
will the Horde need to produce 20 juggernauts at the Flint plant? How much of each input will the
Horde need to produce 20 juggernauts at the Inkster plant? Label with an a on the graph, the point
representing the total amount of each of the two inputs that the Horde needs to produce a total of
40 juggernauts, 20 at the Flint plant and 20 at the Inkster plant.
Date: Jan 19th, 2009
Page 3
Instructor: A. N.
962201 Intermediate Microeconomics
Khon Kaen University
Suggested Solution 1
Solution:
min{x1, 2x2} = 20 means the Flint plant will use x1 = 20 and x2 = 10 to produce 20 Juggernauts.
min{2x1, x2} means the Inkster plant will use x1 = 10 and x2 = 20 to produce 20 Juggernauts.
c. Label with a b on your graph the point that shows how much of each of the two inputs is needed
in total if the Horde is to produce 10 juggernauts in the Flint plant and 30 Juggernauts in the
Inkster plant. Label with a c the point that shows how much of each of the two inputs that the
Horde needs in total if it is to produce 30 juggernauts in the Flint plant and 10 juggernauts in the
Inkster plant. Use a black pen to draw the Horde’s isoquant for producing 40 units of output if it
can split production in any manner between the two plants. Is the technology available to the Horde
convex?
Solution:
To produce 10 juggernauts at the Flint plant means that min{x1, 2x2} = 10 so that x1 = 10 and x2 =
5. To produce 30 juggernauts at the Inkster plant means that min{2x1, x2} = 30 so that x1 = 15 and
x2 = 30. Hence, the point b is the coordinates (x1, x2) = (10 + 15, 5 + 30) = (25, 35). To produce 30
juggernauts at the Flint plant means that min{x1, 2x2} = 30 so that x1 = 30 and x2 = 15. To produce
10 juggernauts at the Inkster plant means that min{2x1, x2} = 10 so that x1 = 5 and x2 = 10. Thus,
the point c is the coordinates (x1, x2) = (30 + 5, 15 + 10) = (35, 25)
11. A firm has the production function f(x, y) = min{2x, x +y}. On the graph below, use red ink
to sketch a couple of production isoquants for this firm. A second firm has the production function
f(x, y) = x + min{x, y}. Use black ink to draw a couple of isoquants of the second firm. What kind
of returns to scale do these firms have?
Solution:
y
x=5
x = 15
x = 10
x=y
min{2x, x + y} = x + min{x, y} = 30
min{2x, x + y} = x + min{x, y} = 20
min{2x, x + y} = x + min{x, y} = 10
x + y = 30
x + y = 10
5 10 15
Date: Jan 19th, 2009
x
x + y = 20
20
Page 4
30
Instructor: A. N.
962201 Intermediate Microeconomics
Khon Kaen University
Suggested Solution 1
Here is the steps to draw the graphs. For f(x, y) = min{2x, x + y}, solve 2x = x + y so that the
combination is x = y. Consider the lower domain where x > y, says x = 2 and y = 1. Then 2x = 4
and x + y = 3. Thus, f(x, y) = min{2x, x + y} = x + y when x > y. Consequently, when x < y,
f(x, y) = min{2x, x + y} = 2x.
To draw the isoquant, suppose the constant value in which that
isoquant equals. For example, f(x, y) = min{2x, x + y} = 10. You will draw the graph x + y = 10
where x > y and you will draw 2x = 10 which means this is just the vertical graph of x = 5 where
x < y.
You use the similar steps to draw f(x, y) = x + min{x, y}. When x < y, the graph will be f(x, y) = x
+ x = 2x. Conseuquently, when x > y, the graph will be f(x, y) = x + y. Thus, f(x, y) = in this case
is exactly similar to what we have drawn before!
Both production functions have constant returns to scale. If we increase all inputs by λ times, the
new output will be
q ′ = f ( λx , λy ) = min {2 ( λx ) , ( λx ) + ( λy )} = λ min {2x , x + y } = λq
q ′ = f ( λx , λy ) = ( λx ) + min {( λx ) , ( λy )} = λx + λ min {x , y } = λ [x + min {x , y }] = λq
Problem Set 6-2. A firm uses a single input to produce a recreational commodity according to a
production function f(x) = 4 x , where x is the number of units of input. The commodity sells for
100 baht per unit. The input costs 50 baht per unit.
a. Write down a function that states the firm’s profit as a function of the amount of input.
