Econ 301 – F07
Answers to PROBLEM SET 5 - due in class on Thursday Nov 1
Wissink
1. Critically evaluate the following statements and explain in what way or ways they are true, false, or
uncertain.
a. Constant returns to scale is incompatible with the law of diminishing marginal productivity.
[Answer] False. Constant returns to scale implies that if you scale all inputs by the same factor,
say t, then output is scales by that same factor, t. On the other hand, the law of diminishing
returns deals with short-run production situations where at least one factor of production is fixed.
For example, x=L1/2K1/2 exhibits constant returns to scale, but also satisfies the law of
diminishing marginal productivity with respect to each input.
b. If an input’s marginal product curve is falling, so must its average product curve.
[Answer] False. When marginal product of an input is less than its average product, its average
product curve should be falling. See the graph below (left).
$
X
MC
MPL
AC
APL
L
X
c. If marginal cost is rising, then average variable cost must also be rising.
[Answer] False. When marginal cost is grater than average cost, the marginal cost is pulling
average cost up. When marginal cost if rising, average cost can be falling or rising. See the
graph above (right).
d. When average total cost is rising, then so is marginal cost and average variable cost.
[Answer] True. Average total cost can only rise if avc is rising and if avc is rising then so must
mc.
e. To cost minimize, the firm should juggle the use of variable inputs until the marginal products of
all the variable inputs are equal to each other.
[Answer] False. To cost minimize, a firm should juggle the use of variable inputs until there is
equal bang per buck, or equivalently, where MRTS=ERTS, i.e., MPL/MPK=w/r. However, if
w=r, then that would be a special case where then you would get equal marginal products at the
cost minimizing solution..
2. x = L1/2K1/2 and that w=$2 and r=$2.
a. Find the 3 long run cost curves and graph them.
b. Find the 7 short run cost curves assuming K=25 and graph them.
3. Suppose x = L2K2 and that w=$2 and r=$2.
a. Find the 3 long run cost curves and graph them.
b. Find the 7 short run cost curves assuming K=25 and graph them.
4. Suppose x = L + 5K and that w=$2 and r=$4.
a. Find the 3 long run cost curves and graph them.
b. Find the 7 short run cost curves assuming you are stuck with K=15 and graph them.
5. Suppose x = min{L, 2K} and that w=$1 and r=$4.
a. Find the 3 long run cost curves and graph them.
b. Find the 7 short run cost curves assuming K=20 and graph them.
[Answers] Q2-Q5
X = L1/2K1/2
w=$2 and r=$2
L*= x & K*= x
4x
4
4
x = L2K2
w=$2 and r=$2
L*=x1/4 & K=x1/4
4x1/4
4x-3/4
x-3/4
X = L + 5K
w=$2 and r=$4
L*=0 & K*=x/5
(4/5)x
4/5
4/5
x = min{L, 2K}
w=$1 and r=$4
L*=x & K*=1/2x
3x
3
3
b. SHORT RUN
K=25
L*= (1/25)x2
K=25
L*=(1/25)x1/2
(1b) srfc(x) = rK
(2b) srvc(x) = wL*(x)
50
(2/25)x2
50
(2/25)x1/2
K=15
L*= 0 for x≤75
L* = x -75 for
x>75
60
0 for x≤75
K=20
L*=x for
L<40(x≤40)
L*=∞ for x>40.
80
x for x≤40
∞ for x>40
(3b) srtc(x) = (1b)+(2b)
50+(2/25)x2
50+(2/25)x1/2
2x - 150 for x>75
60 for x≤75
a. LONG RUN
(1) lrtc(x) = wL*+rK*
(2) lratc(x) = lrtc/x
(3) lrmc(x) = dlrtc/dx
(4) srafc(x) = rK/x
(5) sravc(x) = wL*(x)/x
50/x
(2/25)x
50/x
(2/25)x -1/2
60 + 2x – 150 for
x>75
60/x
undef for x ≤ 75
(6) sratc(x) = (4)+(5)
50/x+(2/25)x
50/x+(2/25)x -1/2
-150/x+2 for x>75
60/x for x≤75
(7) srmc(x) = dsrtc(x)/ dx
(4/25)x
(1/25)x
-1/2
-90/x+2 for x>75
0 for x≤75
x+80 for x≤40
∞ for x>40
80/x
1 for x≤40
∞ for x>40
80/x +1 for x≤40
∞ for x>40
1 for x≤40
∞ for x>40
2 for x >75
2
3
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6. Assuming a two input production function and using both an isoquant/isocost diagram and a total
cost curve diagram, illustrate a situation where you have 2 points on the graphs that are LONG RUN
COST EFFICIENT and 2 additional points on the graphs that are SHORT RUN COST EFFICIENT
but not long run cost efficient. Assume that capital is the input that will become fixed in the short
run.
