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# Homework 2 solutions.pdf

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University of Toronto Mississauga

Economics

ECO204Y5

Kathleen Wong

Fall

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ECO204: Homework 2 Solutions
1. True, False, Uncertain.
a. False. In the short run, when firms want to increase output levels, they cannot always
choose the most efficient combinations of L and K because of the constraints firms face
in the short run: only their use of L is flexible because K is fixed in the short-run. This
may result in decreasing MP Even when costs are minimized, it is not necessary that
L.
their use of L and K will be at the max of where MLand MP K.
b. False. Each individual firm faces a horizontal demand curve. This is because perfectly
competitive firms are price takers, where price is a constant. This is their demand curve.
If firms try to sell their output at any price other than the market price, they will not
attract any buyers. Only the market demand curve is downward sloping.
c. False. Firms will shut down when their AVC is above the market price since this implies
they are incurring negative profits for each unit of output produced.
d. False. Long-run average cost curves are U-shaped to illustrate the different returns of
scale firms experience. When firms are small, they initially experience increasing returns
to scale, which lead to economies of scale (costs decrease as q increase) which cause the
downward sloping portions of the long-run average cost curve. When firms are large,
they experience decreasing returns to scale (costs increase as q increases) which cause the
long-run average cost curve to have a positive slope.
e. True. Assuming that the increase in output levels is incremental, (for example, q=100,
q=200, q = 300), firms with decreasing returns to scale require more and more K and L to
produce higher levels of output. So their isoquants will be drawn further and further apart
from one another.
f. Uncertain – For the firm to increase output in the short-run, it must increase their use of L
since the level of K is fixed. Whether their AC is increasing with output depends on
whether the firm is experiencing diminishing marginal product of labor.
If the firm sees that MPL is decreasing, the statement is true. As the MPL decreases, it
increases their MC and thus causes AC to increase as output increases.
However, if the firm’s MPL is increasing, the statement is false. An increasing or
constant MPL will cause the MC to either decrease or remain constant (respectively), so
the firm’s AC will also decrease or remain constant as output increases.
2. Firm A: qA L 2/ K 1/, Firm B: qB L K/ 2 1/ 2
1/ 3 2/ 3
a. Firm A:MPL qA 2K , MPK qA L
L 3L1/ 3 K 3K 2/ 3
1/ 2 1/ 2
q B K qB L
Firm B: MPL L 2L1/ 2 MPK K 2K 1/ 2
b. You can substitute values of L and K into the production function:
qA(2, 2) = 2, A (4, 4) = 4
qB(2, 2) = 2, B (4, 4) = 4 2 1 1 1
Or you can sum the exponents to determine 1 and 1. Both methods
3 3 2 2
indicate that this technology exhibits constant returns to scale.
c. In the short run, K = 64, so the production function simplifies to:
q 8
Firm A: qA L2/ (64)/ 4L 2/ MPL A A
L 3L1/ 3
Firm B: q L (64) 1/ 8L1/ MPL q B 4
B B L L1/ 2
To determine whether Firm A and Firm B will have the larger marginal product of labor,
you can either substitute in values of L intA MPLMPL B or a more efficient way is to
set MPLA = MPLB to solve for L. At this level of L, the MPL is the same for both firms:
8 4
3L1/3 L1/ 2
L1/ 2 4(3)
1/3
L 8
L1/6 3/2
L 11.4
So at L 11.4, MPLA= MPL FB. L < 11.4 and L >11.4 however, one firm will have a
larger marginal product of labor. Suppose L = 8, then APL
8 8 8 1.33 and MPL = 4 4 1.41. So of L < 11.4, MPL <
3L1/3 3(8)/3 3(2) B L1/ 2 (8)1/ 2 A
MPL B If L > 11.4, MPLA> MPL . B
2
3. F(L, S) = 0.5LS
( , ,) P L S (0.5LS 2 q)
L S
2 2PL
0 PL 0.5S 0 2
L S
P
0 P S 0.5 )2SL 0 S
S SL
0 0.5LS 2 0 q 0.5S 2
From the first two first order conditions: PS 2P SL P S 2
S 2 SL L S
2
2P L S 2PLL
P S S P
S S
P S 2 P S
L S S
2PLS 2PL
To produce q gallons of lemonade, substitute the S into the third first-order condition: 2
q 0.5LS
2
2 PLL
q 0.5L
PS
2 2 3 2
q 0.5L 4 PL L 2 L PL
P 2 PS2
S
qPS2 3
2 L
2 PL
2 1/3
qPS
2P 2 L
L
Substitute L into the third first-order condition:
2
q 0.5LS
q 0.5 PSS S 2
2P
L
P S 3
q
4P L
4 P q
L S 3
PS
1/3
4PLq
S
PS
2
4. C(q) = 50 + 0.5q + 0.08q
a. If P = 8.50, set MC = P
C(q)
MC = 0.50.16 q
q
0.5 + 0.16q = 8.50
0.16q = 8
q = 50
π = TR(q) – TC(q)
2
= P(q) × q – [50 + 0.5q + 0.08q ] 2
= (8.50)(50) – [50 + 0.5(50) + 0.08(50)]
= 425 – 275
$150 =
b. Firms will enter the market in the long run because existing firms are currently earning
positive profits. In the LR, firms will continue to enter the market until each firm will
earn zero profits. c. In the long run, firms will produce where the MC curve intersects the long-run average
cost curve at it’s minimum: so set MC = AC
2
AC = TC/q = (50 + 0.5q + 0.08q )/q = 50/q + 0.5 + 0.08q
MC = 0.5 + 0.16q
AC = MC
50/q + 0.5 + 0.08q = 0.5 + 0.16q
(50/q) = 0.5 – 0.5 + 0.16q – 0.08q

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