Managerial Economics Assignment

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We tabulate production data as follows:

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By substituting the respective values of independent variable into the estimated quadratic production function:

Q = 997.728155 + 0.026213K – 3.022041L +0.002814L2 – 0.000013LK

The equation indicates that:

- Every 1%increase in capital will increase production by 0.026213 tons.

- Every 1% increase in labor will reduce production hours by 3.02241 hours.

- Is the estimated production function “good”? Why or why not?

Standard error of estimation is 71.47. The prediction error is too large.

Using t-test at the k = 5% level of significant with degree of freedom 12, t.025,12 = 2.179.

Because each of independent variables t-ratiosnot less than -2.160 or greater than

+2.160. Therefore, all of the coefficients are statistically in significant at the 5% level ofsignificance.

Correlation coefficient, r = 0.9643

The regression model has 96.43% degree of association between variables.

Coefficient of determination, r2 = 0.9299

The regression model, explain approximately 92.99% of the variation in the sample.

Use F-test to test the hypothesis that all the regression coefficients are zero.

The decision is to reject the null hypothesis of no relationship between X and Y at the k=5% level of significance if F-ratio is greater than the F0.05,1,13 = 4.67

From the summary output, F-ratio = 33.1907 is greater than critical value of 4.67.

Therefore, we reject the null hypothesis that there is no relationship between X and Y at the 5% level of significance. We can conclude that the regression model does explain a significant proportion of the variation in the sample.

Conclusion:

The regression model is good but has a large prediction error.

b. Cobb-Douglas Production Function

- Estimate the Cobb-Douglas production function Q = αLβ1Kβ2, where Q = output; L = labor input; K = capital input; and α, β1, and β2 are the parameters to be estimated.

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Transform Cobb-Douglas production function Q= αLβ1Kβ2 into logarithm turn equation into the following equation:

Ln(Y) = α + β1ln(L) + β2ln(K)

α = - 4.7547; β1= 1.0780 and β2 = 0.4152

Y = - 4.7547 + 1.0780ln(L) + 0.4152ln(K)

The antilog is: Q = 0.0086L1.0780K0.4152

- Use t-test at the 5 percent level ofsignificance with degree of freedom 12,

t.025,12 = 2.179.From the summary output, each independent variable t-ratios exceed 2.179. Therefore,both of the coefficients are statistically significant at the 5% level of significant.

- The percentage of the variation in output that is explained by the capital (K) and labor

(L) variables is R2 = 94.81%

- EK = β1 = 0.4152;1% increase in K yields a 0.4152% increase in Q

EL = β2 = 1.0780;1% increase in L yields a 1.0780% increase in Q

- β1 + β2= 1.0780 + 0.4152 = 1.4932

Since the sum of the exponent of the L and K is greater than 1, the production function exhibits increasing return to scale.

PART 3

a. Impact of price cut of model A from $30 to $27.

i). ED = ΔQ/ΔPA

= {(Q2 – Q1)/ [(Q2 + Q1)/2]}/{(P2 – P1)/[(P2 + P1)/2]}

= [(Q2 – Q1)/ (Q2 + Q1)] × [(P2 + P1)/ (P2 – P1)]

–2.5 = [(Q2 – 15,000)/ (Q2 + 15,000)] × [(27 + 30)/ (27 – 30)]

Q2 =19,545 units, therefore model A new total revenue,

TR2 = P2xQ2 =27x19,545 = $527,715, and

ΔTR = $527,715 – $450,000 = +$77,715

ii). Margin per unit, M2 = P2 – Variable Cost/unit = $27 – $15 = $12

Contributing Margin, CM2 = M2xQ2 = 12 x 19,545 = $234,540

ΔCM = $234,540 – $225,000 = +$9,540

The change in the contribution margin is positive,price cut for model A would be good.

b. Impact of price cut of model A from $30 to $27 to other model.

i). EAxB= 0.5 = ΔQB/ΔPA

= [(QB2– Q B1)/(QB2 + Q B1)] × [(PA2 + PA1)/(PA2 – PA1)]

= [(QB2 – 5,000)/ (QB2 + 5,000)] × [(27 + 30)/(27 – 30)] = 0.5

QB2 = 4,744 units

EAxC = 0.2 = ΔQC/ΔPA

= [(QC2 - QC1)/(QC2 + QC1)] × [(PA2 + PA1)/(PA2 – PA1)]

= [(QC2 – 10,000)/ (QC2 + 10,000)] × [(27 + 30)/(27 – 30)] = 0.2

QC2 = 9,792 units

Total Revenue:

A: 27 x 19,545 = $527,715

B: 35 x 4,744 = $166,040

C: 45 x 9,792 = $440,640

Total Revenue = $1,134,395

ΔR = $1,134,395–$1,075,000=+$59,395.