Solution:
π = pf ( x ) − wx = 100 × ( 4 x ) − 50 × x = 400 x − 50x .
b. What is the profit-maximizing amount of input and output? How much profit does it make when
it maximizes profits?
Solution:
d π 200
200
=
− 50 = 0 →
= 50 → x = 4 → x = 16 . The
dx
x
x
maximum profit can be found by substituting x = 2 into the profit function above. That is,
Profit maximizing output can be found by
400 16 − 50 (16 ) = 800 .
Date: Jan 19th, 2009
Page 5
Instructor: A. N.
962201 Intermediate Microeconomics
Khon Kaen University
Suggested Solution 1
c. Suppose that the firm is taxed 20 baht per unit of its output and the price of its input is
subsidized by 10 baht. What is its new input level? What is its new output level? How much profit
does it make now? (Hint: A good way to solve this is to write an expression for the firm’s profit as a
function of its input and solve for the profit-maximizing amount of input.)
Solution:
The commodity price is fallen from 100 to 80 and the input price has fallen from 50 to 40. The new
d π 160
profit is then π = 80 × 4 x − 40x = 320 x − 40x . Then,
=
= 40 → x = 4 → x = 16 . The
dx
x
new output level is 4 16 = 16 . The profit becomes 320 16 − 40 (16 ) = 640 .
d. Suppose that instead of these taxes and subsidies, the firm is taxed at 50% of ts profits. Write
down its after-tax profits as a function of the amount of input.
What is the profit-maximizing
amount of output? How much profit does it make after taxes?
Solution:
1
( 400 x − 50x ) . It is easy to show that the profit maximizing inputs is
2
similar to part b but the profit will be half so that it equals just 400.
The new profit will be π new =
Problem Set 6-3. A profit-maximizing firm produces one output, y, and uses one input x, to product
it. The price per unit of the factor is denoted by w and the price of the output is denoted by p.
You observe the firm’s behavior over three periods and find the followings:
Period
y
x
w
P
1
1
1
1
1
2
2.5
3
0.5
1
3
4
8
0.25
1
a. Write an equation that gives the firm’s profits, π , as a function of the amount of input x it uses,
the amount of output y it produces, the per-unit cost of the input w, and the price of output p.
Solution:
This is just π = py − wx .
Date: Jan 19th, 2009
Page 6
Instructor: A. N.
962201 Intermediate Microeconomics
Khon Kaen University
Suggested Solution 1
b. In the diagram below, draw an isoprofit line for each of the three periods, showing combinations of
input and output that would yield the same profits that period as the combination actually chosen.
What are the equations for these three lines? Using the theory of revealed profitability, (WAPM),
shade in the region on the graph that represents input-output combinations that could be feasible as
far as one can tell from the evidence that is available. How would you describe this region in words?
y=x
Output
y =1+
1
x
2
y =2+
4
1
x
4
2.5
1
1
3
Input
8
c. You are hired by the Lawsuit firm, which has a purpose to find the evidence of NOT maximizing
profit, and then punish the manager who does not produce at profit-maximizing output. Is there any
evidence from this firm?
Solution: No. this firm’s input and output combinations satisfy WAPM. It is clear from the graphs
if you follow me carefully. If this firm uses other input-output combination, its isoprofit curves will
fall. The input and output chosen in each period maximize profit. For example, if we use x = 3 to
produce y = 2.5 when w = P = 1. Then, profit becomes (1)(2.5) – (1)(3) = – 0.5< 0 = profit in the
first period. You can convince yourself more by comparing profits in any other period, but the graph
should be enough.
d. In your diagram, shade in the region that would prove the misbehavior of not maximizing profit
for this firm.
Solution: See above.
Date: Jan 19th, 2009
Page 7
Instructor: A. N.
