Question

Part 1: Graphing Production and Variable Cost Functions 1) Complete this part on the worksheet named Cost Functions. 2) I started a table for you. All the column headings have the same definitions as they do in our notes. IL CAP MPVC 1 0 0 -- -- 2 3) We will use a cubic production function such that: q=xL + yL² + z In this case statistical estimation of the firm's production function reveals the values of the coefficients to be: x= 1, y = 35, and z=-1. Use this information to fill in the values of q, AP, and MPin the table from L=0 to L = 26. Use formulas in Excel to do the calculations. For MP, calculate exact MP. In other words, do not simply take the change in q divided by change in L. Find a function for MP and use that. 4) Build a graph of q. I should be on the horizontal axis and q (output) on the vertical. Use a scatter graph with smooth lines. Clearly label both axes. This is a production function. 5) Build a graph of AP and MP. Per-unit measures should be on the vertical axis and L on the horizontal. Use a scatter graph with smooth lines. Clearly label all lines and both axes.

Answer 1

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q = xL + yL2 + zL3 where, x = 1, y = 35, z = -1

q = L + 35L2 - L3

Marginal product = d(q)/d(L) = d(l + 35L2 - L3)/d(L) = 1+ 70L -3L2

Marginal product function = 70L - 3L2 +1

Average Product = q/L = (L + 35L2 - L3)/L = 1 + 35L - L2

Average Product function = 1 + 35L - L2

labour	output	marginal product	Average Product
0	0	-	-
2	134	67	67
4	500	183	125
6	1050	275	175
8	1736	343	217
10	2510	387	251
12	3324	407	277
14	4130	403	295
16	4880	375	305
18	5526	323	307
20	6020	247	301
22	6314	147	287
24	6360	23	265
26	6110	-125	235

4). Production Function

The Production Function 7000 6000 5000 Output o 5 10 20 15 Labour 25 30

5). Average Product (AP) and Marginal Product (MP)

AP and MP AP & MP 10 15 20 25 Labour marginal product - Average Product -

6). Variable Cost (VC)

wage (w) = 12

Total Variable Cost = w*L

labour	Variable Cost
0	-
2	804
4	1500
6	2100
8	2604
10	3012
12	3324
14	3540
16	3660
18	3684
20	3612
22	3444
24	3180
26	2820

7). Variable Cost Graph

Variable Cost Variable Cost 1 5 10 20 25 30 15 Labour

8). Stage of Production

labour	output	marginal product	stage
0	0		stage 1
1	35	35
2	134	99
3	291	157
4	500	209
5	755	255
6	1050	295
7	1379	329
8	1736	357
9	2115	379
10	2510	395
11	2915	405
12	3324	409
13	3731	407	stage 2
14	4130	399
15	4515	385
16	4880	365
17	5219	339
18	5526	307
19	5795	269
20	6020	225
21	6195	175
22	6314	119
23	6371	57
24	6360	-11	Stage 3
25	6275	-85
26	6110	-165

Stage of production

Stage 1 - Increasing returns - where output increases with the increasing rate it means Marginal product (MP) positive and increases with additional labour. in the above table, up to 12 unit of labour MP increase with increasing it is stage 1

stage 2 - Decreasing returns  - where output increase with the decreasing rate it means Marginal product (MP) positive but decreases as additional labour employed. in the above table, from 13 to 23 unit of labour, the marginal product is positive but decrease as labour employed.

stage 3 Negative returns - where output starts to decline as more labour employed and the marginal product becomes negative. in the above table, after 23 units of labour, the marginal product becomes negative and total output starts to decline. it is stage 3