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^{k} f(K, L) \\] a. Show that if a production function is homogencous of degree \(k\), its marginal productivity 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 production function depends only on the ratio \(K / L\) d. More generally, show that the \(R\) TS 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. 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 homogencous function is homothetic.

Step by step solution

01

Calculate the Marginal Productivities

To calculate the marginal productivities of a production function f(K, L), we need to find the partial derivatives of f(K, L) with respect to K and L: 1. Marginal Product of Capital (MPK) = \(\frac{\partial f(K, L)}{\partial K}\) 2. Marginal Product of Labor (MPL) = \(\frac{\partial f(K, L)}{\partial L}\)

02

Prove that Marginal Productivity Functions are Homogeneous of Degree k-1

Now, let's find the marginal productivities of the production function \(f(tK, tL)\) and show that they are homogeneous of degree k-1. 1. MPK of \(f(tK, tL)\) = \(\frac{\partial f(tK, tL)}{\partial K}\) 2. MPL of \(f(tK, tL)\) = \(\frac{\partial f(tK, tL)}{\partial L}\) We know that f(tK, tL) = \(t^k * f(K, L)\). Now let's differentiate f(tK, tL) with respect to K and L. 1. \(\frac{\partial f(tK, tL)}{\partial K} = \frac{\partial (t^kf(K, L))}{\partial K} = t^{k-1} \frac{\partial f(K, L)}{\partial K}\) 2. \(\frac{\partial f(tK, tL)}{\partial L} = \frac{\partial (t^kf(K, L))}{\partial L} = t^{k-1} \frac{\partial f(K, L)}{\partial L}\) Thus, the marginal productivity functions are homogeneous of degree k-1. #b. Marginal Productivities Based on K/L Ratio#

03

Introduce the Ratio K/L

We know that for a constant returns-to-scale (CRS) production function, the degree of homogeneity is 1. So, for the CRS production function, the marginal productivities are homogeneous of degree 0. This means that the marginal productivities of a CRS production function are independent of K and L and depend on the ratio K/L.

04

State the Result

The marginal productivities for any constant returns-to-scale production function depend only on the ratio K/L. #c. RTS Depending on K/L Ratio#

05

Define the Returns to Scale

Returns to scale (RTS) is defined as: RTS = \(\frac{f(tK, tL)}{f(K, L)}\) Since we know from part (b) that the marginal productivities for a CRS production function depend only on the ratio K/L, the RTS will also depend only on the ratio K/L.

06

State the Result

The returns to scale (RTS) for a constant returns-to-scale production function depend only on the ratio K/L. #d. Homothetic Functions#

07

Prove the Homogeneity of RTS

For a homogeneous production function of degree k, we proved in part (a) that its marginal productivity functions are also homogeneous. This means that the RTS is also homogeneous. Since all isoquants are radial expansions of the unit isoquant, such a function is homothetic.

08

State the Result

The returns to scale (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. #e. Homothetic Monotonic Transformation#

09

Define a Monotonic Transformation

A monotonic transformation is a function M: R → R that is either entirely non-increasing (i.e., M'(x) <= 0 for all x) or non-decreasing (i.e., M'(x) >= 0 for all x).

10

Prove that Homogeneous Function's Monotonic Transformation is Homothetic

Let's consider a homogeneous function H(K, L) of degree k and its monotonic transformation M(H(K, L)). We want to show that this transformation is homothetic. Since H(K, L) is homogeneous of degree k, we know that H(tK, tL) = \(t^k * H(K, L)\). Let's apply the monotonic transformation to H(tK, tL): M(H(tK, tL)) = M(\(t^k * H(K, L)\)) Now let's differentiate M(H(tK, tL)) with respect to K and L. Since H(K, L) is a homogeneous function, and the partial derivatives of the transformation (dM/dK and dM/dL) are nonnegative or nonpositive (depending on whether the transformation is increasing or decreasing), the result will be homothetic.

11

State the Result

Any monotonic transformation of a homogeneous function is homothetic.

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