Chapter 7—The Production Function

Chapter 7—The Production Function

defn.—a mathematical relationship showing the effect of changes in inputs:

K and L (capital) (labor)

on output:Y

(real GDP)

Chapter 7—The Production Function

defn.—a mathematical relationship showing the effect of changes in inputs:

K and L (capital) (labor)

on outputs:Y

(real GDP)

General form:

Y = F(K, L)

Cobb-Douglas production fcn:

Y = A x Kα x L1-α

Chapter 7—The Production Function

A simple Cobb-Douglas example showing:a. Constant returns to scale (to increases in

all inputs)b. Diminishing returns to increases in any

single input

Y = 2 x K0.5 x L0.5

Chapter 7—The Production Function

A simple Cobb-Douglas example showing:a. Constant returns to scale (to increases in

all inputs)b. Diminishing returns to increases in any

single input

Y = 2 x K0.5 x L0.5

Let K = 100, L = 25

Y = 2 x 1000.5 x 250.5

= 2 x 10 x 5

= 100 widgets

Chapter 7—The Production Function

A simple Cobb-Douglas example showing:a. Constant returns to scale (to increases in

all inputs)b. Diminishing returns to increases in any

single input

Now double all inputs, i.e., let K = 200, L = 50

Y = 2 x 2000.5 x 500.5

= 2 x 14.1 x 7.07

= 200 widgets

So—double all inputs → double output

Chapter 7—The Production Function

A simple Cobb-Douglas example showing:a. Constant returns to scale (to increases in

all inputs)b. Diminishing returns to increases in any

single input

Now double labor inputs only, i.e., let K = 100, L = 50

Y = 2 x 1000.5 x 500.5

= 2 x 10 x 7.07

= 141 widgets

So—double one input only → less than double output

Chapter 7—The Production Function

A simple Cobb-Douglas example showing:a. Constant returns to scale (to increases in

all inputs)b. Diminishing returns to increases in any

single input

Now let’s show that returns are actually diminishing—let’s add another 25 units of labor, i.e., let K = 100, L = 75

Y = 2 x 1000.5 x 750.5

= 2 x 10 x 8.67

= 173 widgets

Chapter 7—The Production Function

A simple Cobb-Douglas example showing:a. Constant returns to scale (to increases in

all inputs)b. Diminishing returns to increases in any

single input

Now let’s make a little table to show what we’ve done— K L Y ∆L ∆Y ∆Y/ ∆L

100 25 100 - - - - - - - - -100 50 141 25 41 1. 66100 75 173 25 32 1. 27100 100 200 25 27 1. 07100 125 224 25 24 0. 94100 150 245 25 21 0. 85100 175 265 25 20 0. 79100 200 283 25 18 0. 73

Chapter 7—The Production Function

A simple Cobb-Douglas example showing:a. Constant returns to scale (to increases in

all inputs)b. Diminishing returns to increases in any

single input

Now let’s plot this—

Chapter 7—The Production Function

Another way to look at this—Capital to Labor ratio (K/L)

Think of this as tools per worker

As L↑, K/L ratio decreases—fewer tools per worker—and so each extra worker produces less extra output . . .

. . . but each unit of capital—each tool—becomes more productive!

On the other hand—if we increase K—K/L ratio increases, and each worker becomes more productive, but each unit of capital becomes less productive.

Chapter 7—The Production Function

Hmm . . . let’s see . . .

Simplifying a bit, a high K/L ratio—lots of capital per worker—would lead to:

And so a low K/L ratio—little capital per worker, or lots of workers per unit of capital—would lead to:

• high labor productivity• high wage rate• low capital productivity• low return on capital investment

• high capital productivity• high return on capital investment• low labor productivity• low wage rate

Chapter 7—The Production Function

Now—a little fable . . .

Chapter 7—The Production Function

Continuing with this—what is the effect of immigration on wage rates?

Chapter 7—The Production Function

Continuing with this—what is the effect of immigration on wage rates?

Chapter 7—The Production Function

Continuing with this—what is the effect of immigration on wage rates?

Unskilled and semi-skilled occupations: lots of immigration of workers increaselabor supply; these workers are also consumers, but don’t purchase as muchgoods as they produce in this industry, so there’s only a small increase in labordemand—hence wages FALL

Skilled and professional occupations—just the opposite, so wages rise.

This explains much of the politics of immigration as well—low-skilled workers tend to oppose immigration, because they compete with immigrants for jobs.Professionals are more supportive of immigration, because they don’t—andthey do benefit from being able to have their house built, or landscaping done,for a lower price as a result.

Note that this isn’t universal—MD’s incomes would be (even) higher were it notFor immigration of doctors from overseas.