2.LounchPad - Suppose the production func- tion in medieval Europe is Y=K^0.5L^0.5 where K is the amount of land and L is the amount of labor. The economy begins with 100 units of land and 100 units of labor. Use a calculator and equations in the chapter to find a numerical answer to each of the following questions. a. How much output does the economy produce? b. What are the wage and the rental price of land? c. What share of output does labor receive? d. If a plague kills half the population ,what is the new level of output? e. What is the new wage and rental price of land? f. What share of output does labor receive now?

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2.LounchPad - Suppose the production func- tion in medieval Europe is Y=K^0.5L^0.5 where K is the amount of land and L is the amount of labor. The economy begins with 100 units of land and 100 units of labor. Use a calculator and equations in the chapter to find a numerical answer to each of the following questions. a. How much output does the economy produce? b. What are the wage and the rental price of land? c. What share of output does labor receive? d. If a plague kills half the population ,what is the new level of output? e. What is the new wage and rental price of land? f. What share of output does labor receive now?

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Answer 1

Let's solve each part of the problem step by step.

Given:

Production Function: \( Y = K^{0.5}L^{0.5} \)
Initial values: \( K = 100 \), \( L = 100 \)

a. How much output does the economy produce?
Substitute the initial values into the production function:

\[ Y = 100^{0.5} \times 100^{0.5} \]
\[ Y = 10 \times 10 \]
\[ Y = 100 \]

So, the economy produces 100 units of output.

b. What are the wage and the rental price of land?
In the Cobb-Douglas production function \( Y = K^{0.5}L^{0.5} \), the exponents (0.5) represent the share of output attributed to each input.

- Share of output attributed to land (K) = 0.5
- Share of output attributed to labor (L) = 0.5

Since the total output is 100 units:

- Rental price of land (R) = 0.5 * 100 = 50
- Wage (W) = 0.5 * 100 = 50

c. What share of output does labor receive?
As calculated above, the share of output attributed to labor (L) is 0.5, or 50%.

d. If a plague kills half the population, what is the new level of output?
If half the population is killed, the new amount of labor (L) is halved:

\[ L_{\text{new}} = \frac{100}{2} = 50 \]

Substitute the new values into the production function:
\[ Y_{\text{new}} = 100^{0.5} \times 50^{0.5} \]
\[ Y_{\text{new}} = 10 \times 7.07 \]
\[ Y_{\text{new}} \approx 70.7 \]

So, the new level of output is approximately 70.7 units.

e. What is the new wage and rental price of land?
With the new amount of labor (L = 50):

- Share of output attributed to land (K) = 0.5
- Share of output attributed to labor (L) = 0.5

Total output is now 70.7 units:
- Rental price of land (R) = 0.5 * 70.7 = 35.35
- Wage (W) = 0.5 * 70.7 = 35.35

f. What share of output does labor receive now?
As calculated above, with the new level of output, the share of output attributed to labor (L) remains 0.5, or 50%.

Summary:

a. Output: 100 units
b. Wage: 50, Rental price of land: 50
c. Share of output for labor: 50%
d. New output: 70.7 units
e. New wage: 35.35, New rental price of land: 35.35
f. New share of output for labor: 50%

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