Producer Theory:
(a) 10 harvested acres requires 50 man-hours and 50 machine-hours, for a total cost
of 50 ($15) + 50 ($45) = $3,000. Given that total cost, the equation for the farm’s
isocost equation is $3000 = $15L + $45C, where L = man-hours/labor and C =
machine-hours on the combine. The isocost line is drawn below and labeled C0.
C (hours)
133
100
67
q1 = 20
q0 = 10
50
C0
C1
L (hours)
50 100
200
400
(b) Average cost = total cost/output = $3000/10 = $300 per acre. Marginal cost of
another acre is found by noting that an additional acre requires 5 more man-hours
at $15 per hour + 5 more machine-hours at $45 per hour = $75 + $225 = $300.
(c) See the isoquant q0 above.
(d) 100 hours of labor means that 20 acres of corn can now be harvested (100 hours/5
man-hours per acre). The new isoquant is depicted in q1 above.
(e) 20 harvested acres requires 100 man-hours and 100 machine-hours, for a total
cost of 100 ($15) + 100 ($45) = $6,000. Given that total cost, the equation for the
farm’s isocost equation is $6000 = $15L + $45C, where L = man-hours/labor and
C = machine-hours on the combine. The isocost line is drawn above and labeled
C1 .
(f) Average cost = total cost/output = $6000/20 = $300 per acre. Marginal cost of
another acre is found by yet again noting that an additional acre requires 5 more
man-hours at $15 per hour + 5 more machine-hours at $45 per hour = $75 + $225
= $300.
(g) Given this information, we know that AC = MC = $300 no matter the level of
output (since both MC and AC are constant). Thus, C(q) = 300q is the total cost
function.
Consumer Theory
(a) Under this plan, Otis’ budget constraint is $3500 = $70V + $1 AOG. The budget
constraint is below in black.
(b) Under the employer’s plan, Otis’ budget constraint is depicted in red below. You
can see that without the 40-visit constraint, the budget constraint would go all the
way out to 62.5 visits. However, with the 40-visit constraint, the budget
constraint stops at 40 visits.
The equation for this second budget constraint after he pays the lump-sum $1000
AND ignoring the 40-visit constraint is $2500 = $40V + $1 AOG. We need this
to find where the two constraints cross. To find this point, first begin by rewriting
the budget constraint in (a) to get AOG on one side of the equation and V on the
other side. We get:
AOG = 3500 – 70V
Rewriting the budget constraint in (b) to get AOG on one side of the equation
again, we get:
AOG = 2500 – 40V
Now that we’ve done this, we can find where the constrainst cross. We set the
budget constraints equal to each other:
3500 – 70V = 2500 – 40V
Solving for the number of visits tells us where they cross:
1000 = 30V OR V = 33 visits
This is also depicted on the graph below.
(c) Here, you need to discuss under what circumstances Otis would be likely to go
with his own plan in (a) versus his employers’ plan. For example, if Otis has very
steep indifference curves, that shows that he’s very interested in having AOG
versus doctors’ visits (he’s less willing to trade AOG for visits). This would be
the type of indifference curve under which we’d see him choose plan (a). Thus, if
Otis didn’t go to the doctor very often (less than 33 times), he would likely go it
on his own. On the other hand, a flat indifference curve would signify that he is
very willing to trade AOG to get more visits, or that he would probably go to the
doctor more often than 33 times – here, we’d be more likely to see him choose the
plan in part (b).
AOG
3500
2500
V
33
40
50
Market Structure
(a) The firms will have a joint industry profit maximization problem to find the
collusion outcome, as they will share the market. Industry total revenues will be:
TR = 250Q – Q2
So industry marginal revenue will be:
MR = 250 – 2Q
Profit max occurs when MR = MC, so here that will be:
250 – 2Q = 30
OR Q = 110. If A and B split the market, qA = qB = 55.
Market price will therefore be P = 250 – 110 = $140, and each firm’s profit will
be:
πA = πB = $140(55) - $30(55) = $6120.
(b) If qB = 40, then Firm A’s residual demand is Pres = 250 – qA – 40 = 210 – qA.
Thus, A’s total revenue function is TRA = 210qA – qA2, and its marginal revenue
function is MRA = 210 – 2qA. Thus, profit max, MR = MC, is:
210 – 2qA =30
SO qA = 90. Market price is therefore P = 250 – 90 – 40 = $120.
Firm A’s profits will be:
πA = $120(90) - $30(90) = $8100
And Firm B’s profits will be:
πB = $120(40) - $30(40) = $3600
(c) If the firms are Bertrand rivals, they will compete price down to marginal cost.
Thus, in the end, P = $30. This will mean that market output will be:
Q = 250 – 30 = 220.
If they split the market, qA = qB = 110.
Profits for each firm will be πA = πB = 0.
(d) The game will look as follows. L = large, S = small output.
L
L
AA
S
B
πA,
πB
$0,
$0
$8100, $3600
S
L
$3600, $8100
S
$6120, $6120
B
Using backward induction, if we start at B’s top subgame, we see that he
compares producing a lot (Large) and getting $0 or producing a little (Small) and earning
$3,600. Thus, B will choose Small at that subgame. At B’s lower subgame, we see he
can produce a lot (Large) and earn $8,100 or produce a little (Small) and earn $6120.
Here, he’ll obviously choose to produce Large. Now, move back to A’s subgame. If he
chooses to produce a lot (Large), he will earn $8100 (since B chose S at that point). On
the other hand he can produce a little (Small), he will earn $3600 (since B chose S at that
point also). Thus, A will choose to produce a lot. In the end, the subgame perfect
outcome is for A to produce Large, B to produce Small, and the market goes to the
Cournot outcome.