Question

Suppose life expectancy in years (L) is a function of two inputs, health expenditures (H) and nutrition expenditures (N) in hundreds of dollars per year. The production function is LE CH0.40 0.60 Beginning with c= 1, a health input of $300 per year (H = 3) and a nutrition input of $500 per year (N = 5), show that the marginal product of health expenditures and the marginal product of nutrition expenditures are both decreasing. Life expectancy (L) , The marginal product of health expenditures when H goes from 3 to 4 is and the marginal product of health when H goes from 5 to 6 is . (Round your answers to three decimal places.) The marginal product of nutrition expenditures when N goes from 5 to 6 is | , and the marginal product of nutrition when N goes from 7 to 8 is . (Round your answers to three decimal places.) Health expenditures (H)

Answer 1

Solution:

We are given the required function as follows: L = c*H^0.4*N^0.6

With c = 1, H = 3, and N = 5, we have life expectancy as:

L = 1*3^0.4*5^0.6 = 4.076

On substituting for c = 1, we have L = H^0.4*N^0.6

Now, marginal product of health expenditure = dL/dH

MPH = 0.4*H^0.4-1*N^0.6 = 0.4*(N/H)^0.6, so clearly in this case, if we increase health expenditure (with everything else constant), MPH will fall. In other words, MPH is decreasing

Similarly, marginal product of nutrition = dL/dN

MPN = 0.6*H^0.4*N^0.6-1 = 0.6*(H/N)^0.4

Again if we increase nutrition, MPN will fall. In other MPN is decreasing in N.

Marginal product is the amount by which total product change, when any input (here, either health expenditure (H) or nutrition (N)) increases by 1. Then,

1. With N = 5, if H goes from

i) 3 to 4:

L(H = 3) = 1*3^0.4*5^0.6 = 4.076

L(H = 4) = 1*4^0.4*5^0.6 = 4.573

Thus, marginal product of health expenditures when H goes from 3 to 4 is (4.573 - 4.076 =) 0.497

ii) 5 to 6:

L(H = 5) = 1*5^0.4*5^0.6 = 5

L(H = 6) = 1*6^0.4*5^0.6 = 5.377

Thus, marginal product of health expenditures when H goes from 5 to 6 is (5.377 - 5 =) 0.377

(Note how the marginal product has decreased with increase in factor health expenditure)

2. With H = 3, if N goes from

i) 5 to 6:

L(N = 5) = 1*3^0.4*5^0.6 = 4.076

L(N = 6) = 1*3^0.4*6^0.6 = 4.547

Thus, marginal product of health expenditures when N goes from 5 to 6 is (4.547 - 4.076 =) 0.547

ii) 7 to 8:

L(N = 7) = 1*3^0.4*7^0.6 = 4.988

L(N = 8) = 1*3^0.4*8^0.6 = 5.404

Thus, marginal product of health expenditures when N goes from 7 to 8 is (5.404 - 4.988. =) 0.416