Econ 301 – F06
PROBLEM SET 5 - ANSWERS
Wissink
1. Critically evaluate the following statements and explain in what way or ways they are true, false, or
uncertain.
a. Increasing returns to scale is incompatible with the law of diminishing marginal productivity.
False. Increasing returns to scale has to do with proportional changes in all inputs and the effect
of that on output. On the other hand, the law of diminishing returns deals with short-run
production situations where at least one factor of production is fixed. You could easily have
I.R.T.S. and still have the law of diminishing marginal returns.
b. If the average product of an input is falling, then the average product must exceed the marginal
product at that input level.
True, remember that if average product is falling, then marginal product must be BELOW the
average product, pulling average product down.
c. A production function with increasing returns to scale produces nothing but decreasing cost
curves.
False. It would produce a declining lratc relationship, but, srtc, srvc, lrtc would all be positively
sloped. There are lots of cost curves to consider. Beware of exactly which one.
d. Knowing a firm's production function is sufficient information for determining the firm's
efficient combination of inputs.
False. Need to know input prices as well (unless the technology is characterized by perfect
complements).
e. If marginal cost is greater than average cost, then average cost must be falling.
False. If marginal cost is greater than average cost, then marginal cost is pulling average cost
up.
f. If marginal cost is rising, then average cost must also be rising.
False. Marginal cost can easily start to rise BEFORE average cost is rising. As long as marginal
cost is still below average cost, average cost would still be falling.
2. Fill in the blanks in the following table that describes the short-run production function of labor in
the production of wine, holding capital constant at K=10.
labor
tp of labor
ap of labor
mp of labor
3hrs.
(1) 30*3 = 90
30
n.a.
4
(2) 90+20 = 110
(3) 110/4 = 27.5
20
5
130
(4) 130/5 = 26
(5) [130-110]/1 = 20
6
(6) 130+5 = 135
(7) 135/6 = 22.5
5
7
(8) 19.5*7 = 136.5
19.5
(9) [136.5-135]/1 = 1.5
a. Does the tabular production function exhibit diminishing marginal returns, and if so, where do
they "set in”?
After L=5.
b. Does the production function exhibit increasing, decreasing, or constant returns to scale?
Don’t have enough information to tell.
3. Calculate the marginal productivity of L (px/pL), the marginal utility of K (px/pK), and the MRTS
for each of the following production functions (the "p" stands for the partial derivative).
See answers to PS#2 - exact same functions except replace x & y with L & K and replace u
with x.
x = 2L + 3K;
x = 4L + 6K;
x = v(L) + K;
x = LK;
x = aL + bK;
x = LaKb;
x = 2(L).5 + K;
x = (L+2)(K+1);
x = ln(L) + K
x = (L+a)(K+b);
x = La + Kb
4. Suppose the production function for widgets is: x = 5L + K. Suppose the price of labor and capital
are both equal to $1. What combination of K and L should be used to produce 1 unit of output? 2
units? 3 units? Assuming there are no fixed costs, algebraically determine the total cost function; the
average total cost function; and the marginal cost function. Now re-do the cost functions assuming
exogenous fixed costs are equal to $2.00.
x = 5L + K  mpL = 5 & mpK = 1. Since $r = $w then only use labor - it's as cheap as K and 5
times as productive!
So ... x = 1  L = 1/5; x = 2  L = 2/5; x = 3  L = 3/5  L*(x) = (1/5)x and K*(x) = 0 
tc = $wL*(x) + $rK*(x)  tc = vc = $(1/5)x  atc = avc = $1/5  mc = $1/5.
Now, if fc = $2 we get...fc = $2; vc = $(1/5)x
tc = $(1/5)x + $2  atc = $1/5 + $2/x  avc = $1/5  mc = $1/5.
IMPORTANT NOTE OF CAUTION: The assumed fixed cost in this problem is exogenous –
something like a licensing fee. If we changed the problem to ask what would happen if, let us say, K
was already fixed in the short run at some value K=10 (let us say), then it would be a totally different
problem to solve. Just beware of that.
