### Production Function and Equilibrium Real Wage Calculation**Problem Statement:**Suppose that the production function for the economy is given by the equation: \[ Y = A K^{0.5} L^{0.5} \]Where:- $ K $ represents the capital stock,- $ L $ represents the quantity of labor,- $ A $ is the efficiency index.Given the following values:- Capital stock $ K = 40,000 $- Quantity of labor $ L = 10,000 $- Efficiency index $ A = 3 $Determine the equilibrium real wage.**Multiple Choice Options:**- $ \text{A. } \$3 $- $ \text{B. } \$4.50 $- $ \text{C. } \$9 $- $ \text{D. } \$16.67 $**Explanation:**To find the equilibrium real wage, we need to calculate the marginal product of labor (MPL), which is derived from the production function. In this case, the marginal product of labor is given by the partial derivative of $ Y $ with respect to $ L $:\[MPL = \frac{\partial Y}{\partial L} = A \cdot K^{0.5} \cdot \frac{\partial}{\partial L} L^{0.5} = A \cdot K^{0.5} \cdot 0.5 \cdot L^{-0.5}\]Substitute the given values $ A = 3 $, $ K = 40,000 $, and $ L = 10,000 $:\[MPL = 3 \cdot (40,000)^{0.5} \cdot 0.5 \cdot (10,000)^{-0.5}\]\[MPL = 3 \cdot 200 \cdot 0.5 \cdot \frac{1}{100}\]\[MPL = 3 \cdot 200 \cdot 0.5 \cdot 0.01\]\[MPL = 3 \cdot 2\]\[MPL = \$6\]However, if we consider getting our options as per the choices, the closest option is **Option C:** \$9 which **Production Function and Equilibrium Real Rental Price Calculation**In this example, we are given a production function for the economy represented by the equation:\[ Y = AK^{0.5}L^{0.5} \]Where:- $ Y $ is the output,- $ A $ is the efficiency index,- $ K $ is the capital stock,- $ L $ is the quantity of labor.Given:- Capital stock ($ K $) = 40,000- Quantity of labor ($ L $) = 10,000- Efficiency index ($ A $) = 3The problem asks us to determine the equilibrium real rental price of capital. The answer choices are:- A. $0.33- B. $0.75- C. $1.00- D. $2.22To find the equilibrium real rental price of capital, we need to use the marginal product of capital (MPK). The MPK can be derived from the production function as follows:\[ \text{MPK} = \frac{\partial Y}{\partial K} = \frac{0.5 \cdot AK^{0.5}L^{0.5}}{K} = 0.5 \cdot A \cdot \frac{L^{0.5}}{K^{0.5}} \]Substitute the given values:- $ A = 3 $- $ K = 40,000 $- $ L = 10,000 $\[ \text{MPK} = 0.5 \cdot 3 \cdot \frac{10,000^{0.5}}{40,000^{0.5}} \]Calculate the values:- $ 10,000^{0.5} = 100 $- $ 40,000^{0.5} \approx 200 $\[ \text{MPK} = 0.5 \cdot 3 \cdot \frac{100}{200} = 0.5 \cdot 3 \cdot 0.5 = 0.75 \]Therefore, the equilibrium real rental price of capital is:**B. $0.75**This example illustrates the steps to calculate the equilibrium real rental price of capital using a given production function, labor, capital stock, and

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Suppose that the production function for the economy is Y = AKPLOS. If the capital stock = 40,000, the quantity of labor = 10,000, and the efficiency index = 3, the equilibrium real wage is O A. $3. O B. $4.50. O C. $9. O D. $16.67.

Suppose that the production function for the economy is Y = AKPLOS. If the capital stock = 40,000, the quantity of labor = 10,000, and the efficiency index = 3, the equilibrium real wage is O A. $3. O B. $4.50. O C. $9. O D. $16.67.

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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