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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
100
100
100
100
100
100
100
25
50
75
100
125
150
175
200
100
141
173
200
224
245
265
283
--25
25
25
25
25
25
25
--41
32
27
24
21
20
18
--1.66
1.27
1.07
0.94
0.85
0.79
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—
Y = F(K, L)
300
Y = F(K, L)
250
200
150
100
50
0
0
25
50
75
100
125
150
175
200
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:
high labor productivity
high wage rate
low capital productivity
low return on capital investment
•
•
•
•
And so a low K/L ratio—little capital per worker, or lots of workers per
unit of capital—would lead to:
•
•
•
•
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 increase
labor supply; these workers are also consumers, but don’t purchase as much
goods as they produce in this industry, so there’s only a small increase in labor
demand—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—and
they 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 not
For immigration of doctors from overseas.