Consider the production function: Q = Bo + B1VKL + B2K + B3L where: 0 < B; < 1 (i) If this function were to exhibit constant returns to scale, what restrictions should be placed on the parameters Bo, B1, B2, and B3? (ii) A production function q f(L,K) is homogeneous of degree k if for any t > 0, the following holds: f(tL,tK) = tf(L,K). Show that in the constant returns to scale case, this function is homogeneous of degree 1. (iii) Show that in the constant returns-to-scale case, this function exhibits diminishing marginal productivities and that the marginal productivity functions are homogeneous of degree 0. (iv) In the constant returns-to-scale case, calculate MRT SLK.
The Correct Answer and Explanation is:
We are given the production function:Q=B0+B1KL+B2K+B3LQ = B_0 + B_1 \sqrt{K L} + B_2 K + B_3 LQ=B0+B1KL+B2K+B3L
with the condition: 0<B1<10 < B_1 < 10<B1<1
(i) Restrictions for Constant Returns to Scale (CRS)
A production function exhibits constant returns to scale if multiplying all inputs by a positive constant ttt multiplies output by exactly ttt. That is:f(tK,tL)=tf(K,L)f(tK, tL) = t f(K, L)f(tK,tL)=tf(K,L)
Let us apply this to the given function:f(tK,tL)=B0+B1tK⋅tL+B2(tK)+B3(tL)=B0+B1tKL+tB2K+tB3Lf(tK, tL) = B_0 + B_1 \sqrt{tK \cdot tL} + B_2 (tK) + B_3 (tL) = B_0 + B_1 t\sqrt{KL} + tB_2 K + tB_3 Lf(tK,tL)=B0+B1tK⋅tL+B2(tK)+B3(tL)=B0+B1tKL+tB2K+tB3L
Now compare this with:tf(K,L)=t(B0+B1KL+B2K+B3L)=tB0+tB1KL+tB2K+tB3Lt f(K, L) = t(B_0 + B_1 \sqrt{KL} + B_2 K + B_3 L) = tB_0 + tB_1 \sqrt{KL} + tB_2 K + tB_3 Ltf(K,L)=t(B0+B1KL+B2K+B3L)=tB0+tB1KL+tB2K+tB3L
These are equal only if B0=0B_0 = 0B0=0. Hence, for constant returns to scale, we must have:
- B0=0B_0 = 0B0=0
There are no further restrictions on B1,B2,B3B_1, B_2, B_3B1,B2,B3 as long as they are constants and not functions of ttt.
(ii) Homogeneity of Degree 1
To prove homogeneity of degree 1, we again test:f(tK,tL)=tf(K,L)f(tK, tL) = t f(K, L)f(tK,tL)=tf(K,L)
From above:f(tK,tL)=B1tKL+tB2K+tB3L=t(B1KL+B2K+B3L)=tf(K,L)f(tK, tL) = B_1 t\sqrt{KL} + tB_2 K + tB_3 L = t(B_1 \sqrt{KL} + B_2 K + B_3 L) = t f(K, L)f(tK,tL)=B1tKL+tB2K+tB3L=t(B1KL+B2K+B3L)=tf(K,L)
This confirms the function is homogeneous of degree 1 in the constant returns to scale case (with B0=0B_0 = 0B0=0).
(iii) Diminishing Marginal Product and Degree 0 Homogeneity of MP
Marginal Product of Labor (MPL):∂Q∂L=B1⋅12⋅K1/2L1/2+B3\frac{\partial Q}{\partial L} = B_1 \cdot \frac{1}{2} \cdot \frac{K^{1/2}}{L^{1/2}} + B_3∂L∂Q=B1⋅21⋅L1/2K1/2+B3
MPL diminishes with respect to LLL since K1/2L1/2\frac{K^{1/2}}{L^{1/2}}L1/2K1/2 decreases as LLL increases. So, MPL is diminishing.
Similarly, Marginal Product of Capital (MPK):∂Q∂K=B1⋅12⋅L1/2K1/2+B2\frac{\partial Q}{\partial K} = B_1 \cdot \frac{1}{2} \cdot \frac{L^{1/2}}{K^{1/2}} + B_2∂K∂Q=B1⋅21⋅K1/2L1/2+B2
This decreases as KKK increases, so MPK also diminishes.
To check homogeneity of degree 0, we evaluate:MPL(tL,tK)=B1⋅12⋅(tK)1/2(tL)1/2+B3=B1⋅12⋅K1/2L1/2+B3=MPL(L,K)\text{MPL}(tL, tK) = B_1 \cdot \frac{1}{2} \cdot \frac{(tK)^{1/2}}{(tL)^{1/2}} + B_3 = B_1 \cdot \frac{1}{2} \cdot \frac{K^{1/2}}{L^{1/2}} + B_3 = \text{MPL}(L, K)MPL(tL,tK)=B1⋅21⋅(tL)1/2(tK)1/2+B3=B1⋅21⋅L1/2K1/2+B3=MPL(L,K)
So MPL and MPK are homogeneous of degree 0.
(iv) Marginal Rate of Technical Substitution (MRTSLK_{LK}LK)
MRTS is the negative ratio of marginal products:MRTSLK=−∂Q/∂L∂Q/∂K=−B12⋅K1/2L1/2+B3B12⋅L1/2K1/2+B2\text{MRTS}_{LK} = -\frac{\partial Q/\partial L}{\partial Q/\partial K} = -\frac{\frac{B_1}{2} \cdot \frac{K^{1/2}}{L^{1/2}} + B_3}{\frac{B_1}{2} \cdot \frac{L^{1/2}}{K^{1/2}} + B_2}MRTSLK=−∂Q/∂K∂Q/∂L=−2B1⋅K1/2L1/2+B22B1⋅L1/2K1/2+B3
This expression shows how many units of capital can replace one unit of labor while keeping output constant. It depends on the ratio of KKK to LLL, not their absolute levels, consistent with CRS.
Summary:
- CRS requires B0=0B_0 = 0B0=0
- The function is homogeneous of degree 1 under CRS
- It shows diminishing marginal productivity
- Marginal products are homogeneous of degree 0
- MRTS is based on the ratio K/LK/LK/L and is consistent with standard production theory
