A) The following data are available for output (Q) and Long Run Total Cost (LTC)
ID: 1177801 • Letter: A
Question
A) The following data are available for output (Q) and Long Run Total Cost (LTC) for a firm. Using appropriate calculations determine the range of outputs over which the firm%u2019s technology exhibits Increasing, Decreasing or Constant Returns to Scale.
Q
LTC
1
33
2
54
3
75
4
100
5
150
6
228
7
350
Provide economic reasoning
B) A monopolist%u2019s demand function is given by
P = 80 %u2013 3Q
(with MR = 80 %u2013 6Q).
Its total cost function is
TC = 20Q + 200
(with MC = 20).
(i) Using algebra determine the profit maximizing output, price and optimal profit for the firm.
(ii) Suppose that instead of maximizing profit, the firm wants to maximize total revenue. Using algebra determine the optimal output, price, profit and revenue for the firm.
Q
LTC
1
33
2
54
3
75
4
100
5
150
6
228
7
350
Explanation / Answer
%uF0B7 If we find the points of tangency between isoquants [defined by the long-run
production function, Q(K,L)] and isocosts [defined by w and r], these points of
efficiency make up the curve that is the expansion path. A formula for this curve
can be determined by the relationship
r
MP
w
MPL K
%uF03D
%uF0B7 If you know the intercepts of the isocost line, and the price of capital & labor, you
can find the total cost represented by that line. Or, if you know the intercepts and
the total cost, you can solve for w and r.
L-intercept=
w
TC
K-intercept=
r
TC
Relationships between long-run total cost (LTC), long-run average cost (LAC) and longrun marginal cost (LMC) are the same as for short-run STC, ATC and SMC:
%uF0B7 LAC=LTC/Q
%uF0B7 LMC= %uF0B6Q
%uF0B6
LTC
%uF0B7 A line from the origin to the LTC curve has a slope equal to the LAC at that point.
%uF0B7 A line tangent to the LTC curve has a slope equal to the LMC at that point.
%uF0B7 The comments about points B & D on the STC curve (prior page) are true about
the LTC curve.
Finding the long-run cost function, knowing r, w and the long-run production function
Q(K,L). NOTE: Finding a LT cost function won%u2019t be on the exam!
%uF0B7 First, find total cost in terms of K and L: TC=r K + w L
%uF0B7 Find the expansion path using
r
MP
w
MPL K
%uF03D ; solve for K in terms of L so that you
have a formula like K=g(L).
%uF0B7 Plug this formula for K into the long-run production function, Q(K,L)=Q(g(L),L),
and solve for L to get L=f(Q).
%uF0B7 Now substitute this formula for L into the formula for K to get K in terms of Q:
K=g(L)=g[f(Q)].
%uF0B7 Plug these formulas for K and L into the TC found above: TC=r g[f(Q)] + w f(Q)
Example: A firm faces production function Q=K2
L
2
; the cost of capital is 3 ($/unit) and
the cost of labor is 4 ($/hour).
%uF0B7 TC=r K + w L=3K+4L
%uF0B7 The marginal product wrt capital is MPK=
%uF0B6K
%uF0B6 [ K2
L
2
]=2KL2
The marginal product wrt labor is MPL
=
%uF0B6L
%uF0B6 [ K2
L
2
]=2K
2
L
The expansion path is
r
MP
w
MPL K
%uF03D
3
2
4
2
2 2
K L KL
%uF03D
Solving for K we get K=(4/3)L
%uF0B7 Plug this into the LR production function and solve for L:
Q(K,L)=K2
L
2
=[(4/3)L]
2
L
2
=(16/9)L
4
. Solve for L to get %uF028 %uF029
1/ 4
L %uF03D (9/16)Q .
%uF0B7 Use this to get K in terms of Q: K=(4/3)L=(4/3)%uF028 %uF029
1/ 4
(9/16)Q = %uF028 %uF0291/ 4
(16/9)Q
%uF0B7 Plug this into the cost formula: TC =3K+4L =3%uF028 %uF029
1/ 4
(16/9)Q +4%uF028 %uF0291/ 4
(9/16)Q
=144Q
1/4+144Q
1/4=288 Q1/4
Returns to scale (rts) determines the shape of the LAC curve. A u. A u-shaped LAC curve like
the one above has increasing rts until the minimum LAC, then decreasing rts.
If an industry has increasing rts at all output levels, it will have a continually declining
LAC curve, which will lead to a natural monopoly
Relationships between Short- and Long-Run Costs
When capital is fixed, the firm can only vary labor, so the short-run expansion path is
horizontal.
The LAC curve envelopes, or encloses, the set of ATC curves for each of the possible
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