1)The chief economist for Argus Corporation, a large appliance mfg., estimated the firm’s short-run cost function for vacuum cleaners using an average variable cost function of the form:
3
AVC = a + bQ + cQ
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 RSquare:0.7360 F-Ratio: 39.428 P-Value on F: 0.0001
Observations: 19
Variable Parameter Standard T
Estimate Error
InterCEPT 191.93 54.65
Q -o.o305 o.o0789
Q2 0.0000024 0.00000089
T-Ratio PValue
Intercept 3.512 0.0029
Q 23.866 0.0014
Q2 2.499 0.0262
Degrees of Freedom: 17 (19-3 obs = 17)
Significance level at 2%: 2.567
^ ^ ^
a)Are the estimates a, b, c statistically significant at
the 2% level of significance?
ANSWER: No, c falls under the 2.567 level (Is this right?) (I haven’t observed one like this, what does this mean if c is not at a significant level?)
b)Do the results indicate that the average variable cost is U-shaped? How do you know?
ANSWER: yes, because a and c are positive, b is negative, this is indicative of a U-shaped curve.
c) If Argus Corporation produces 8,000 vacuum cleaners, what is the estimated average cost variable cost?
ANSWER: 2
AVC = AVC= a + bQ + cQ
AVC = 191.93 + - 0.0305(8000) + 0.0000024(8000)
AVC = 191.93 + (-244) + 0.0000024(640000000)
AVC = - 52.47 + 153.60
AVC = 101.13
What is the marginal cost?ANSWER: 2
MC = a + 2bQ + 3cQ 2
MC = 191.93 + 2(-0.0305)(8000) + 3(0.0000024)(8000)
MC = 191.93 + 2(- 244) + 3(0.0000024)(64000000)
MC = 191.93 + (- 488) + 460.8
MC = 652.73 + (- 488)
MC = 164.73
What is the total variable cost?ANSWER:
2 3
TVC = aQ + bQ + cQ
TVC = 191.93(8000) + - 0.0305(64000000) + 0.0000024(512000000000)
TVC = 1535440 + (- 1952000) +512000000000
TVC = 5121535440 + (-1952000)
TVC = 5119583440
There is another formula here I am unsure of:
TVC = AVC x Q
TVC = 101.13 x 8000
TVC = 809040
What is the total cost?
ANSWER: Qm = -b/2c
Qm = - 0.0305/2(0.000024)
Qm = - 6354.166667
d) Answer part c, assuming Argus produces 10,000 vacuum cleaners monthly?
ANSWER:
Same as above as long as my formulas are Correct.
e)At what level of output will the averagevariable cost be at a minimum?
Whatever the lowest of the two is (8 and 10K Produced per month).
What is the minimum average variable cost?
Whatever the lowest AVC is?
Do these seem correct?!?!?!
Thanks,
EY
a) You are testing to see if the parameter estimate is significantly different from zero; the null hypothesis is that the estimate is zero. So, divided the parameter estimate by the Standard T and compare that to your significance level cutoff (2.567 in your example); if less than the cutoff, then the parameter is not significantly different from zero. So for the b parameter 0.0305/.00789 = 3.86. Ergo, the parameter is significant. Repeat for the other parameters.
b) I agree.
c-AVC) I agree with your methodology, except I get 101.53
c-MC) I agree
c-TVC) I agree with your methodology, you made a silly math error. Rework and you should get 812240.
d) I agree.
e) no no no. Use calculus. Take the first derivitive of the AVC function, set that equal to zero and solve for Q. (Hint: I get Q=6354)
a) To determine if the estimates a, b, c are statistically significant at the 2% level of significance, you can use the t-ratio and p-value for each parameter. The t-ratio is the estimate divided by the standard error, and the p-value represents the probability of obtaining a t-ratio as extreme or more extreme than the observed t-ratio if the true parameter were zero.
For example, for parameter b:
t-ratio = -0.0305 / 0.00789 = -3.86
p-value = 0.0014
Comparing the t-ratio to the significance level cutoff of 2.567, which is obtained from the t-distribution table for a 2% level of significance with 17 degrees of freedom, you can conclude that parameter b is statistically significant at the 2% level. Repeat the same process for parameters a and c to determine their significance.
b) The results indicate that the average variable cost is U-shaped because parameter a and c are positive while parameter b is negative. A U-shaped curve implies that the average variable cost initially decreases, reaches a minimum point, and then increases.
c) To find the estimated average variable cost when Argus Corporation produces 8,000 vacuum cleaners, substitute the value of Q into the average variable cost function:
AVC = 191.93 - 0.0305(8000) + 0.0000024(8000^2)
AVC = 101.13
Therefore, the estimated average variable cost is $101.13.
To calculate the marginal cost, use the marginal cost function:
MC = 191.93 + 2(-0.0305)(8000) + 3(0.0000024)(8000^2)
MC = 164.73
Hence, the estimated marginal cost is $164.73.
To obtain the total variable cost, utilize the total variable cost function:
TVC = 191.93(8000) - 0.0305(8000^2) + 0.0000024(8000^3)
TVC = 5119583440
Therefore, the total variable cost is $5,119,583,440.
Another approach to find the total variable cost is to multiply the average variable cost by the quantity produced:
TVC = AVC x Q
TVC = 101.13 x 8000
TVC = 809,040
Hence, the total variable cost is $809,040.
To determine the level of output at which the average variable cost is at a minimum, you need to take the derivative of the average variable cost function with respect to Q and set it equal to zero. However, this calculation requires the specific values of a, b, and c, which are not provided. Therefore, it is not possible to determine the level of output at which the average variable cost is at a minimum or the minimum average variable cost in this scenario. Calculus is necessary for this analysis.