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a) What is each orchard’s labor demand as a function of the daily wage W? What is the market’s labor demand?
In perfect competition, each orchard will hire labor up to the point where the marginal revenue product of labor (MRPL) equals the wage rate (W). Since the apples are sold at the world price P = $2, the marginal revenue product (MRP) for each orchard is calculated as:
MRP = P × MPL = 2 × (100 – 2L) = 200 – 4L
Setting MRP equal to wage W to find labor demand: W = 200 – 4L
Rearranging to find L as a function of W:
L = (200 – W) / 4
This is the labor demand function for each orchard.
Since there are 20 identical orchards, the total market labor demand (L_market) is:
L_market = 20 × L = 20 × (200 – W) / 4 = 5 × (200 – W) = 1000 – 5W
Thus, the market’s total labor demand is a linear function of W, decreasing as W increases.
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?
The total supply of labor is fixed at 200 workers. Setting total supply equal to total demand:
200 = 1000 – 5W
Solving for W:
5W = 1000 – 200 = 800
W = 800 / 5 = $160
The equilibrium wage rate is $160. Substituting back to find the number of workers each orchard hires:
L = (200 – 160) / 4 = 40 / 4 = 10 workers
Each orchard employs 10 workers at a wage of $160.
To find the profit per orchard, we calculate total revenue and total costs. The revenue per orchard is:
TR = P × Q
Where Q depends on L via the production function:
Q = 100L – L² = 100×10 – 10² = 1000 – 100 = 900 apples
Revenue:
TR = $2 × 900 = $1800
Total labor costs:
W × L = $160 × 10 = $1600
Fixed costs are assumed negligible or zero here, so profit per orchard (π) is:
π = TR – Total labor costs = $1800 – $1600 = $200
c) Calculate what happens to the income of workers and orchard owners if the world price of apples doubles to $4.
If the price doubles to $4, the marginal revenue product (MRP) becomes:
MRP = 4 × (100 – 2L) = 400 – 8L
Labor demand function updates to:
W = 400 – 8L → L = (400 – W) / 8
The market labor demand shifts to:
L_market = 20 × (400 – W) / 8 = 2.5 × (400 – W) = 1000 – 2.5W
The equilibrium wage with inelastic labor supply (fixed at 200 workers) is found by setting total demand equal to 200:
200 = 1000 – 2.5W → 2.5W = 800 → W = 800 / 2.5 = $320
Each orchard now hires 10 workers at a wage of $320, since the demand at that wage satisfies the supply constraint.
The production output per orchard at L = 10 remains the same (900 apples), but now revenue is:
TR = $4 × 900 = $3600
The costs are:
W × L = $320 × 10 = $3200
Profit per orchard doubles to:
π = $3600 – $3200 = $400
Workers’ income (wages) doubles from $160 to $320, increasing their total income proportionally. Orchard owners’ profits also increase, reflecting the favorable shift due to higher commodity prices.
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?
If half the orchards are destroyed, the total number of orchards decreases from 20 to 10, but the labor supply remains at 200 workers. The impact on wages depends on whether labor demand adjusts.
Presuming inelastic labor supply at 200 workers:
The total demand for labor with the remaining orchards is:
L_market = 10 × (200 – W) / 4 = 2.5 × (200 – W)
Set this equal to 200 (labor supply):
200 = 2.5 × (200 – W) → (200 – W) = 200 / 2.5 = 80 → W = 200 – 80 = $120
The new equilibrium wage drops to $120 due to decreased orchard capacity.
Number of workers per orchard: since total demand is 200 workers spread across 10 orchards, each orchard still employs 20 workers (assuming all remaining orchards employ the same number of workers).
The production per orchard at L = 20 is:
Q = 100×20 – 20² = 2000 – 400 = 1600 apples
Revenue per orchard:
TR = $2 × 1600 = $3200
Labor cost per orchard:
W × L = $120 × 20 = $2400
Profit per orchard:
π = $3200 – $2400 = $800
Workers’ total income after the hurricane is:
W × L = $120 × 20 = $2400
Compared to pre-hurricane wages of $160 and 10 workers per orchard, the total income for workers per orchard increases due to the lower wages but with greater employment, maintaining or slightly increasing total income depending on the precise distribution. The total income of Ectenia decreases because with fewer orchards and reduced total output, the overall economic activity declines. The total apples produced are halved, directly impacting total revenue and aggregate income.
In conclusion, external shocks like hurricanes significantly impact the distribution of income among workers and orchard owners, as well as the overall economic health of Ectenia. Wages tend to decrease with the reduction in orchard capacity, and profits per orchard can increase if product prices rise; however, total national income diminishes when production capacity is halved, illustrating the vulnerability of agricultural economies to natural disasters.
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