Smelling of Tulips, Inc., a perfume company, estimated its short-run costs using a U-shaped average variable...

Smelling of Tulips, Inc., a perfume company, estimated its short-run costs using a U-shaped average variable cost function of the form and obtained the following results. Total fixed cost (TFC) at S.T. Inc. is $1,250.

Adjusted R Square	0.758
	Coefficients	Standard Error	t Stat	P-value
Intercept	32.52	2.33	13.95	0.0008
Q	-1.39	0.51	-2.72	0.0146
Q^2	0.10	0.02	4.23	0.0006

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 MC. Graph one scatterplot of Q vs. TC, and another scatterplot of Q vs. ATC, AVC, and MC.

c. When output is 9, how much is TC, AVC, ATC, and MC?

d. At what amount of output does labor change from exhibiting increasing returns to decreasing returns?

Solutions

Expert Solution

a) The regression equation for AVC is
AVC = 32.52 - 1.39Q + 0.10Q2

To obtain the minimum AVC, we differentiate the AVC wrt Q
dAVC/dQ = -1.39 + 0.20Q = 0
=> Q* = 6.95 or Q* 7

b) We know TFC = $1,250. Therefore, AFC is 1250/Q.

ATC = Average Fixed Cost (AFC) + Average Variable Cost (AVC)
ATC = 1250/Q + 32.52 - 1.39Q + 0.10Q2

TC = TFC + Total Variable Cost (TVC)
TC = 1250 + AVC*Q
TC = 1250 + 32.52Q - 1.39Q2 + 0.10Q3

MC = dTC/dQ = 32.52 - 2.78Q + 0.30Q2

The y-axis scale (Cost) in the graph is logarithmic

c) When Q = 9
TC = 1502.99; ATC = 166.99; AVC = 28.11 and MC - 31.8.
You can check from the Excel sheet above.

d) The production of S.T. Inc. changes from increasing returns to scale to decreasing returns to scale where the MC curve changes its slope. MC is a U-shaped curve.

Q	MC	Change in MC
1	30.04
2	28.16	-1.88
3	26.88	-1.28
4	26.2	-0.68
5	26.12	-0.08
6	26.64	0.52
7	27.76	1.12
8	29.48	1.72
9	31.8	2.32
10	34.72	2.92

When the marginal cost decreases, there is increasing returns to scale. When the marginal cost increases, there is decreasing returns to scale. From the table above, we see that the change occurs at output, Q = 6.

Therefore, at output level, Q = 6, the production of S.T. Inc with labor input changes from from incresaing returns to scale to decreasing returns to scale.

Rahul Sunny answered 2 weeks ago

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