2. Straker Industries estimated its short-run costs using a U-shaped average var
ID: 1111541 • Letter: 2
Question
2. Straker Industries estimated its short-run costs using a U-shaped average variable cost function of the form AVC= a + bQ + cQ2 and obtained the following results. Total fixed cost (TFC) at Straker Industries is $1,030.
Adjusted R Square
0.815
Coefficients
Standard Error
t Stat
P-value
Intercept
34.62
1.99
17.40
0.0033
Q
-3.65
0.44
-8.36
0.0000
Q^2
0.18
0.02
8.92
0.0000
a. What level of output (Q) is associated with the minimum AVC? What is the value of AVC at this minimum?
b. Determine equations for ATC, TC, and SMC. Graph one scatterplot of Q vs. TC, and another scatterplot of Q vs. ATC, AVC, and SMC.
c. When output is 14, how much is TC, AVC, ATC, and SMC?
d. At what amount of output does labor change from exhibiting increasing returns to decreasing returns?
Adjusted R Square
0.815
Coefficients
Standard Error
t Stat
P-value
Intercept
34.62
1.99
17.40
0.0033
Q
-3.65
0.44
-8.36
0.0000
Q^2
0.18
0.02
8.92
0.0000
Explanation / Answer
Estimated equation of Total cost (TC) = TFC + (Q x AVC) = 1,032 + [Q x (34.62 - 3.65Q + 0.18Q2)]
TC ($) = 1,032 + 34.62Q - 3.65Q2 + 0.18Q3
(a)
AVC = 34.62 - 3.65Q + 0.18Q2
AVC is minimum when dAVC / dQ = 0
- 3.65 + 0.36Q = 0
0.36Q = 3.65
Q = 10.14
AVC = 34.62 - (3.65 x 10.14) + (0.18 x 10.14 x 10.14) = 34.62 - 37.01 + 18.51 = 16.12
(b)
TC = 1,032 + 34.62Q - 3.65Q2 + 0.18Q3
ATC = TC / Q = (1,032 / Q) + 34.62 - 3.65Q + 0.18Q2
SMC = dTC / dQ = 34.62 - 7.3Q + 0.54Q2
Data table for graphs:
Graph:
(c) When Q = 14, from the data table,
TC = $1,295.2
AVC = $18.8
ATC = $92.51
SMC = $38.26
(d) During transition from increasing returns to scale to decreasing returns to scale, ATC is minimum at point of transition.
ATC is minimum when dATC / dQ = 0
(- 1,032 / Q2) - 3.65 + 0.36Q = 0
- 1,032 - 3.65Q2 + 0.36Q3 = 0
0.36Q3 - 3.65Q2 - 1,032 = 0
Solving this polynomial equation using online solver tool,
Q = 18.51 (Other two roots are imaginary, hence dismissed)
Q TC ATC AVC SMC 1 1063.15 1063.15 31.15 27.86 2 1088.08 544.04 28.04 22.18 3 1107.87 369.29 25.29 17.58 4 1123.60 280.90 22.90 14.06 5 1136.35 227.27 20.87 11.62 6 1147.20 191.20 19.20 10.26 7 1157.23 165.32 17.89 9.98 8 1167.52 145.94 16.94 10.78 9 1179.15 131.02 16.35 12.66 10 1193.20 119.32 16.12 15.62 11 1210.75 110.07 16.25 19.66 12 1232.88 102.74 16.74 24.78 13 1260.67 96.97 17.59 30.98 14 1295.20 92.51 18.80 38.26 15 1337.55 89.17 20.37 46.62Related Questions
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