The chief economist for Argus Corporation, a large appliance manufacturer, estimated the firm’s short- run cost function for vacuum cleaners using an average variable cost function of the form
AVC = a + bQ + cQ2
where AVC dollars per vacuum cleaner and Q number of vacuum cleaners produced each month. Total fixed cost each month is $180,000. The following results were obtained:
DEPENDENT VARIABLE:
AVC R-SQUARE 0.7360 F-RATIO 39.428 P-VALUE ON F 0.0001
OBSERVATIONS:19 0
pARAMETER STANDARD
VARIABLE ESTimate error
INTERCEPT 191.93 54.65
-0.0305 0.00789
0.0000024 0.00000098
T-RATIO P-VALUE
3.512 0.0029
23.866 0.0014
2.449 0.0262
a. Are the parameter estimates statistically significant at the 2 percent level of significance?
b. Do the results indicate that the average variable cost curve is U-shaped? How do you know?
c. If Argus Corporation produces 8,000 vacuum cleaners per month, what is the estimated average variable cost? Marginal cost? Total variable cost? Total cost?
d. Answer part c, assuming that Argus produces 10,000 vacuum cleaners monthly.
e. At what level of output will average variable cost be at a minimum? What is minimum average variable cost?
a. Well, let's take a look at those p-values. The p-values for the parameter estimates are all less than 0.02, which means they are statistically significant at the 2 percent level of significance. So, yes, the parameter estimates are statistically significant.
b. Ah, the U-shaped average variable cost curve question. The coefficient on the Q-squared term, which is c in the equation, is positive and statistically significant. This indicates that the average variable cost curve is indeed U-shaped. Oh, the joys of U-shaped curves!
c. Alrighty then! Let's plug in Q = 8,000 into the average variable cost function and calculate away. AVC = a + bQ + cQ^2 = 191.93 - 0.0305 * 8,000 + 0.0000024 * (8,000)^2. That gives us an estimated average variable cost of... *drumroll please*... $39,320! As for the marginal cost, it is just the derivative of the average variable cost function with respect to Q, so take the derivative and evaluate it at Q = 8,000. And don't forget about the total variable cost, which is just AVC times Q, and the total cost, which is the sum of the total fixed cost and the total variable cost.
d. Alright, let's switch gears and use Q = 10,000 this time. Plug that into the average variable cost function and we get AVC = 191.93 - 0.0305 * 10,000 + 0.0000024 * (10,000)^2 = $75,680. The marginal cost, total variable cost, and total cost can be calculated just like in part c.
e. Ah, the minimum average variable cost quest! To find the level of output at which the average variable cost is at a minimum, we need to find the vertex of the U-shaped curve. The vertex occurs at Q = -b / (2c), so plug in the values for b and c from the parameter estimates and we'll get the level of output. And the minimum average variable cost is just the value of the average variable cost at that level of output. So, put a smile on our face and calculate away!
a. To determine if the parameter estimates are statistically significant at the 2 percent level of significance, we need to compare the p-values of the estimates to the significance level.
For the intercept, the p-value is 0.0029, which is less than 0.02 (2 percent level of significance). Therefore, the intercept is statistically significant at the 2 percent level.
For the coefficient estimates of bQ and cQ^2, their respective p-values are 0.0014 and 0.0262. Both of these values are less than 0.02, so the coefficient estimates are also statistically significant at the 2 percent level.
b. The results indicate that the average variable cost curve is U-shaped because the coefficient estimate for the cQ^2 term is positive (0.0000024) and statistically significant. A positive coefficient for the quadratic term indicates a U-shaped curve.
c. To find the estimated average variable cost, we substitute the given values into the average variable cost function.
AVC = a + bQ + cQ^2
Substituting a = 191.93, b = -0.0305, c = 0.0000024, and Q = 8,000:
AVC = 191.93 - 0.0305(8,000) + 0.0000024(8,000)^2
AVC = 191.93 - 244 + 0.0000024(64,000,000)
AVC ≈ $102.49 (estimated average variable cost)
To find the marginal cost, we take the derivative of the average variable cost function with respect to Q:
MC = d(AVC)/dQ = b + 2cQ
Substituting the values b = -0.0305, c = 0.0000024, and Q = 8,000:
MC = -0.0305 + 2(0.0000024)(8,000)
MC = -0.0305 + 0.0384
MC ≈ $0.0079 (estimated marginal cost)
To find the total variable cost, we multiply the average variable cost by the quantity produced:
Total Variable Cost = AVC × Q
Total Variable Cost = $102.49 × 8,000
Total Variable Cost ≈ $819,920
To find the total cost, we sum the total variable cost and the total fixed cost:
Total Cost = Total Variable Cost + Total Fixed Cost
Total Cost = $819,920 + $180,000
Total Cost ≈ $999,920
d. Using the same process as in part c, we can find the estimated average variable cost, marginal cost, total variable cost, and total cost for Q = 10,000.
Estimated Average Variable Cost ≈ $89.99
Marginal Cost ≈ $0.0091
Total Variable Cost ≈ $899,920
Total Cost ≈ $1,079,920
e. To find the level of output at which average variable cost is at a minimum, we take the derivative of the average variable cost function with respect to Q and set it equal to zero:
d(AVC)/dQ = b + 2cQ = 0
Solving for Q:
-0.0305 + 2(0.0000024)Q = 0
-0.0305 = -2(0.0000024)Q
Q ≈ 6,375.42
Therefore, the level of output at which average variable cost is at a minimum is approximately 6,375 vacuum cleaners per month.
The minimum average variable cost can be found by plugging this value of Q back into the average variable cost function:
AVC = 191.93 - 0.0305(6,375) + 0.0000024(6,375)^2
AVC ≈ $87.17 (minimum average variable cost)
a. To determine if the parameter estimates are statistically significant at the 2 percent level of significance, we need to check the p-values associated with each parameter estimate. A p-value less than 0.02 would indicate statistical significance at the 2 percent level.
b. To determine if the average variable cost curve is U-shaped, we can examine the coefficient of the quadratic term (cQ2). If the coefficient is positive, it suggests a U-shaped curve. If it is negative, it suggests an inverted U-shaped curve.
c. To calculate the estimated average variable cost when Argus Corporation produces 8,000 vacuum cleaners per month, we can substitute Q = 8,000 into the average variable cost function (AVC = a + bQ + cQ2) using the parameter estimates obtained from the regression analysis. The estimated average variable cost is the value of AVC at Q = 8,000.
To calculate the estimated marginal cost, we need the derivative of the average variable cost function with respect to Q. Differentiating AVC with respect to Q gives us the expression for the marginal cost function (MC). To calculate MC at Q = 8,000, substitute Q = 8,000 into the marginal cost function using the parameter estimates obtained from the regression analysis.
Total variable cost is the product of average variable cost (AVC) and the quantity produced (Q). To calculate the estimated total variable cost when Argus Corporation produces 8,000 vacuum cleaners per month, multiply the estimated average variable cost (AVC at Q = 8,000) by 8,000.
Total cost is the sum of total fixed cost and total variable cost. Since the fixed cost is given as $180,000 per month, to calculate the estimated total cost when Argus Corporation produces 8,000 vacuum cleaners per month, add the estimated total variable cost (TVQ at Q = 8,000) to $180,000.
d. Repeat the same calculations as in part c, but substitute Q = 10,000 instead of Q = 8,000.
e. The level of output at which average variable cost is at a minimum can be determined by finding the value of Q that corresponds to the vertex of the quadratic function, which represents the average variable cost curve in this case. The minimum average variable cost is the value of AVC at the level of output (Q) that minimizes the average variable cost function.