Indifference curves and budget constraints

Let's examine how indifference curves and budget constraints may be used to illustrate the optimal combination of labor and leisure. An indifference curve is a graph of alternative combinations of goods that provide a given level of satisfaction (utility). In the simple neoclassical model of labor supply, it is assumed that the individual's utility level is a function of two goods: real income (Y), and leisure time (L). In mathematical terms, this utility function may be expressed as:

U = U(Y,L)

where U = the level of utility associated with alternative combinations of L and Y. In this case, an indifference curve provides a graph of all of the combinations of income and leisure that provides a given level of utility to an individual. The diagram below illustrates a possible indifference curve.

This indifference curve is downward sloping because an individual is willing to give up some income to receive additional leisure (or vice versa).

A few points have been added to the diagram below. Due to the definition of the indifference curve, this individual would be just as happy with the combination of real income and leisure represented by point A as he or she would be with the combination of Y and L represented by point B. Points that lie above and to the right of the indifference curve, such as point C, provide a higher level of utility.

Each point in this diagram provides a particular level of utility. The indifference curve that passes through point C provides a higher level of utility. This means that U' corresponds to a higher level of utility in the diagram below.

Utility maximization

In the diagram below, three indifference curves have been added to the diagram containing the budget constraint. Each point on the budget constraint is a feasible combination of income and leisure. It is assumed that the individual will select the combination of income and leisure that provides the highest possible level of utility. As indicated by the diagram below, this optimal combination of L and Y occurs at a point of tangency between the budget constraint and an indifference curve. In the diagram below, this optimal point occurs when real income equals Y* and hours of leisure equals L*. At this point, the individual chooses to work H* hours.

Topic 5, 6: Demand for Labor (1,2 part)

Profit maximization

Economists assume that firms attempt to maximize their profits. One question that might be asked is whether the employment of an additional unit of labor raises or lowers a firm's profits. To analyze this, recall that:

economic profits = total revenue - total costs

When an additional worker is hired, total revenue will rise (under most practical situations). On the other hand, total costs rise as well. The increase in revenue results in an increase in profits while the increase in costs lowers the level of profits. Thus, the addition of an additional worker will increase profits only if the additional revenue resulting from this labor is greater than the additional costs. Profits will decline if costs increase by more than revenue.

To examine this issue, economists rely on two measures:

• the marginal revenue product (MRP) of labor, and

• the marginal factor cost (MFC) of labor.

The marginal revenue product of labor is defined to be the additional revenue that results from the use of an additional unit of labor. In a similar manner, the marginal factor cost of labor is defined to be the additional cost associated with the use of an additional unit of labor. (Your textbook defines this using the somewhat less conventional term of marginal expense (ME) of labor.) In this course, I'll use the term "marginal factor cost" since this is the term you probably saw in your micro principles course and are likely to see in any subsequent economics classes.

A little bit of reflection should convince you that a profit-maximizing firm will:

• increase the use of labor if MRP > MFC, and

• reduce the use of labor if MRP < MFC.

Marginal revenue product

The marginal revenue product of labor can also be expressed as:

MRP = MR x MP

where MR (marginal revenue) equals the additional revenue resulting from the sale of an additional unit of output and MP (marginal product, also known as marginal physical product or MPP in many micro principles texts) is the additional output resulting from the use of an additional unit of labor, holding the use of other inputs constant. Suppose, for example, that you wished to compute the marginal revenue product of labor when MR = 4 and MPP = 5. In this case, the employment of an additional worker results in a 5 unit increase in output (holding other inputs constant) while revenue increases by $4 when an additional unit of output is sold. In this case, the marginal revenue product of labor will equal $20 (= $4 x 5).

Using a little bit of algebra, the marginal revenue product can be defined as:

Similarly, marginal revenue and marginal product are defined as:

and:

The relationship among MRP, MR, and MP can also be seen quite clearly in an algebraic manner:

Since MRP is equal to the product of MR and MP, to determine the relationship between MRP and the level of labor use, we need to understand how MR and MP change when the level of labor changes.

As you may recall from your micro principles course, the law of diminishing returns can be stated as:

law of diminishing returns - as additional units of a variable input are added to a production process in which other inputs are fixed, the marginal product of the variable input will ultimately decline.

While it is possible that MP may initially increase, a profit-maximizing firm will never hire workers in the range in which this occurs. (To see this, note that if it is profitable to hire the second worker and the third worker has a higher marginal product, it will always be optimal to hire the 3rd worker. A profit-maximizing firm would never hire only two workers in this case.) Thus, marginal product declines over the range of labor use that will be considered by a firm. The diagram below illustrates the relationship between marginal product and the level of labor use.

As this diagram indicates, it is possible that the marginal product of labor will become negative beyond some level of labor use.