Question

Производственная функция кондитерского цеха Q = корень (L*K) , где Q – выпуск конфет (кг), L – затраты труда в часах рабочего времени, K – затраты капитала в часах работы оборудования. Исходные затраты труда – 8 ч., капитала – 2 ч. Цена труда – 10 руб./ч., капитала 50 руб./ч., конфет – 100 руб./кг. Найдите: 1) предельный продукт труда; 2) предельный продукт капитала; 3) выручку, прибыль и общие издержки цеха

Answer 1

Держи, готово :) Удачи :)

Answer 2

Proizvodstvennaya Funktsiya (Production Function) of a Confectionery Workshop

The production function of a confectionery workshop, denoted as Q = √(L * K), where Q represents the output of candies in kilograms (kg), L represents labor input in hours of working time, and K represents capital input in hours of equipment operation.

Given the following initial inputs: - Labor input: 8 hours - Capital input: 2 hours - Labor cost: 10 rubles per hour - Capital cost: 50 rubles per hour - Price per kilogram of candies: 100 rubles

We need to find the following: 1) Marginal product of labor 2) Marginal product of capital 3) Revenue, profit, and total costs of the workshop

1) Marginal Product of Labor

The marginal product of labor (MPL) represents the additional output produced by adding one more unit of labor while keeping the capital input constant.

To calculate MPL, we need to differentiate the production function with respect to labor (L) and evaluate it at the given labor input.

Differentiating the production function Q = √(L * K) with respect to L, we get:

dQ/dL = (1/2) * (K/L)^(1/2)

Substituting the given labor input (L = 8 hours) and capital input (K = 2 hours) into the derivative, we can find the MPL.

Calculating the MPL: dQ/dL = (1/2) * (2/8)^(1/2) = (1/2) * (1/2) = 1/4 = 0.25

Therefore, the marginal product of labor is 0.25.

2) Marginal Product of Capital

The marginal product of capital (MPK) represents the additional output produced by adding one more unit of capital while keeping the labor input constant.

To calculate MPK, we need to differentiate the production function with respect to capital (K) and evaluate it at the given capital input.

Differentiating the production function Q = √(L * K) with respect to K, we get:

dQ/dK = (1/2) * (L/K)^(1/2)

Substituting the given labor input (L = 8 hours) and capital input (K = 2 hours) into the derivative, we can find the MPK.

Calculating the MPK: dQ/dK = (1/2) * (8/2)^(1/2) = (1/2) * (4)^(1/2) = (1/2) * 2 = 1

Therefore, the marginal product of capital is 1.

3) Revenue, Profit, and Total Costs of the Workshop

To calculate the revenue, profit, and total costs of the workshop, we need to consider the following:

- Revenue: The revenue is calculated by multiplying the output (Q) by the price per kilogram of candies. In this case, the price per kilogram is 100 rubles.

Revenue = Q * Price per kilogram Revenue = √(L * K) * Price per kilogram

- Total Costs: The total costs include both labor costs and capital costs. Labor costs are calculated by multiplying the labor input (L) by the labor cost per hour. Capital costs are calculated by multiplying the capital input (K) by the capital cost per hour.

Total Costs = Labor Costs + Capital Costs Labor Costs = L * Labor Cost per hour Capital Costs = K * Capital Cost per hour

- Profit: Profit is calculated by subtracting the total costs from the revenue.

Profit = Revenue - Total Costs

Substituting the given values into the equations, we can calculate the revenue, profit, and total costs of the workshop.

Revenue = √(8 * 2) * 100 = √16 * 100 = 4 * 100 = 400 rubles

Labor Costs = 8 * 10 = 80 rubles Capital Costs = 2 * 50 = 100 rubles

Total Costs = 80 + 100 = 180 rubles

Profit = 400 - 180 = 220 rubles

Therefore, the revenue of the workshop is 400 rubles, the profit is 220 rubles, and the total costs are 180 rubles.

Please note that the calculations provided are based on the given information and assumptions.