Marginal and Average Products (optional): (a) Consider a production function q = f(KL) such that its marginal product of labor (MPL), equals to the average product of labor… What function is this? (b) What production function would let MPL be proportion to the average product of labor i.e. 1 = L for some constant az? (c) What production function would let marginal product of capital (MPK) be propor- tion to the average product of capital i.e. = ax for some constant ax? (d) Show that the Cobb-Douglas production function q = S(KL) = AKK LoL satisfies these three properties: i. Zero output when one lacks any of the inputs: q = 0 if K = 0 or L = 0. ii. MPL is proportion to the average product of labor: iii. MPK is proportion to the average product of capital: (e) Show that the Cobb-Douglas production function is the only production function that satisfies the above three properties.
The correct answer and explanation is:
(a) Production function where MPL equals the Average Product of Labor (APL):
The marginal product of labor (MPL) is the derivative of output with respect to labor (L), and the average product of labor (APL) is output divided by the number of units of labor.
Given the condition that MPL = APL, this implies: ∂q∂L=qL\frac{\partial q}{\partial L} = \frac{q}{L}
If the production function is q=f(K,L)q = f(K, L), we can assume that the production function has the form q=A⋅Ka⋅Lbq = A \cdot K^a \cdot L^b for some constants AA, aa, and bb.
Taking the partial derivative of qq with respect to LL: MPL=∂q∂L=A⋅Ka⋅b⋅Lb−1MPL = \frac{\partial q}{\partial L} = A \cdot K^a \cdot b \cdot L^{b-1}
The average product of labor is: APL=qL=A⋅Ka⋅Lb−1APL = \frac{q}{L} = A \cdot K^a \cdot L^{b-1}
Equating MPL and APL: A⋅Ka⋅b⋅Lb−1=A⋅Ka⋅Lb−1A \cdot K^a \cdot b \cdot L^{b-1} = A \cdot K^a \cdot L^{b-1}
This simplifies to: b=1b = 1
Thus, the production function is of the form: q=A⋅Ka⋅Lq = A \cdot K^a \cdot L
(b) MPL proportional to APL:
If MPL is proportional to APL, then: MPLAPL=constant\frac{MPL}{APL} = \text{constant}
This implies: ∂q∂LqL=constant\frac{\frac{\partial q}{\partial L}}{\frac{q}{L}} = \text{constant}
For the production function q=A⋅Ka⋅Lbq = A \cdot K^a \cdot L^b, we find: MPLAPL=b1=b\frac{MPL}{APL} = \frac{b}{1} = b
So, for proportionality, b=1b = 1. Therefore, the production function is: q=A⋅Ka⋅Lq = A \cdot K^a \cdot L
(c) MPK proportional to APK:
The marginal product of capital (MPK) is: MPK=∂q∂K=A⋅a⋅Ka−1⋅LbMPK = \frac{\partial q}{\partial K} = A \cdot a \cdot K^{a-1} \cdot L^b
The average product of capital (APK) is: APK=qK=A⋅Ka−1⋅LbAPK = \frac{q}{K} = A \cdot K^{a-1} \cdot L^b
If MPK is proportional to APK, then: MPKAPK=a1=a\frac{MPK}{APK} = \frac{a}{1} = a
Thus, a=1a = 1. The production function is: q=A⋅K⋅Lq = A \cdot K \cdot L
(d) Cobb-Douglas Production Function:
The Cobb-Douglas production function is: q=A⋅Ka⋅Lbq = A \cdot K^a \cdot L^b
- Zero output when no input is available: If either K=0K = 0 or L=0L = 0, q=0q = 0.
- MPL proportional to APL: From earlier calculations, b=1b = 1 ensures that MPL equals APL.
- MPK proportional to APK: Similarly, a=1a = 1 ensures that MPK is proportional to APK.
Thus, q=A⋅K⋅Lq = A \cdot K \cdot L satisfies all the conditions.
(e) Cobb-Douglas as the Only Production Function:
The Cobb-Douglas production function is the only one that satisfies all three properties because it naturally ensures that the marginal products are proportional to the average products through the specific form of the production function q=A⋅Ka⋅Lbq = A \cdot K^a \cdot L^b. Other functional forms would not necessarily lead to these proportional relationships, making Cobb-Douglas the unique solution.