Question

1.You collect the following production data for your firm 241 L.189 459 64 L.102 1.514 755 4 1,117 L550 10 Question a Which do vou consider to be the best trendline for this data? Select one a. A graph showing a trendline connecting the L,Q dots from the previous question; L is measured on the horizontal axis and Q on the vertical axis. The estimated trendline is y-25.758x2+450.09x-519.01 b. A graph showing a trendline connecting the LQ dots from the previous question; L is measured on the horizontal axis and Q on the vertical axis. The estimated trendline is y-4.9908x3+63.101x2+14.309x. c. A graph showing a trendline connecting the LQ dots from the previous question; L is measured on the horizontal axis and Q on the vertical axis. The estimated trendline is y-4.8793x3+60.936x2+26.524x-18.613. d. A graph showing a trendline connecting the LQ dots from the previous question; L is measured on the horizontal axis and Q on the vertical axis, The estimated trendline is y-137.77x+153.91 Question b Use the regression results below R Square: 0.9976 Adjusted R Square: 0.990 Error t Stat P-value answer the following two questions: Intercept 0 N/A N/A N/A L 14.309 L2 63.101 8.350 7.557 0.000 890 0,479 0,647 L3 -4.991 0.545 -9.166 0.000 Question Using these results, calculate the Q, AP, and MP for L=8 workers l483. AP= 185 4: MP-51.4 b. Q-2,280; AP-214.5: MP-83.2 c. Q 1,598; AP=199.7; MP- 65.7. d. Q1,555; AP=193.4; MP - 61,1, Question d At 8 workers, is SMC rising falling, and how you know? Select one orkers, SMC is rising since MP is falling. b. At 8 workers, SMC is falling since MP is falling c. At 8 workers, SMC is falling since MP is rising d. At 8 workers, SMC is rising since MP is rising

Answer 1

(a)

Q	L	Trendline a y=25.758x2+450.09x-519.01	Trendline b y=4.9908x3+63.101x2+14.309x	Trendline c y=4.8793x3+60.936x2+26.524x-18.613	Trendline d y=137.77x+153.91	Error (trendline a)	Error (trendline b)	Error (trendline c)	Error (trendline d)
241	2	484.202	320.9484	317.2134	429.45	-243.202	-79.9484	-76.2134	-188.45
1189	6	3108.818	3435.503	3388.156	980.53	-1919.82	-2246.5	-2199.16	208.47
459	3	1063.082	745.5876	741.1241	567.22	-604.082	-286.588	-282.124	-108.22
64	1	-43.162	82.4008	73.7263	291.68	107.162	-18.4008	-9.7263	-227.68
1539	9	5618.198	8878.255	8712.929	1393.84	-4079.2	-7339.26	-7173.93	145.16
1102	5	2375.39	2272.92	2247.32	842.76	-1273.39	-1170.92	-1145.32	259.24
1514	7	3893.762	4903.956	4826.519	1118.3	-2379.76	-3389.96	-3312.52	395.7
755	4	1693.478	1386.263	1374.734	704.99	-938.478	-631.263	-619.734	50.01
117	11	7548.698	14435.37	14140.76	1669.38	-7431.7	-14318.4	-14023.8	-1552.38
1550	10	6557.69	11443.99	11219.53	1531.61	-5007.69	-9893.99	-9669.53	18.39
					Total Error	-23770.2	-39375.2	-38512	-999.76

Above table provides the predicted values using trendlines in option a, b, c and d along with their errors (the difference between actual Q and predicted Q based on the formulae). The sum of these errors is minimum for d which means that trendline d has the best fit.

Note: If the sum of errors were the same for two or more trendlines, one with the minimum sum of square of errors would have the best fit.

(c) The correct answer is c

Based on the information available in (b)

Q=14.309 L +63.101 L² – 4.991 L³

If L=8, then Q= 1597.544 [putting L=8 in above formula]

Average Productivity of Labour, AP=Q/L= 199.693 and Marginal Productivity of Labour, MP= dQ/dL=14.309+126.202 L - 14.973 L² = 65.653

(d) Short-run marginal cost (SMC) increases when the marginal productivity falls (we shall assume that the cost of one unit of labour remains the same). Falling marginal productivity (MP) means that more labour is required to produce the same quantity of additional good. To check whether MP is falling or rising at L=8, we check the maxima of the MP function. If the maxima is before L=8 then MP is falling. If the maxima is after L=8, then MP is rising at L=8. If maxima is at L=8, then it is neither falling nor rising.

Based on the previous question, we know that MP= 14.309+126.202 L - 14.973 L²

Where dMP/dL= 126.202 - 29.946 L

Solving the above function when dMP/dL= 126.202 - 29.946L= 0 (the first order condition for maxima), we will get L=4.214319

Since the second order derivative of MP is negative, MP is maximum at L=4.214319

Since maxima is at L=4.214319, MP is falling at L=8. It means that SMC is rising at L=8.

The correct answer is a