Question 5 (HPW11.5, Problem 80, p. 527)A monopolist who employs m workers finds that they produce:๐ = 2๐(2๐ + 1) *&Units of product per day. The total revenue, r (in dollars), is given by:๐ = 50๐:1000 + 3๐a) What is the price per unit (to the nearest cent) when there are 12 workers?b) Determine the marginal revenue when there are 12 workersc) Determine the marginal-revenue product when m =12.Note: round your answers to 2 decimal places
Question
Question 5 (HPW11.5, Problem 80, p. 527)A monopolist who employs m workers finds that they produce:๐ = 2๐(2๐ + 1) *&Units of product per day. The total revenue, r (in dollars), is given by:๐ = 50๐:1000 + 3๐a) What is the price per unit (to the nearest cent) when there are 12 workers?b) Determine the marginal revenue when there are 12 workersc) Determine the marginal-revenue product when m =12.Note: round your answers to 2 decimal places
Solution
a) To find the price per unit when there are 12 workers, we first need to find the quantity of product produced by 12 workers.
Using the given production function, ๐ = 2๐(2๐ + 1), we substitute ๐ = 12:
๐ = 212(212 + 1) = 2425 = 600 units
The total revenue, r, is given by ๐ = 50๐:1000 + 3๐. Substituting ๐ = 600, we get:
๐ = 50600/1000 + 3600 = 30 + 1800 = 1830 dollars
The price per unit is then the total revenue divided by the quantity, or ๐/๐ = 1830/600 = 3.05 dollars per unit, or $3.05 to the nearest cent.
b) The marginal revenue is the derivative of the total revenue with respect to the quantity. The total revenue function is ๐ = 50๐/1000 + 3๐. Taking the derivative, we get:
๐๐/๐๐ = 50/1000 + 3 = 0.05 + 3 = 3.05 dollars per unit
c) The marginal-revenue product is the product of the marginal revenue and the marginal product of labor. The marginal product of labor is the derivative of the production function with respect to labor, or ๐๐/๐๐. The production function is ๐ = 2๐(2๐ + 1). Taking the derivative, we get:
๐๐/๐๐ = 4๐ + 2
Substituting ๐ = 12, we get ๐๐/๐๐ = 4*12 + 2 = 50 units per worker
The marginal-revenue product is then ๐๐/๐๐ * ๐๐/๐๐ = 3.05 * 50 = 152.5 dollars per worker.
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