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Diminishing marginal returns and optimal inputs

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Diminishing Marginal Returns and Optimal Inputs

Economic concepts

Marginal product, real wage, diminishing returns, optimal input level

Problem

A farmer's output of corn is given by the production function, Q = AF(L), where Q is the output of corn (in bushels), L is the number of workers, and A is the level of technology used in production. The farmer hires workers in a competitive labor market. The market for corn is also assumed to be perfectly competitive. Thus, the wage rate (w) and the price of corn (p) are taken as fixed by the farmer. The farmer's objective is to maximize profits.

Questions

1. Illustrate the law of diminishing marginal returns.

2. How many workers should the farmer hire in order to maximize profits?

3. At the optimal point, what is the relationship between marginal product and real wage?

4. How does each of the following affect the optimal number of workers?

- An increase in the wage rate

- An increase in the market price of corn

- An improvement in technology

Data

A = 30, F(L) = L^(1/2), p = 2, w = 10.

Steps

1. Define the production function and assign values to the parameters.

> A:=30; p:=2; w:=10; Q:=A*L^(1/2);

A := 30

p := 2

w := 10

Q := 30*sqrt(L)

2. To obtain the marginal product of labor, differentiate Q w.r.t L.

> MPL:=diff(Q,L);

MPL := 15*1/(sqrt(L))

3. To depict diminishing marginal returns, differentiate MP_L w.r.t L. A negative sign for the resulting derivative (for positive values of L) confirms the existence of diminishing marginal returns: As more workers are added, the additional output produced by a worker decreases.

> MPL_L:=diff(MPL,L);

MPL_L := -15/2*1/(L^(3/2))

4. Plot the production function. Note that, as L increases, so does Q but at a decreasing rate. The slope of the curve at a point measures the MPL at that point. As L increases, the slope of the curve decreases, implying that the marginal product of labor falls as additional workers are added.

> plot(Q,L=0..20,'Q',axes=BOXED, title='Production_function');

[Maple Plot]

5. To find the optimal value of L, begin by defining the profit function.

> profit:=p*Q-w*L;

profit := 60*sqrt(L)-10*L

6. Differentiate the profit function w.r.t L.

> profit_1:=diff(profit,L);

profit_1 := 30*1/(sqrt(L))-10

7. At the optimal point, the first derivative of the profit function is zero. Accordingly, set the derivative equal to zero and solve for the optimal value of L. ( Note: L cannot be negative. )

> solution:=solve(profit_1,L): solution:=evalf(%);

solution := 9.

8. The optimal value of L is 9 . Call it L_opt .

> L_opt:=solution;

L_opt := 9.

9. To confirm that L_opt does indeed yield maximum profits, verify that the second-order derivative of the profit function w.r.t L. evaluated at L_opt is negative.

> profit_11:=diff(profit_1,L);

profit_11 := -15*1/(L^(3/2))

10. Compute maximum profit by substituting L_opt in the profit function.

> profit_max:=subs({L=L_opt},profit): profit_max:=evalf(%);

profit_max := 90.0000000

11. At the optimal point, marginal product will equal the real wage (w/p). To show this, obtain MPL at L_opt, and verify that it equals w/p.

> MPL:=subs(L=L_opt,MPL): MPL:=evalf(%);

MPL := 5.000000000

> Real_wage:=w/p: Real_wage:=evalf(%);

Real_wage := 5.

12. In order to investigate how changes in the wage rate, price of corn, and technology affect the optimal number of workers and maximum profit, assign suitable values for the parameters in Step 1 and follow Steps 2-10.

Show the following:

(a) An increase in w will lead the farmer to hire fewer workers, ceteris paribus *.

(b) An increase in p will lead the farmer to hire more workers, ceteris paribus .

(c) An increase in A will lead the farmer to hire more workers, ceteris paribus .

* ceteris paribus means "others things remaining equal."

__________

May 2000

Dr. Sanjay Paul, St. Norbert College