SBU Econ · All tools

ECO 314 · PRODUCTION & DISTRIBUTION

Cobb–Douglas explorer

Y = A Kα L1−α

Change the economy. See what happens to production and to the people who earn its income.

Each example starts from the original economy: K = 100, L = 100, A = 1, α = 0.50, P = 1.

The economy

100
1200
100
1200
1.00
0.252.00
0.50
0.100.90

Also capital’s income share in this model. Labor’s share is 1 − α.

Land / capital per worker 1.00

Production & factor prices

All values below are real

Total output Y

100.00

Original: 100.00

Wage per worker W/P

0.500

Original: 0.500

Rent per unit R/P

0.500

Original: 0.500

Holding K = 100 and A = 1.00 fixed

Current slopeMPL = 0.500
Output as labor changesA production curve with the original point, current point, and marginal-product tangent.
Original economyCurrent economyCurrent slope

The slope shows how much extra output a small increase in labor produces, with land / capital held fixed.

WHAT CHANGED?

Start with 100 workers and 100 units of land.

Move a slider or try an example. Compare income per worker with total labor income below.

Who receives the output?

Real income = units of output

Bar length shows total output. The two colored portions show how it is divided.

Labor incomeLand / capital income
Income amounts and shares before and after the change
Income measureOriginalCurrentChange
Show the math: from output to wages and rents

Competitive firms pay each input its marginal product. Differentiating with respect to one input means holding the other input, technology, and the exponents fixed.

For labor, hold K fixed

MPL = (1 − α) A Kα L−α

W/P = MPL = (1 − α)Y/L

For capital, hold L fixed

MPK = α A Kα−1 L1−α

R/P = MPK = αY/K

The power rule: bring down the exponent, then subtract 1 from it. For labor, (1 − α) − 1 = −α. For capital, α − 1 remains α − 1.

The tangent is the derivative: the output change per unit for an arbitrarily small input change. Adding one whole worker gives a slightly different finite change.

What if all nominal prices double?
1.00

Keep the real economy fixed and change its price level. Nominal wages and rents adjust with the output price.

Nominal wage 0.500W = P × MPL

Nominal rent 0.500R = P × MPK

Real wage 0.500W/P = MPL

Real rent 0.500R/P = MPK

Doubling W, R, and P leaves W/P and R/P unchanged. The money amounts double; the quantities of goods those payments buy stay the same.