ECON 100B Lecture Notes - Lecture 7: Average Variable Cost, Marginal Cost

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26 Mar 2017

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Lagrangian method to find long run cost minimization. Problem: min l, k wl + rk such that f (l, k) = q. = wl + rk + [ q f (l, k) ] / l = w - f (l, k) / l = 0. / k = r - f (l, k) / k = 0. / = q f (l, k) = 0. W / r = ( f (l, k) / l) / ( f (l, k) / k) W / r = mp l / mp k. Q = f (l, k) plus the second-order condition: mrts diminishing hidden condition, l 0, k 0. Example of cost minimization using lagrangian f (l, k) = 2l 1/3 k 2/5. Given w > 0, r > 0, calculate c lr (q) = wl + rk + [q 2l 1/3 k 2/5] / l = w - 2/3 l -2/3 k 2/5 = 0.

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3. A firm has a long-run production function:

(a) In the short run, K = 81 is fixed. Find the short-run production function. What is the marginal product of labor in the short-run?

(b) If w = 10 and r = 15.24, find the short-run cost function. To do this, first, figure out how many workers are needed to produce Q units of output, and then add up the (fixed) SR cost of capital and the cost of labor as a function of Q. How many workers are needed to produce Q = 10?

(c) Write down equations for an average total cost (ATC) and average variable cost (AVC) as a function of Q. What is ATC when Q = 10? What as AVC when Q = 10?

(d) Write down the equation for the marginal cost (MC) as a function of Q. What is the marginal cost of producing Q = 10 in the short run?

(e) Verify that MC = w*MPL, evaluating the marginal product of labor from part (3a) at the number of workers needed to produce 10 units of output.

(f) In the long run, both capital and labor are variable. Find the marginal product of labor and the marginal product of capital. Find the marginal rate of technical substitution. On a properly-labeled graph, draw an isoquant for Q = 10.

(g) Does the production function exhibit increasing, decreasing, or constant returns to scale?

(h) Find the lowest cost method of producing Q = 10 (two unknowns: K and L). To do this, equate the MRTS with the input price ratio w r (one equation) and use the isoquant for Q = 10 as the second equation.

(i) Use the fact that the cost, in the long run, is wL + rK to find the LR cost of producing Q = 10. What is the long-run average cost (LRAC) at Q = 10?

(j) (Hard) More generally, use the MRTS and the production function to find the cost of producing any quantity Q. To do this, repeat the above exercise but keep Q as an (unknown) variable, and solve for K and L as functions of Q.

(k) On a properly labeled graph, depict ATC from part (3b) and LRAC from part (3j). Why is LRAC everywhere below ATC?

(l) What is the production level at which SRAC is minimized?

salmonhyena380

37.Use the following setup for the next question.

A manufacturing firm is deciding whether or not to invest in a new printer that needs an initial investment of $150,000. The investment would increase cash flows in the first year by $80,000 and in the second year by $75,000.

?If the interest rate is 10% then the net present value of the investment is

?a. $5,000

b. ?- $9,091

?c. -$15,290

?-$21,901

33.

Table 13-16
Listed in the table are the long-run total costs for three different firms.

Quantity	1	2	3	4	5
Firm A	100	100	100	100	100
Firm B	100	200	300	400	500
Firm C	100	300	600	1,000	1,500

Refer to Table 13-16. Which firm is experiencing constant returns to scale?

a.	Firm C only

b. Firm A only

c.	Firm B only

Firm A and Firm B only

27.

Table 13-7
The Flying Elvis Copter Rides

Quantity	Total Cost	Fixed Cost	Variable Cost	Marginal Cost	Average Fixed Cost	Average Variable Cost	Average Total Cost
0	$50	$50	$0	--	--	--	--
1	$150	A	B	C	D	E	F
2	G	H	I	$120	J	K	L
3	M	N	O	P	Q	$120	R

Refer to Table 13-7. What is the value of E?

ECON 100B Lecture Notes - Lecture 7: Average Variable Cost, Marginal Cost

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