9b. From Production Functions to Cost Functions

Another main objective for the seller is to minimize the cost of production to enable profit maximization. But first, lets clear up the differences between economic costs and accounting costs.

Accounting costs: includes historical costs, depreciation, and other bookkeeping entries

Economic Costs: the cost of any input is given by the amount of payment necessary to keep the resource employed (includes the normal rate of return of an alternative choice of investment instead – opportunity costs are part of the economic costs)

Labor Costs: explicit, denoted w, which is the hourly wage rate. Also assumed to be what the labor services would earn in their best alternative employment.

Capital Costs: implicit; whereas the accountant would include the historical cost of the input, economists consider it a sunk cost. Thus, the cost is the hourly rental rate, v, (can be seen as the energy costs required to run a machine)

Costs of entrepreneurial services: included under capital costs when viewed from an Economist’s perspective

Profit

  1. Accountants view: the extra revenue left after paying all prices of inputs
  2. Economists view: extra revenues – opportunity costs; the view used in our economic analysis

Short Run vs. Long Run

Also, the difference between short run and long run economic decisions must be cleared up.

Short Run:

  1. A fixed scale (or a fixed factor of production) – cannot change scale of production
  2. No entry or exit from an industry

Long Run:

  1. No fixed factors of production – they can increase by however much they want

Some more simplifying assumptions

  1. Only inputs are homogenous Labor (L)
  2. Entrepreneurial costs are included in capital costs
  3. Inputs are hired in perfectly competitive markets – firms must pay the going wage rate and rental costs)

Economic Profits and Cost Minimization

Profits = Total Revenues – Total Costs

(Price of output)*(quantity of output sold) – Total costs

Total costs = wL + vK (assume K is constant)

Also, quantity of output sold is the production function q = f(L)

So,

Profits = P*f(L) – wL + C (where C is a constant)

As mentioned, the Total Cost Function for a one input production is the (wage)*(number of labor employed)

TC = wL

Study Questions

1. At a wage of $6.75, a Fast Food restaurant hires 18 workers with a maximum amount of hours each employee can work being 20 hours a week. If the workers choose to maximize their hours at work, what would be the total labor cost of the restaurant for a week?

2. A firm decides to switch production from candy to cake. The revenues from candy were steady over the last 10 years, averaging $3 million in total revenues with an average profit of $1.2 million. Assuming that the cost of making cakes is equal to the cost of making candy and the average over the 10 years is the actual case, at a price of $10 a cake, how much cake must the company sell to make a profit?

3. An umbrella firm whose production function is q = 9L2 – L3 decides to hire 2 more workers to a workforce of about 6, making the total 8 workers. If the price of the good is $5 and the going wage rate is $4, was this sound managerial decision making?

4. Consider a production function q = 2L2 - L3 for a hammer factory. If the wage rate is $5 and the price of each hammer is $10, what is the profit function of the factory?

Answers:

  1. The total cost is w*L*(number of hours per week) = 6.75*18*20 = $2,507
  2. Since profits = TR – TC, it can be deduced that TC = TR – profits = $1.8 million. The company must make at least $1.8million in revenue to make a profit because the opportunity costs are included (the alternative is continuing to make candy). So, at a price of $10, the company must sell 1,800,000/10 = 180,000 cakes to make a profit
  3. At L = 6, the output is q = 9(6)2 – (6)3 =108 umbrellas. The profits are TR – TC, which is 108*5 – 6*4 = $516

At L = 8, the output is q = 9(8)2 – (8)3 = 64 umbrellas. The profits are TR – TC, which is equal to 64*5 – 8*4 = 288.

In reality, the company is still making profits ($288) with 8 workers, but it is making (516-288) dollars less profit than if it had 6 employees. (Actually, the firm would have been operating at an optimal level with 6 employees).

  1. The profit function is in the following form: TR – TC = p*f(L) – wL = 10(2L2 – L3) – 5*L

20L2 – 10L3 – 5L