Cobb-Douglas Production Function

Production is a function of capital \( K \) and labor \( L \):
\[ P(K,L) = \gamma K^{\alpha} L^{1-\alpha}, \]
a form that gives constant returns to scale.

cd = makeFun(K^a * L^(1 - a) ~ K & L, a = 0.3)
cd.dK = D(cd(K, L) ~ K)
cd.dL = D(cd(K, L) ~ L)
plotFun(cd(K = K, L = L) ~ K & L, K.lim = range(0, 100), L.lim = range(0, 
    100))

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plotFun(cd.dK(K = K, L = L) ~ K & L, K.lim = range(0, 100), L.lim = range(0, 
    100))

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plotFun(cd.dL(K = K, L = L) ~ K & L, K.lim = range(0, 100), L.lim = range(0, 
    100))

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Often, there is a budget constraint. We'll come to that later.

Treat this as a modeling problem.

Diminishing marginal returns.

Suppose we increase capital but hold labor constant. How will output increase? Fast at first, then slower. Same for labor. These are partial derivatives (one way to look at things) but we can also just plot out P against K for constant L. Ask students to do that.

plotFun(cd(K = K, L = 50) ~ K, K.lim = range(0, 100))

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plotFun(cd(K = 30, L = L) ~ L, L.lim = range(0, 100))

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Constant returns to scale.

Double both capital and labor, e.g. build a new factory. This should double output. NOTE: It is not a partial derivative.

plotFun(cd(K = inc, L = inc) ~ inc, inc.lim = range(0, 100))

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plotFun(cd(K = inc, L = 2 * inc) ~ inc, inc.lim = range(0, 100))  ## spend twice as much on labor as on capital

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The value of labor and of capital

The value of labor can be thought of as the number of bicycles produced per unit of labor. Similarly for capital.