962201 Intermediate Microeconomics
Khon Kaen University
Suggested Solution 1
Problem Set 6-6. Nadine sells user-friendly software. Her firm’s production function is f(x1, x2) = x1
+ 2x2, where x1 is the amount of unskilled labor and x2 is the amount of skilled labor that she
employs.
a. In the graph below, draw a production isoquant representing input combinations that will produce
20 units of output.
Draw another isoquant representing input combinations that will produce 40
units of output.
Solution:
x2
20
10
x1 + 2x2 = 40
x1 + 2x2 = 20
20
x1
40
b. Does this production function exhibit increasing, decreasing, or constant returns to scale? Show
your argument.
Solution:
Clearly, the production function exhibits constant returns to scale. If we increase the all inputs by
the factor λ , the new output will be q ′ = ( λx 1 ) + 2 ( λx 2 ) = λ ( x 1 + 2x 2 ) = λq .
c. If Nadine uses only unskilled labor, how much unskilled labor would she need in order to produce
y units of output?
Solution:
She will use x1 = y.
Date: Jan 19th, 2009
Page 8
Instructor: A. N.
962201 Intermediate Microeconomics
Khon Kaen University
Suggested Solution 1
d. If Nadine uses only skilled labor to produce output, how much skilled labor would she need in
order to produce y units of output?
Solution:
She will use 2x2 = y, which means she uses just x2 = y/2. The skilled labor has twice marginal
productivity to that of unskilled labor.
e. If Nadine faces factor prices w1 = w2 = 1, what is the cheapest way for her to produce 20 units of
output, i.e., how many units of x1 and x2 will be used?
Solution:
If she uses only x1, she must use x1 = 20 units so she must pay 20. If she uses only x2, she must use x2
= 20/2 = 10 units so she must pay just 10. Thus, she will use x1 = 0 and x2 = 10.
f. If Nadine faces factor prices w1 = 1 and w2 = 3, what is the cheapest way for her to produce 20
units of output?
Solution:
If she uses x1, she pays 20. If she uses x2, she will use x2 = 5, but pay 3 for each unit. Thus, she will
end up paying 15 if she uses only x2. Therefore, she will use x1 = 20 and x2 = 0.
g. If Nadine faces factor prices, in general terms, w1 and w2, what will be the minimal cost of
producing 20 units of output?
Solution:
The minimal cost will be min{w1, w2/2}(20)
h. If If Nadine faces factor prices, in general terms, w1 and w2, what will be the minimal cost of
producing y units of output? (Hint: The question asks you to find the cost function, c(w1, w2, y).)
Solution:
Everyone should be able to do all these if you follow me in the class carefully. If she wants to produce
y units of output, she will compare the amount she pays for using x1 and x2. If she use x1 alone, she
use x1 = y so she pays w1y if she uses only x1 .
If she uses only x2 to produce y units, she uses x2 =
y/2 units. Hence, she pays w2y/2. She compares which one is lower between w1y and w2y/2. Thus,
w
the minimal cost of producing y units of output will be C(w1, w2, y) = min{w1, 2 }y.
2
Date: Jan 19th, 2009
Page 9
Instructor: A. N.
962201 Intermediate Microeconomics
Khon Kaen University
Suggested Solution 1
Problem Set 6-7. The Ontario Brassworks produces brazen effronteries. As you know brass is an
alloy of copper and zinc, used in fixed proportions. The production function is given by f(x1, x2) =
min{x1, 2x2}, where x1 is the amount of copper it uses and x2 is the amount of zinc that it uses in
production.
a. Illustrate a typical isoquant for this production function in the graph below.
Solution:
x2
x1 = 2x2
x1
b. Does this production function exhibit increasing, decreasing, or constant returns to scale?
You must show your argument why that is so.
Solution:
Constant returns to scale. q ′ = min {λx 1, 2 ( λx 2 )} = λ min {x 1, 2x 2 } = λq
c. If the firm wanted to produce 10 effronteries, how much copper would it need? How much zinc
would it need?
Solution:
It must use x1 = 2x2 = 10. Thus, x1 = 10 and x2 = 5.
Date: Jan 19th, 2009
Page 10
Instructor: A. N.
