313 Econ F07 Answers to PS5XtraQ - due in class on Thursday Nov 1 Wissink 1. 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.
2. 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.
3. 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.
4. 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 K=25 L*= (1/25)x2 x = L2K2 w=$2 and r=$2 L*=x1/4 & K=x1/4 4x1/4 4x-3/4 x-3/4 K=25 L*=(1/25)x1/2 X = L + 5K w=$2 and r=$4 L*=0 & K*=x/5 (4/5)x 4/5 4/5 K=15 L*= 0 for x75 L* = x -75 for x>75 60 0 for x75 2x - 150 for x>75 60 for x75 60 + 2x 150 for x>75 60/x undef for x 75 -150/x+2 for x>75 60/x for x75 -90/x+2 for x>75 0 for x75 2 for x >75 x = min{L, 2K} w=$1 and r=$4 L*=x & K*=1/2x 3x 3 3 K=20 L*=x for L<40(x40) L*= for x>40. 80 x for x40 for x>40 x+80 for x40 for x>40
a. LONG RUN (1) lrtc(x) = wL*+rK* (2) lratc(x) = lrtc/x (3) lrmc(x) = dlrtc/dx b. SHORT RUN
(1b) srfc(x) = rK (2b) srvc(x) = wL*(x)
50 (2/25)x2 50+(2/25)x2
50 (2/25)x1/2 50+(2/25)x1/2
(3b) srtc(x) = (1b)+(2b)
(4) srafc(x) = rK/x (5) sravc(x) = wL*(x)/x
50/x (2/25)x
50/x (2/25)x -1/2 50/x+(2/25)x -1/2
-1/2
80/x 1 for x40 for x>40 80/x +1 for x40 for x>40 1 for x40 for x>40
(6) sratc(x) = (4)+(5)
50/x+(2/25)x
(7) srmc(x) = dsrtc(x)/ dx
(4/25)x
(1/25)x
2
3
5. 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
$
C B D
srtc at K1 lrtc
11000 8000
K
D 1
B C
6000 5000
A
K
0
A
x=200 x=100
isocost=$5,000 $6,000 isocost=$8,000 $11,000
L
100
200
X
(see the graph on the left) Points A & B would both be on the lratc lrtc and 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.
6. 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]
4
200 190 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 0 5 10 15 20 25 30
mc-f irm srat c sravc
4000
3500
3000
srtotal costs f ixed costs
2500
variable costs
$
2000
1500
1000
500
0 0 5 10 15 gals 20 25 30
5
MARKET w ith N=100
290 270 250 230 210 190 Market Supply Market Demand
$
170 150 130 110 90 70 50 0 500 1000 1500 gal 2000 2500
3000
3500
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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