# What is the firm’s long-run marginal cost curve?

For each of the following production functions (a and b)
find the following equations (i-iii) in terms of Q0, w and
r.
i) MRT SL,K
ii) Long-run capital and labor demand curve.
iii) Long-run total cost curve.
iv) Short-run capital and labor demand curve if the firm
is stuck with K¯ = 3.
v) Short-run total cost if the firm is stuck with K¯ = 3.
(a) Q = 5L
1/2K
1/2
(b) Q = LK + 6L
2. Smoothies must be produced with 6 bananas and 1
pineapple for each smoothie. For example, if you have
7 bananas and 1 pineapple then you can only produce
one smoothie (and you have one extra bananas that gets
wasted). Let Q be the number of smoothies, B be the
number of bananas, and P be the number of pineapples. The price of one banana is PB and the price of one
pineapple is PP .
(a) Determine three (B, P) points on the Q=10 isoquant (what combinations of inputs lead to production of 10 smoothies).
(b) What is the long-run total cost function for producing smoothies, T C (Q, PB, PP )? (Hint: think of this
intuitively rather than using formulas. For example,
what is the cheapest way to produce one smoothie?
Two smoothies? Q smoothies?)
(c) What is the production function for smoothies
Q (B, P)? (This tells how many smoothies can be
produced for each combination of B and P)
3. The processing of Q tax returns can be done using hours
of computer time (denoted by K) and hours of human
time (denoted L). The processing of Q = 120 tax returns
requires either 4 hours of computer time (L = 0 and K =
4) or 12 hours of human time (L = 12 and K = 0). Hours
of computer time and hours of human time are perfect
substitutes; for example, the processing of Q = 120 tax
returns could also be done using K = 3 hours of computer
time and L = 3 hours of human time.
(a) Determine 4 points on the Q = 240 isoquant.
(b) Write out the equation for the production function
Q (L, K).
(c) Determine the labor and capital demand in terms
of Q0, w and r.
(d) Determine the total cost in terms of Q0, w and r.
4. A firm’s long-run total cost curve is T C(Q) = 47Q −
40Q2 + 4Q3
.
(a) What is the firm’s long-run marginal cost curve?
(b) What is the firm’s long-run average cost curve?
(c) Over what range of output does the production
function exhibit economies of scale?
(d) Over what range of output does the production
function exhibit diseconomies of scale?
5. The following incomplete table shows a firm’s various
costs of producing up to 6 units of output. Fill in as
much of the table as possible. If you cannot determining
the number in a box, explain why it is not possible to do
so.
Q TC TVC AFC AC MC AVC TFC
1 40
2
3 75 27
4 252 30
5 150
6 168