Lindt’s chocolate factory produces dark chocolate (q) using capital K (buildings

Question

Lindt’s chocolate factory produces dark chocolate (q) using capital K (buildings, machinery), labor L and materials M (cocoa solids, cocoa butter, vanilla extract, soy lecithin and packaging). In the short run, capital is fixed and labor and materials are variable. In the very short run, capital and labor are fixed but materials are variable.

Assume that the amount of capital is K = 100 in the short and very short runs. The amount of labor is L = 256 in the very short run. Lindt’s long run production function is f(K,L,M) = 4K^1/2 L^1/4 M^1/4 .

Does the long run production function exhibit constant returns to scale? Decreasing returns to scale? Increasing returns to scale?

Derive an expression for the marginal product of labor in the short run. (Hint: Your answer should depend on L and M.)

Does the marginal product of labor increase or decrease with the amount of materials used?

Does the short run production function exhibit diminishing marginal returns to labor?

Does the short run production function exhibit constant returns to scale? Decreasing returns to scale? Increasing returns to scale?

Derive an expression for the marginal product of materials in the very short run.

Does the very short run production function exhibit diminishing marginal returns to materials?

Does the very short run production function exhibit constant returns to scale? Decreasing returns to scale? Increasing returns to scale?

Explanation / Answer

Y = f(K, L, M) = 4K0.5L0.25M0.25

(1) If we increase all inputs N times, new production function becomes

Y* = 4 x (N x K)0.5(N x L)0.25(N x M)0.25

= 4 x N(0.5 + 0.25 + 0.25) x K0.5L0.25M0.25 = N x Y

Y* / Y = N

So, production function shows constant returns to scale.

(2) MPL = dY / dL

Short run Y = 4 x (100)0.5L0.25M0.25 = 4 x 10x L0.25M0.25 = 40 x L0.25M0.25

MPL = dY / dL = 40 x 0.25 x M0.25 / L0.75 = 10 x M0.25 / L0.75

(3) From the MPL function derived, as M increases, MPL increases.

(4) From MPL function, as L increases, MPL decreases. This reflects diminishing marginal returns to labor.

(5) Short run Y = 40 x L0.25M0.25

When we increase both inputs by N, new production function becomes:

Y* = 40 x (N x L)0.25(N x M)0.25 = 40 x N(0.25 + 0.25) x L0.25M0.25 = N0.5 x Y

Y* / Y = N0.5 < N

So, short run production function shows decreasing returns to scale.

Note: First 5 sub-parts are answered.

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