Chapter 7—The Production Function

defn.—a mathematical relationship showing the effect of changes in inputs: Kand L (capital)(labor) on output: Y (real GDP)

Chapter 7—The Production Function defn.—a mathematical relationship showing the effect of changes in inputs: Kand L (capital)(labor) on outputs: Y (real GDP) General form: Y = F(K, L) Cobb-Douglas production fcn: Y = A x K α x L 1- 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 K 0.5 x L 0.5

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 K 0.5 x L 0.5 Let K = 100, L = 25 Y = 2 x x = 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 x = 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 x = 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 x = 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—

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 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.