962201 Intermediate Microeconomics
Khon Kaen University
Suggested Solution 1
d. If the firm faces prices w1 = w2 = 1, what is the cheapest way for it to produce 10 effronteries?
How much will this cost?
Solution:
To produce 10 effronteries, x1 = 10 and x2 = 5 so the firm pays (1)(10) + (1)(5) = 15.
e. If the firm faces factor prices in general terms w1, and w2, what is the cheapest cost to produce 10
effronteries?
Solution:
To produce 10 effronteries, the firm pays 10w1 + 5w2 .
f. If the firm faces factor prices in general terms w1 and w2, what will be the minimal cost of
producing y effronteries? (Hint: Again, find the cost function c(w1, w2, y).)
Solution:
See class explanation. The minimal cost is C(w1, w2, y) = {w1 + w2/2}y.
Problem Set 6-10. The prices of inputs (x1, x2, x3, x4) are (4, 1, 3, 2).
a. If the production function is given by f(x1, x2) = min{x1, x2}, what is the minimum cost of
producing one unit of output? (A numerical answer) If the prices of inputs are given by (w1, w2, w3,
w4), what is the minimum cost of producing y units of output? (An analytical cost function, c(w1,
w2, y).)
Solution:
The analytical cost function for f(x1, x2) = min{x1, x2} is C(w1, w2, y) = {w1 + w2}y = {4 + 1}(1) = 5.
b. If the production function is given by f(x3, x4) = x3 + x4, what is the minimum cost of producing
one unit of output? (A numerical answer) If the prices of inputs are given by (w1, w2, w3, w4), what
is the minimum cost of producing y units of output? (An analytical cost function, c(w3, w4, y).)
Solution:
The analytical cost function for f(x3, x4) = x3 + x4 is C(w3, w4, y) = min{w3, w4}y = min{3, 2}(1) = 2.
Date: Jan 19th, 2009
Page 11
Instructor: A. N.
962201 Intermediate Microeconomics
Khon Kaen University
Suggested Solution 1
c. If the production function is given by f(x1, x2, x3, x4) = min{x1 + x2, x3 + x4}, what is the
mininum cost of producing one unit of output? ? (A numerical answer) If the prices of inputs are
given by (w1, w2, w3, w4), what is the minimum cost of producing y units of output? (An analytical
cost function, c(w1, w2, w3, w4, y).)
Solution:
We might need some explanation rather than the final answer. Suppose A = x1 + x2 and B = x3 + x4.
Then, min{x1 + x2, x3 + x4} = min{A, B} so the cost function must be of the form (wA + wB)y, where
wA is like the input price for group A and wB is like input prices for group B. But A is of the form x1
+ x2 and B is of the form x3 + x4 . Hence, the cost function on A must be wA = min{w1, w2}y and
that on B must be min{w3, w4}y. Therefore,
C(w1, w2, w3, w4, y) = (min{w1, w2} + min{w3, w4})y = (min{4, 1}+min{3, 2})(1) = 1+2 = 3.
d. If the production function is given by f(x1, x2, x3, x4) = min{x1, x2} + min{x3, x4}, what is the
mininum cost of producing one unit of output? ? (A numerical answer) If the prices of inputs are
given by (w1, w2, w3, w4), what is the minimum cost of producing y units of output? (An analytical
cost function, c(w1, w2, w3, w4, y).)
Solution:
Suppose A = min{x1, x2} and B = min{x3, x4}. Then the production function looks like A + B so
that the cost function must be of the form min{wA, wB}y. The cost function for A must be of the
form wA = (w1 + w2)y and the cost function for B must be of the form wB = (w3 + w4)y. Hence,
C(w1, w2, w3, w4, y) = min{w1 + w2, w3 + w4}y = min{4 + 1, 3 + 2}(1) = 5.
