The quantity produced (in thousands) and the average cost (in dollars) of produc

Question

The quantity produced (in thousands) and the average cost (in dollars) of producing a toy at different plants are shown here.

Plant - Average Cost - Quantity - (Quantity)^2

1 $0.75 100 10,000

2 0.40 200 40,000

3 0.50 140 19,600

4 0.60 260 67,600

5 0.45 160 25,600

6 0.55 120 14,400

7 0.70 280 78,400

8 0.45 180 32,400

9 0.40 220 48,400

10 0.45 240 57,600

Use regression analysis to estimate average cost as al inear function of quantity and quantity squared (e.g., the quantity and quantitty squares data for plant 1 would be 100 and 10,000, respectively). Determine the equation, t-statistics, and coefficient of determination. At what quantity is average cost a minimum?

Explanation / Answer

Below are regression estimates:

The estimated equation is:
AC = 1.830152 -0.014920*Q + 0.000039*Q^2

To find the minimum average cost, the first order condition will be -

dAC/dQ = 0
-0.014920 + 2*0.000039*Q = 0
Q_min = 2* 0.000039/0.014920 = 0.005227882 i.e approx 0.

d2AC/dQ2= 2*0.000039 > 0 which means indeed AC is minimized at above stationary point.

Dependent var: Average Cost Coefficients Standard Error t Stat P-value Intercept 1.830152 0.1260 14.5193 0.0000 Q -0.014920 0.0014 -10.5919 0.0000 Q2 0.000039 0.0000 10.5660 0.0000

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