Macroeconomic Theory: Econ 2220

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Math For Econ 2220 B 1
1.1

Shifts of a Curve vs. Movement along a Curve
Parameter and variable y = f (x, a) (1)

Suppose y is a function of x, and

where a is a parameter, which represents the impact of other exogenous variables. Given a certain value of a, we can plot the function y = f (x, a) as a curve in a two-dimension graph. A change in x implies a movement along the curve, while a change in a means a shift of the curve.

1.2

An example

Suppose the firm’s daily output, Y , depends on both the number of workers that the firm employs, N , and the number of machines that the firm uses, K. The relation between input and output is specified as the following, √ √ Y =2 K N (2)

Output Y is a function of both the number of machines, K, and the number of workers, √ √ N . For example, if there are 100 machines and 4 workers, the output Y = 2 100 4 = 2 × 10 × 2 = 40. Now suppose we are concerned with the relation between output and the number of workers employed. That means that we hold the number of machines constant and study the relation between Y and N . Graphically, we plot Y against N , where N is on the horizontal axis. If we hold K constant at 100, the relation reduces to √ √ √ Y = 2 100 N = 20 N (3)

It is shown as the solid curve in Figure 1. If the firm increases the number of workers employed from 4 to 9, the output would increase from 40 to 60. In the graph, the combination of Y and N moves along the solid curve from point C to point D.

Macroeconomic Theory: Econ 2220

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Figure 1: Shifts of a Curve vs. Movement along a Curve

Now suppose that the firm increases the number of machines used in the production from 100 to 225. The relation becomes √ √ √ Y = 2 225 K = 30 K (4)

It is the dashed curve in Figure 1. Note that the increase in K has shifted the curve up. That is, for any number of