Problem Set 6-11. The T-bone chicken firm is concerned about its behavior whether it satisfy
WACM. The firm feeds its chickens on a mixture of soybeans and corn, depending on the prices of
each. According to the data submitted by its managers, when the price of soybeans was $10 a bushel
and the price of corn was $10 a bushel, it used 50 bushels of corn and 150 bushels of soybeans for
each coop of chickens. When the price of soybeans was $20 a bushel and the price of corn was $10 a
bushel, it used 300 bushels of corn and no soybeans per coop of chickens. When the price soybeans
was $10 and the price of corn was $20 a bushel, it used 250 bushels of soybeans and no corn for each
coop of chickens
Date: Jan 19th, 2009
Page 12
Instructor: A. N.
962201 Intermediate Microeconomics
Khon Kaen University
Suggested Solution 1
a. Graph these three input combinations and isocost lines.
Solution:
x2 = Corn
300 B
200
125
100
A
150
200
C
250
x1 = Soybeans
b. How much money did this firm spend per coop of chickens when the prices were (w1, w2) = (10,
10), when the prices were(w1, w2) = (10, 20), and when the prices were (w1, w2) = (20, 10)?
Solution:
When (w1, w2) = (10, 10), (x1, x2) = (150, 50) so the firm spent (10)(150) + (10)(50) = 2,000.
When (w1, w2) = (10, 20), (x1, x2) = (250, 0) so the firm spent (10)(250) + (20)(0) = 2,500.
When (w1, w2) = (20, 10), (x1, x2) = (0, 300) so the firm spent (20)(0) + (10)(300) = 3,000.
c. Is there any evidence that T-bone chicken were not minimizing costs? Why or why not?
Solution: It should be clear from the graph that all input combinations satisfy WACM.
d. The firm wonders whether there are any prices of corn and soybeans at which it will use 150
bushels of corn and 50 bushels of soybeans to produce a coop of chickens. How much would this
production method cost per coop of chickens if the prices were (w1, w2) = (10, 10), (w1, w2) = (10,
20), and (w1, w2) = (20, 10)?
Solution:
Date: Jan 19th, 2009
Page 13
Instructor: A. N.
962201 Intermediate Microeconomics
Khon Kaen University
Suggested Solution 1
When (w1, w2) = (10, 10), (x1, x2) = (50, 150) so the firm spent (10)(50) + (10)(150) = 2,000.
When (w1, w2) = (10, 20), (x1, x2) = (50, 150) so the firm spent (10)(50) + (20)(150) = 3,500.
When (w1, w2) = (20, 10), (x1, x2) = (50, 150) so the firm spent (20)(50) + (10)(150) = 2,500.
Problem Set 6-6.
You manage a plant that mass produces engines by teams of workers using
assembly machines. The technology is summarized by the production function.
q = 5KL
where q is the number of engines per week, K is the number of assembly machines, and L is the
number of labor teams. Each assembly machine rents for r = ฿10,000 per week and each team costs
w = ฿5,000 per week. Engine costs are given by the cost of labor teams and machines, plus ฿2,000
per engine for raw materials. Your plant has a fixed installation of 5 assembly machines as part of
its design.
a. What is the short-run cost function, c(q), for your plant – namely, how much would it cost to
produce q engines? What are average and marginal costs for producing q engine? How do average
costs vary with output?
Solution:
K is fixed at 5. The short-run production function then becomes q = 5(5)L = 25L. This implies that
for any level of output q, the number of labor teams hired will be L = q/25. The total cost function
is thus given by the sum of the costs of capital, labor, and raw materials:
TC(q) = rK +wL +2,000q = (10,000)(5) + (5,000)(
q
) + 2,000q
25
TC(q) = 50,000 +2,200q.
TC (q ) 50, 000 + 2, 200q
=
. Average cost will
q
q
d
50, 000
AC (q ) = −
decrease as output increases since
< 0 . The marginal cost function is given by
dq
q2
d
MC (q ) = TC (q ) = 2, 200 .
dq
The average cost function is then given by AC (q ) =
Date: Jan 19th, 2009
Page 14
Instructor: A. N.
962201 Intermediate Microeconomics
Khon Kaen University
Suggested Solution 1
b. How many teams are required to produce 250 engines? What is the average cost per engine?
Solution:
To produce q = 250 engines we need labor teams L = q/25 or L = 10. Average costs are given by
AC (q = 250) =
50, 000 + 2, 200 ( 250 )
= 2, 400.
