This problem is based on the specific factors model. Consider the economy of the Shire that produces cereals and swords. Each industry employs Hobbits, but land is specific to the production of cereals, and metal is a specific factor in the sword industry. The amounts of land (L) and metal (M) are L=10,000 and M=400, while the population size of the Shire is 1,000 Hobbits. The production functions and the marginal products of Hobbits' labor are as follows: C = (HC)0.5(L)0.5, MPHC=0.5(L)0.5/(HC)0.5 S = (HS)0.5(M)0.5, MPHS=0.5(M)0.5/(HS)0.5 where C is the cereal output in pounds, S is the number of swords produced, and HC and HS are the numbers of Hobbits employed in the cereal and sword sectors, respectively.

If the price of 1 pound of cereals is $10 and the price of a sword is $100, what is the wage rate paid to Hobbits in the Shire? How many Hobbits work in the cereal sector? In sword sector?

Because of the War of the Ring, the price of a sword went up to $150. Calculate new wage and employment level in each sector.

Do Hobbits gain or lose if the price of a sword goes up to $150? (To answer the question, calculate their real wages before and after the price change).

help me plz

To find the wage rate paid to Hobbits in the Shire, we need to calculate the marginal product of Hobbits' labor in each sector and equate it to the price of the output.

1. Wage rate calculation:
- In the cereal sector, the marginal product of Hobbits' labor (MPHC) is given by 0.5(L)0.5/(HC)0.5. Let's substitute L = 10,000 and MPHC = 0.5(L)0.5/(HC)0.5 into the equation.
MPHC = 0.5(10,000)0.5/(HC)0.5
MPHC = 0.5(100)/(HC)0.5
MPHC = 5/(HC)0.5

- In the sword sector, the marginal product of Hobbits' labor (MPHS) is given by 0.5(M)0.5/(HS)0.5. Let's substitute M = 400 and MPHS = 0.5(M)0.5/(HS)0.5 into the equation.
MPHS = 0.5(400)0.5/(HS)0.5
MPHS = 0.5(20)/(HS)0.5
MPHS = 10/(HS)0.5

Now, let's equate the marginal product of labor in each sector to the respective output prices:

For the cereal sector:
MPHC = P(C) / Wage, where P(C) is the price of 1 pound of cereals.
5/(HC)0.5 = 10 / Wage

For the sword sector:
MPHS = P(S) / Wage, where P(S) is the price of a sword.
10/(HS)0.5 = 100 / Wage

2. Calculation for the initial wage and employment level in each sector:
- In the cereal sector:
5/(HC)0.5 = 10 / Wage
Simplify and solve for Wage:
Wage = (10*(HC)0.5) / 5

- In the sword sector:
10/(HS)0.5 = 100 / Wage
Simplify and solve for Wage:
Wage = (100*(HS)0.5) / 10

To find the number of Hobbits working in each sector, we need to solve for HC and HS using the given information, specifically that the population size of the Shire is 1,000 Hobbits:

HC + HS = 1,000

Now we can substitute the initial wage equations into the population equation and solve for HC and HS.

3. Calculation for new wage and employment level after the price change:
If the price of a sword increases to $150, we need to recalculate the wage and employment levels in each sector using the new price.

For the sword sector:
10/(HS)0.5 = 150 / Wage
Simplify and solve for Wage:
Wage = (150*(HS)0.5) / 10

Again, substitute the new wage equation into the population equation and solve for HC and HS.

4. To determine if Hobbits gain or lose due to the price change, we need to compare their real wages before and after the price change. The real wage is the wage adjusted for changes in the price level.

- For the initial wage and employment level, calculate the real wages before the price change.
Real Wage = Wage / P(C)

- For the new wage and employment level after the price change, calculate the real wages after the price change.
Real Wage = Wage / P(S)

Now, compare the real wages before and after the price change to determine if Hobbits gain or lose. If the real wage increases, they gain, and if it decreases, they lose.