[Answer]
srtc at K0
K
$
srtc at K1
C
11000
8000
K
D
D
6000
C
0
B
B
1
K
lrtc
A
5000
A
x=200
x=100
isocost=$5,000
$6,000
isocost=$8,000
L
100
200
X
$11,000
(see the graph on the left) Points A & B would both be on the lratc and lrtc curves and each would be
on its respective short-run average and total cost curves. Points A & C would be on the same sratc
and the same srtc curve - drawn assuming you have K0 capital. Points D & B would be on the same
sratc and the same srtc curve - drawn assuming you have K1 capital.
SUGGESTION: Try drawing the average total cost curve picture. It should have one lratc curve
and two sratc curves.
7. Assume a competitive market for pumpkin pie filling. Suppose that Jack-the-pumpkin-king’s short
run total cost curve for pumpkin pie filling is as follows: (where q is in gallons)
c(q) = 100q – 4q2 + 0.2q3 + 450
a. What are the functions for his seven short cost curves?
[Answer]
fc(q)=450 and srvc(q)= 100q – 4q2 + 0.2q3 and srtc(q)= 100q – 4q2 + 0.2q3 + 450
srafc(q)= 450/q
sravc(q)= 100 – 4q + 0.2q2
sratc(q)= 100 – 4q + 0.2q2 + 450/q
srmc(q)= 100– 8q + 0.6q2
b. Graph all of them BUT average fixed costs in appropriate graphs.
[Answer]
5
200
190
180
170
160
150
140
130
120
110
100
90
80
mc-f irm
70
srat c
60
sravc
50
40
30
20
10
0
0
5
10
15
20
25
30
4000
3500
srtotal costs
3000
f ixed costs
variable costs
$
2500
2000
1500
1000
500
0
0
5
10
15
20
25
30
gals
6
MARKET w ith N=100
290
270
250
230
210
Market Supply
$
190
Market Demand
170
150
130
110
90
70
50
0
500
1000
1500
2000
2500
3000
3500
gal
c. Suppose there are N=100 firms in the market it the short run. What is the market short run
supply curve? Graph it.
[Answer] You need to horizontally sum over N=100 firms each firm’s marginal cost curve for
points where marginal cost is greater than or equal to average variable cost. You can do this by
brute force (with something like Excel) or do the math, which would give you the following:
PS = 100 - .08Q + .00006Q2 for P≥$80.
d. If the market demand is: P = 360 – (18/200)Q, where P is the per gallon price of pumpkin pie
filling, what is P*. Q* and q* in the pumpkin pie filling market. What are Jack’s profits at this
equilibrium?
[Answer] There are several different ways this could be solved, all yielding the same answer;
here is one way…
Demand: Note that Q = 100q and PD=360-(18/200)*(100q) = 360-9q
Firm supply: P=SRMC in equilibrium so for the firm we know  PS = 100-8q+0.6q2
At the equilibrium, PD=PS 360-9q = 100-8q+0.6q2
Now solve for q.
0.6q2+q-260=0.
(6q+130)(0.1q-2)=0
Thus, q*=20
Q*=100q*=2,000
P*=360-9q*=$180
Jack’s profits = total revenue - total costs = $3600 - $2450 = $1150
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8. Show that when apples are produced with only capital and labor and the production function exhibits
constant returns to scale, then “output per laborer” is a function of only the capital to labor ratio.
[Answer] Suppose: X = f(K, L) where the function f(.) exhibits constant returns to scale. This
implies that f(sK, sL) = sX. So, let s = 1/L. Now you have: X/L = f(K/L, L/L) or X/L = f(K/L).
9. You own a firm that has two active production plants: one on the east side of town and one on the
west side of town. You make widgets, q, in both these plants. The short run total cost curves for
each of these plants is as follows: $srtcEast = 100 + qe2 and $srtcWest = 50 + 2qw2. Suppose you want
to produce q = 300. How many units should be produced in the east side plant and how many in the
west side plant?
[Answer] $srmcEast= 2qe and $srmcWest=4qw. You would want to produce in such a way that the east
side plant and the west side plant have the same marginal costs for their last unit produced.
So set 2qe=4qw. ( qe=2qw) .
At the same time, you want a total of Q=300. so qe+qw=300(*). Plugging qe=2qw into (*), you get
qw=100 and qe=200.
10. You are an efficiency expert hired to consult the manufacturing firm XYZ. XYZ uses two inputs,
labor (L) and capital (K) to produce widgets. The firm is currently producing 5000 widgets, and you
know the following information:
PL = $4 per unit, PK = $100 per unit, MPL = 4, and MPK = 40
a. Is the firm producing efficiently? Why or why not?
[Answer] No. Compare MPL/PL (=4/4) and MPK=PK (=40/400). So, 1 is GREATER THAN
40/100, so you have not equated the bang/buck across the variable inputs. Note that you could
also set up MRTS = 4/40 (when L is on the horizontal axis) with ERTS =4/100 and make the
same observation.
b. What should the firm do?
[Answer] SIMULTANEOUSLY use more labor and less capital.
c. Graph the situation both before and after your advice.
[Answer] Refer point D and point A on the graph(left) at Q6. Point D describes the situation
BEFORE you advice. By simultaneously using more labor and less capital, you will be at A,
assuming you want to make 100 units of x.
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