250
c. You are asked to make recommendations for the design of a new production facility.
capital/labor,
What
K
, ratio should the new plant accommodate if it wants to minimize the total cost of
L
producing any level of output q?
Solution:
We no longer assume that K is fixed at 5. We need to find the combination of K and L that
minimizes costs at any level of output q. The cost-minimization rule is given by
MPK
r
=
MPL
w
To find the marginal product of capital, observe that increasing K by 1 unit increases q by 5L, so
MPK = 5L.
Similarly, observe that increasing L by 1 unit increases Q by 5K, so MPL = 5K.
∂Q
∂Q
Mathematically, MPK =
= 5L and MPL =
=5K. Using these formulas in the cost∂L
∂K
5L
r
K w
5, 000
1
minimization rule, we obtain
= →
= =
= . The new plant should accommodate
5K w
L
r 10, 000 2
a capital to labor ratio of 1 to 2. Note that the current firm is presently operating at this capital-labor
ratio.
1
Problem Set 7-9. Consider a firm with the production function Y = L2 . Note that because there is
only one factor of production, the long run cost curves are identical to the short run cost curves.
a. Solve for the conditional factor demand function L*. Note: this part is really easy.
Solution:
The production function is Y = L1/2, so to find L*, we simply solve for L. L* = Y2.
b. Solve for the LRTC, LRAC, and LRMC, functions. Illustrate three curves on a single graph.
Note: it is important that you graph the functions correctly.
Date: Jan 19th, 2009
Page 15
Instructor: A. N.
962201 Intermediate Microeconomics
Khon Kaen University
Suggested Solution 1
Solution:
In this case, LRTC = wL* = wY2. Therefore, LRAC = LRTC/Y = wY, and LRMC = dLRTC/dY =
2wY. The graph is shown below. The LRTC is a quadratic because the LRMC increases with Y, and
the LRMC and LRAC are both straight lines. All of the curves go through the origin (because there
are no variable costs), and the LRMC is twice as steep as the LRAC
.c. Repeat a. and b., using the production function Y = L.
Solution:
In this case, the production function is Y = L. So, L* = Y. Therefore, LRTC = wL* = wY. LRAC
= LRTC/Y = w, and LRMC = dLRTC/dY = w. The graph is shown below. The LRTC is a straight
line through the origin. The LRMC and LRAC are the same line, and are horizontal.
Date: Jan 19th, 2009
Page 16
Instructor: A. N.
962201 Intermediate Microeconomics
Khon Kaen University
Suggested Solution 1
d. Repeat a. and b., using the production function Y = L2.
Solution:
In this case, the production function is Y = L2. So, L* = Y1/2. Therefore, LRTC = wL* = wY1/2.
LRAC = LRTC/Y = w/Y1/2, and LRMC = dLRTC/dY = (1/2)w/Y1/2. The graph is shown below.
The LRTC is increasing, but at an increasing rate. This is because the LRMC is decreasing as Y
increases. The LRAC curve is greater than the LRMC, and therefore is decreasing.
Date: Jan 19th, 2009
Page 17
Instructor: A. N.
962201 Intermediate Microeconomics
Khon Kaen University
Suggested Solution 1
e. Describe the differences in the cost curve graphs for parts b, c, and d. Why do they look different
in the 3 different cases? (Hint: Returns to scale.)
Solution:
The pictures look different because of the differences in returns to scale in the production functions.
In parts a and b, the production function Y = L1/2 has decreasing returns to scale. Therefore the
LRTC is increasing, as are the LRAC and LRMC. In part c, the production function L = Y has
constant returns to scale. Therefore the LRAC curve is flat (and coincides with the LRMC curve). In
the last case, the production function is Y = L2, and there are increasing returns to scale. In this
case, the LRAC curve is downward sloping, and is always greater than the LRMC curve. The LRTC
curve is increasing, but at a decreasing rate. In summary, the shape of the cost curves is dictated by
the features of the production function (i.e. returns to scale).
Date: Jan 19th, 2009
Page 18
Instructor: A. N.