Costs in the Short Run

Read this section about how to calculate costs in the short-run like variable and marginal costs. Make sure to answer the "Try It" questions.

LEARNING OBJECTIVES

Describe the relationship between production and costs, including average and marginal costs
Analyze short-run costs in terms of fixed cost and variable cost

We've explained that a firm's total cost of production depends on the quantities of inputs the firm uses to produce its output and the cost of those inputs to the firm. The firm's production function tells us how much output the firm will produce with given amounts of inputs. A production function can be expressed mathematically as

$Q=f[L, \bar{K}]$

where Q is the firm's output, L is the amount of labor employed, and K is the amount of fixed capital.

Suppose we think about the production function backwards:

$L=g[Q, \bar{K}]$

where the g just means the function f in reverse. This equation tells us how much labor we need to produce a given level of output, with the fixed capital stock we have. If we knew the cost of labor and capital, we could then compute the total cost of producing any level of output. It is to this that we next turn.

For every factor of production (or input), there is an associated factor payment. Factor payments are what the firm pays for the use of the factors of production. From the firm's perspective, factor payments are costs. From the owner of each factor's perspective, factor payments are income. Factor payments include:

Raw materials prices for raw materials
Rent for land or buildings
Wages and salaries for labor
Interest and dividends for the use of financial capital (loans and equity investments)
Profit for entrepreneurship. Profit is the residual, what's left over from revenues after the firm pays all the other costs. While it may seem odd to treat profit as a "cost", it is the payment that goes from total revenues to entrepreneurs or taking the risk of starting a business. You can see this correspondence between factors of production and factor payments in the inside loop of the circular flow diagram.

We now have all the information necessary to determine a firm's costs.

Figure 1. The Circular Flow Diagram is a model of economic activity with firms supplying goods and services (arrow A) to households. In return, households pay for those goods and services (arrow B). The inner circle of arrows shows factors and factor payments. In this figure, household supply labor services (arrow C) to firms, who pay wages, salaries and benefits (arrow D) in return. A more complete model would include all the other factors supplied in arrow C, and the associated factor payments in arrow D.

A cost function is a mathematical equation that shows the cost of producing different levels of output. Table 1 gives an example, which shows the cost of producing different quantities of widgets.

Table 1. Cost Function for Producing Widgets
Q	1	2	3	4
Cost	$32.5	$44	$52	$90

What we observe is that the cost increases as the firm produces higher quantities of output. This is pretty intuitive, since producing more output requires greater quantities of inputs, which cost more dollars to acquire.

What is the origin of these cost figures? They come from the production function and the factor payments. Suppose the production function for widgets is as shown in Table 2:

Table 2. Number of Workers and Widgets Produced
Workers (L)	1	2	3	3.25	4.4	5.2	6	7	8	9
Widgets (Q)	0.2	0.4	0.8	1	2	3	3.5	3.8	3.95	4

We can use the information from the production function to determine production costs. What we need to know is how many workers are required to produce any quantity of output. If we flip the order of the rows, we "invert" the production function so it shows $L=g(Q)$

Table 3. Widgets Produced by Workers
Widgets (Q)	0.2	0.4	0.8	1	2	3	3.5	3.8	3.95	4
Workers (L)	1	2	3	3.25	4.4	5.2	6	7	8	9

Now focus on the whole number quantities of output. We'll eliminate the fractions (or partial widgets) from the table:

Table 4. Number of Widgets Produced
Widgets (Q)	1	2	3	4
Workers (L)	3.25	4.4	5.2	9

Suppose widget workers receive $10 per hour. Multiplying the Workers row by $10 (and eliminating the blanks) gives us the cost of producing different levels of output.

Table 5. Cost of Producing Widgets
Widgets (Q)	1.00	2.00	3.00	4.00
Workers (L)	3.25	4.4	5.2	9
× Wage Rate per hour	$10	$10	$10	$10
= Cost	$32.50	$44.00	$52.00	$90.00

This is same cost function with which we began (shown in Table 1). Figure 2 shows the graph of the cost function.

Figure 2. The Total Cost curve for Widgets. This shows cost increasing at an increasing rate as the firm produces more output.

TRY IT

Consider a firm with the following production schedule:

Assume that each worker gets paid $15 per hour and that other cost of materials is $2 per item produced. Estimate the cost of producing 45 units. It is ________.

Number of Workers	Production (units/hr)
1	10
2	30
3	45
4	55
5	60

$135
$105
$47

Now that we have the basic idea of the cost origins and how they are related to production, let's drill down into the details, by examining average, marginal, fixed, and variable costs.

Source: Lumen Learning, https://courses.lumenlearning.com/wmopen-microeconomics/chapter/costs-in-the-short-run/
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