The nation of Ectenia has 20 competitive apple orchards, which sell apples at the world price of $2. The following equations describe the production function and the marginal product of labor in each orchard: Q = 100L L2.

MPL = 100 2L

where Q is the number of apples produced in a day, L is the number of workers, and MPL is the marginal product of labor.?

a) What is each orchard's labor demand as a ?function of the daily wage W? What is the ?market's labor demand?

b) Ectenia has 200 workers who supply their ?labor inelastically. Solve for the wage W. How many workers does each orchard hire? How much profit does each orchard owner make?

c) Calculate what happens to the income of workers and orchard owners if the world price of apples doubles to $4.

d) Now suppose the price of apples is back at $2, but a hurricane destroys half the orchards. Calculate how the hurricane affects the income of each worker and of each remaining orchard owner. What happens to the income of Ectenia as a whole?

(a)The market labor is =1000 -5 W (b) The profit is $200 (c) If the world price of apples doubles to $4 it becomes, wage is $320 and the profit is $400 (d) The income as a whole before the hurricane is $3600 and income as a whole after the hurricane is $40,000

Explanation:

Solution

Given that:

Production function Q = 100 L-L²

Marginal product of labor = MPl = 100 -2L

(a) Calculate Value of Marginal Product of Labor:

VMPl =P (MPl)

=P (100 - 2L)

Then

We substitute  world price is $2

Thus

VMPl = P (100 -2L)

=2 (100 -2L)

= 200 - 4L

Now

The daily wages are equal to the value of marginal product of labor

Thus

W =200 -4L

-4L = W - 200

4L =200 - W

L = 200- W/4

Hence the market demand for labor in Ectenia is given as follows:

L = 200- W/4 * 20

=(200 - W) *5

= 1000 -5 W

(b) Calculate the wage (W), Total workers and Total profit:

Now

The wage is given below:

200 = 1000 - 5 W

- 5W = 200 - 1000

W =-800/ 5

=$160

Thus

Ectenia supplied 200 labors, the number of worker required is given below:

Number of workers = 200/20

= 10 workers

So

We calculate the profit:

Profit = Total Revenue - Total cost

= (P *Q) - (L*W)

= 2(100 L- L²) -( L*160)

=2 (100 * 10 - (10)² - (10 * 160)

=1800 -1600

=$200

The profit is $200

(c) Here, the price doubles, so the VMPl must be raised.Thus the wage is $320. the profit is $400

VMPl = 2* (200 -4 L)

=400 - 8 L

(d) Now

The price of apples is back to $2. but hurricane destroys half the orchards.

Thus the market demand for labor comes down to half relative before.

Thus

The market demand for labor in Ectenia is given below:

L = 1000 - 5W /2

= 500 -2.5 W

Thus

the average is

200 = 500 - 2.5 W

2.5W =500 -200

W =300/2.5

=$120

The wages of labor after the hurricane is $120

Now,

Calculate the profit after hurricane:

profit = Total Revenue - Total cost

= 2(100 L - L²) - (120 * L)

= 200 L - 2L² - 120 L

= 80L -2L² -120 L

=80L - 2L²

Which is

80 * 20 - 2 * 20²

= 1600 - 800

= 800

The profit of the firm after hurricane is $800

Now,

The income of Ectenia as a whole before the hurricane is given below:

The income (Total) = Total profit  + total wages

= 20 *200 + 200 * 160

= 4000 + 3200

= $3600

The income as a whole before the hurricane is $3600

Now,

income as a whole after the hurricane is given below:

The Income (Total) = Total profit  + total wages

=20 *800 + 200 * 120

= 16000+ 24000

=$40,000

income as a whole after the hurricane is $40,000