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

L = c H0.8N0.2.

a. Beginning with a health input of \(400 per year (H = 4) and a nutrition input of \)4900 per year (N = 49), show that the marginal product of health expenditures and the marginal product of nutrition expenditures are both decreasing.

b. Does this production function exhibit increasing, decreasing, or constant returns to scale?

c. Suppose that in a country suffering from famine, N is fixed at 2 and that c = 20. Plot the production function for life expectancy as a function of health expenditures, with L on the vertical axis and H on the horizontal axis.

d. Now suppose another nation provides food aid to the country suffering from famine so that N increases to 4. Plot the new production function.

e. Now suppose that N = 4 and H = 2. You run a charity that can provide either food aid or health aid to this country. Which would provide a greater benefit: increasing H by 1 or N by 1?

Short answer

a. The mathematical derivation proves that both MPH and MPN are diminishing.

b. The production function shows constant returns to scale.

c. The relevant production function is plotted below.

d. The relevant production function is plotted below.

e. The country would enjoy the greater benefit by increasing H by 1.

Step-by-step solution

The explanation for part a 

The production function is:

L = c (H^0.8 N^0.2)

If labor is N = $49, the marginal product of health expenditure is:

MPH is further differentiated with respect to health expenditure to conclude whether the function is increasing or decreasing.

Thus as health expenditure rises, the marginal productivity of additional health units diminishes. Hence, MPH is decreasing.

If health expenditure is H = 4 units, the marginal product of nutrition expenditure is:

MPN is further differentiated with respect to nutrition expenditure to conclude whether the function is increasing or decreasing.

Thus, as nutrition expenditure rises, the marginal productivity of additional units falls. Hence MPN is decreasing.

The computation of returns to scale  

The production function is

L = c (H^0.8 N^0.2)

Both health and nutrition expenditures increase by λ unit, i.e., H’ = λH and N’ = λN.

So,

$$\begin{gathered} L=c\times {\left(\lambda H\right)}^{0.8}\times {\left(\lambda N\right)}^{0.2} \\ ={\lambda }^{0.8+0.2}\times c\times H^{0.8}\times N^{0.2} \\ =\lambda q \end{gathered}$$

Here, it is observed that the inputs (health and nutrition expenditure) and the production level rise by the same proportion (λ).

Thus, the production function exhibits constant returns to scale (CRS).

The illustration of the production function 

In a country, the nutrition expenditure is fixed at N = 2 and c = 20.

The production function is:

L = c (H^0.8 N^0.2)

Or, L = 20(H^0.8 2^0.2)

The production function is illustrated below.

So, the production function is inverted U-shaped. It shows that as health expenditure rises, the life expectancy also increases up to a certain limit and then decreases.

The illustration of new production function   

In a country, the nutrition expenditure is fixed at N = 4 and c = 20.

The production function is:

L = c (H^0.8 N^0.2)

Or, L = 20(H^0.8 2^0.2)

The production function is illustrated below.

So, the production function is upward sloping. It shows that as health expenditure rises, life expectancy also increases.

The explanation for part e 

The value of marginal productivity of health expenditure is,

So, if health expenditure increases by $1, MPH would rise by $24.2515

The value of marginal productivity of nutrition expenditure is,

So, if nutrition expenditure increases by $1, MPN would rise by $21.11213

So the country would have greater benefit if it increases H by 1 rather than increasing N by 1 unit since the value of MPH is higher than MPN.