5. Repeat the previous exercise assuming x = 5LK.
This time, use MRTS=ERTS and then the production function. So: MRTS=(5K/5L)=K/L. And
ERTS=w/r. So, K/L = w/r  K=Lw/r. Now use the production function to get: x = 5(L)(Lw/r) 
L2 = (r/5w)x  L*(x) = [(r/5w)x]1/2. Now get that K*(x) = [(w/5r)x]1/2 . Now just plug into the cost
equation to get: tc = $wL*(x) + $rK*(x)  tc = vc = $(2/√5)√x  atc = avc = $(2/√5√x) and
mc=$(1)(1/√5√x).
With fc = $2 we get…fc = $2; vc = $(2/√5)√x
tc = $(2/√5)√x + $2  avc = $(2/√5√x); atc = $(2/√5√x) + $2/x; mc=$(1)(1/√5√x).
6. A firm in a perfectly competitive industry has the following production function: x=5K.5L.5.
a. Does the production function exhibit constant, increasing, or decreasing returns to scale?
CONSTANT returns to scale are exhibited.
b. Does it satisfy the law of diminishing returns?
Yes, examine the partial derivatives of x with respect to both K and L and note that these
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marginal products are positive but if you take each 2nd partial, they are each negative
demonstrating that each input here exhibits diminishing marginal productivity.
c. Assume now that capital is fixed at K=100 units. Also assume that PK=$1, and PL=$5. Find the
cost minimizing amount of labor to use to produce any given level of output. Algebraically and
graphically represent all seven short-run cost curves.
K = 100, PK = $1, PL = $5  x = 5√100√L  x = 5×10×√L  x = 50√L  L*(x) = x2/2500 
vc = $5x2/2500 = x2/500; fc = $100; tc = $x2/500 + $100; afc = 100/x; avc = x/500;
atc = 100/x + x/500; mc = 2x/500 = x/250 (You draw them, please!)
7. Fill in the blanks.
x
$tc
$fc
$vc
$atc
$avc
$afc
$mc
0
12
12
0
infinite
n.a.
infinite
n.a.
1
14
12
2
14
2
12
2
2
18
12
6
9
3
6
4
3
24
12
12
8
4
4
6
4
32
12
20
8
5
3
8
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.
Suppose: A = f(K, L) where the function f(.) exhibits constant returns to scale. This implies that
f(sK, sL) = sA. So, let s = 1/L. Now you have: A/L = f(K/L, L/L) or A/L = f(K/L).
9. Repeat exercise #4 assuming x = min{2L, K}
Can’t do MRTS=ERTS, so use intuition to get that: K = 2L at any optimal input combination. Now
use the production function to get: x = K  K*(x) = x. Also note that x=2L  L*(x) = x/2. Now
just plug it all into the cost function assuming that w = r= $1 to get that tc =vc = $(3/2)x;
atc = avc = $3/2; mc=$3/2.
If there are exogeneous fixed costs of $2, then you would get: fc = $2, vc=$(3/2)x, tc = $(3/2)x+$2;
afc = $2/x; avc = $3/2; atc = $3/2 + $2/x; mc = $3/2.
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10. Using an isoquant diagram with appropriate isocost curves, illustrate and match-up 2 points in the
diagram where the value of the firm’s long-run total cost is equal to its short-run total cost. Also
include 2 additional points which would lie on 2 distinct short-run total cost curve but not on the
long-run total cost curve.
K
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
K
you have K0 capital. Points D & B
1
would be on the same sratc and the
same srtc curve - drawn assuming
K
you have K1 capital.
0
SUGGESTION: Try drawing the
average total cost curve picture. It
should have one lratc curve and two
sratc curves
D
B
C
A
x=200
x=100
L
11. 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?
No. Set up “bang/buck” and notice that 4/4 = 1 is GREATER THAN 40/100.
b. What should the firm do? SIMULTANEOUSLY use more labor and less capital.
c. Graph the situation. It’s just like above where you are at D when you should be at A, assuming
you can vary both K and L and want to make 100 units of x.
12. 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 qwidgets, q, in both these plants. The marginal cost curves for each of
these plants is as follows: $srmcEast = 4q and $srmcWest = 2q. 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?
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 4qE=2qW and get that qW = 2qE . At the
same time you want a total of Q=300. So qW + qE = 300. Now plug in and get: 2qE + qE = 300 
3qE = 300  qE = 100 and so qW = 200.
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