Question

N-1 N=2 N-4 N-16 Table 3: Question 4, Part2 Question 4: The Aggregate Production Function (30 Marks) This question focuses on labour productivity, labour demand, and generally on the production function. Assume that the Aggregate Production Function is represented by the following equation: Y stands for output, K stands for the capital stock, N stands for the number of people employed, L stands for the quantity of land used in production, and A stands for a measure of labour efficiency a, B, and γ are parameters whose values are between 0 and 1. Part 1) Derive a formula for - the marginal product of capital (MPK) the marginal product of labour (MPN) - the marginal product of land (MPD). Part 2) N,for N = 1, 2, . . . , 15, 16. α -γ Find the level of employment(N) in the economy, what happens to the level of employment - Using a spreadsheet, fix K-: 9 and A L 1 . Consider a-β-γ and plot the MPN vs. - Suppose the real wage paid to labour is w -1. Again, fix K-9 and A-L-1 and consider (N) if labour becomes more productive, for example if A-47 Use calculus to show that (given K, L and A) the MPN declines as the level of employment (N) " Using a spreadsheet, fix A-4 and L-1. Consider α 0.65, β-0.35, and γ-1 and compute total output for all the pairs shown in Table 3. Report the values in a similar table.

Answer 1

Table 1

N	MPN
0	0
1	2
2	1.414214
3	1.154701
4	1
5	0.894427
6	0.816497
7	0.755929
8	0.707107
9	0.666667
10	0.632456
11	0.603023
12	0.57735
13	0.5547
14	0.534522
15	0.516398
16	0.5

Figure 1

With A=4 and L=1, α=0.65, β=0.35, γ=1, the production function is

$Y=\left (4N \right )^{0.35}\left (K^{0.65}+1 \right )$

Using this function the spreadsheet is calculated for different combinations of K and N below

	K=0	1	2	4	16
L=0	0	0	0	0	0
1	1.624505	3.24901	4.173626	5.624505	11.47366
2	2.07053	4.14106	5.319539	7.168772	14.62388
4	2.639016	5.278032	6.780076	9.137035	18.63902
16	4.287094	8.574188	11.01427	14.84316	30.27917