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Chapter 11: Problem 10

Constant returns-to-scale production functions are sometimes called homogeneous of degree 1 More generally, as we showed in footnote 1 of Chapter \(5,\) a production function would be said to be homogeneous of degree \(k\) if \\[ f(t K, t L)=t y f(K, L) \\] a. Show that if a production function is homogeneous of degree \(k\), its marginal productiv ity functions are homogeneous of degree \(k-1\) b. Use the result from part (a) to show that marginal productivities for any constant returns-to-scale production function depend only on the ratio \(K / L\) c. Use the result from part (b) to show that the \(R T S\) for a constant returns-to-scale pro duction function depends only on the ratio \(K / L\) d. More generally, show that the \(R T S\) for any homogencous function is independent of the scale of operation - all isoquants are radial expansions of the unit isoquant. Hence, such a function is homothetic. e. Show that the results from part (d) apply to any monotonic transformation of a homo geneous function. That is, show that any such transformation of a homogeneous func tion is homothetic.

Short Answer

Expert verified
In summary, we analyzed homogeneous production functions and their properties, such as marginal productivity, returns to scale, and homotheticity. We showed that marginal productivity functions are homogeneous of degree k-1 for homogeneous production functions of degree k. We also demonstrated that for constant returns-to-scale production functions, the marginal productivities and RTS depend only on the ratio K/L. Furthermore, we established that RTS for any homogeneous function is independent of the scale of operation, and the function itself is homothetic. Finally, we proved that any monotonic transformation of a homogeneous function results in a homothetic function.

Step by step solution

01

Define the production function and its homogeneity of degree k

We are given the homogeneous production function of degree k: \\[ f(t K, t L) = t^k f(K, L) \\]
02

Calculate the marginal productivity of K and L

To find the marginal productivities, we need to take the partial derivatives of the production function with respect to K and L: Marginal productivity of K (MPK): \\[ MPK = \frac{\partial f(K,L)}{\partial K} \\] Marginal productivity of L (MPL): \\[ MPL = \frac{\partial f(K,L)}{\partial L} \\]
03

Show that MPK and MPL are homogeneous of degree k-1

Substitute tK and tL into the MPK and MPL formulas: \\[ MPK(tK, tL) = \frac{\partial f(tK, tL)}{\partial (tK)} \\] \\[ MPL(tK, tL) = \frac{\partial f(tK, tL)}{\partial (tL)} \\] Using the chain rule, calculating the marginal productivity of new inputs would result in: \\[ MPK(tK, tL) = t^{k-1} \frac{\partial f(K, L)}{\partial K} = t^{k-1} MPK(K, L) \\] \\[ MPL(tK, tL) = t^{k-1} \frac{\partial f(K, L)}{\partial L} = t^{k-1} MPL(K, L) \\] Hence, the marginal productivity functions MPK and MPL are homogeneous of degree k-1. b. Use the result from part (a) to show that marginal productivities for any constant returns-to-scale production function depend only on the ratio K / L
04

Define constant returns-to-scale production function

A constant returns-to-scale production function is homogeneous of degree 1: \\[ f(tK, tL) = t f(K,L) \\]
05

Use MPK and MPL homogeneous of degree k-1 result

From part (a), we know that MPK and MPL are homogeneous of degree k-1. In this case, k-1 = 0: \\[ MPK(tK, tL) = t^{0} MPK(K, L) = MPK(K, L) \\] \\[ MPL(tK, tL) = t^{0} MPL(K, L) = MPL(K, L) \\]
06

Show marginal productivities depend only on the ratio K/L

We can rewrite the homogeneous production function as: \\[ f(K, L) = f \left( \frac{K}{L} \cdot L, L \right) \\] Since MPK and MPL are homogeneous of degree 0, they depend only on the ratio K/L: \\[ MPK \left( \frac{K}{L} \cdot L, L \right) = MPK(K, L) \\] \\[ MPL \left( \frac{K}{L} \cdot L, L \right) = MPL(K, L) \\] Thus, marginal productivities for any constant returns-to-scale production function depend only on the ratio K/L. c. Use the result from part (b) to show that the RTS for a constant returns-to-scale production function depends only on the ratio K/L
07

Define RTS

RTS for a production function is given by: \\[ RTS = \frac{MPK}{MPL} \\]
08

Show RTS depends only on the ratio K/L

From part (b), we know that MPK and MPL depend only on the ratio K/L. Therefore, RTS must also depend only on the ratio K/L: \\[ RTS = \frac{MPK \left( \frac{K}{L} \cdot L, L \right)}{MPL \left( \frac{K}{L} \cdot L, L \right)} = \frac{MPK(K, L)}{MPL(K, L)} \\] Hence, the RTS for a constant returns-to-scale production function depends only on the ratio K/L. d. More generally, show that the RTS for any homogeneous function is independent of the scale of operation - all isoquants are radial expansions of the unit isoquant. Hence, such a function is homothetic.
09

Define homogeneous function and RTS

Recall that a homogeneous function f of degree k is given by: \\[ f(t K, t L) = t^k f(K, L) \\] RTS is given by: \\[ RTS = \frac{MPK}{MPL} \\]
10

Show RTS is independent of scale of operation

Using the fact that MPK and MPL are homogeneous of degree k-1: \\[ RTS = \frac{t^{k-1}MPK(K, L)}{t^{k-1}MPL(K, L)} = \frac{MPK(K, L)}{MPL(K, L)} \\] As we can see, the factor \(t^{k-1}\) cancels out, which means the RTS is independent of the scale of operation.
11

Connect to homotheticity

Since the RTS is independent of the scale of operation, all isoquants are radial expansions of the unit isoquant, which means the production function is homothetic. e. Show that the results from part (d) apply to any monotonic transformation of a homogeneous function. That is, show that any such transformation of a homogeneous function is homothetic.
12

Define a monotonic transformation

Consider a monotonic transformation g(f(K, L)) of the homogeneous production function f(K, L), where g is a monotonic function.
13

Apply the transformation to the homogeneous function

Apply g to the production function f(tK, tL): \\[ g(f(tK, tL)) = g(t^k f(K, L)) \\]
14

Show that the transformed function is homothetic

Given that g is monotonic, it maintains the same order of the arguments of the homogeneous function f. Since f(tK, tL) = t^k f(K,L) and the order is preserved under the monotonic transformation, g(f(tK, tL)) also satisfies the condition for homotheticity. Hence, any monotonic transformation of a homogeneous function is homothetic.

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Most popular questions from this chapter

As in Problem \(11.8,\) again use Euler's theorem to prove that for a constant returns-to-scale production function with only two inputs \((K \text { and } L), /^{\wedge}\) must be positive. Interpret this result.

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