Suppose you are given the following production function:
,
where y is output and K is capital.
y= 60K + 20.3K2- K3
1.1 what is a production function, what is the real work application of such, and where would you source the data to develop a production of the type given above.
1.2 Derive the average product of capital and the marginal product of capital.
1.3 How would you distinguish the equation above from a constant elasticity production function, a Cobb-Douglas production function, and a Leontief production function?
1.4 Finally, derive the points K and Y in which the MP of capital is equal to the minimum (or maximum) point on the average production curve.
1.1 A production function is a mathematical representation of the relationship between inputs and outputs in the production process. It shows how much output can be produced using various combinations of inputs, such as labor, capital, and technology. The real-world applications of production functions are vast and can be found in industries ranging from manufacturing to services. They help firms understand how to allocate their resources effectively to maximize their production.
To develop a production function like the one given above, you would typically source data from empirical studies or data sets collected by economic research institutes or government agencies. Data on capital stock, labor input, and corresponding levels of output can be used to estimate the parameters of the production function. This data can often be obtained from official statistical sources, business surveys, or academic research databases.
1.2 The average product of capital (APK) measures the average amount of output produced per unit of capital. It is calculated by dividing total output (y) by the amount of capital (K) used:
APK = y / K
The marginal product of capital (MPK) measures the additional output that is generated when an additional unit of capital is used. It is obtained by taking the derivative of the production function with respect to capital (K):
MPK = d(y) / d(K)
To derive the APK and MPK from the given production function, you would need to differentiate the function with respect to K. This would give you the formulas for calculating the APK and MPK.
1.3 The given production function is a general form without any specific functional form mentioned. However, we can differentiate it from different types of production functions:
- Constant Elasticity Production Function: Constant Elasticity (CES) production functions have a constant elasticity of substitution between inputs. They can be represented in the form: y = A(Kα + Lβ)γ, where A is a constant, K and L represent capital and labor inputs respectively, and α, β, and γ are elasticity parameters. The given production function does not fit this form as it does not have the same type of equation structure.
- Cobb-Douglas Production Function: Cobb-Douglas production functions have the form: y = A(K^α)(L^β), where A is a constant, K and L represent capital and labor inputs, and α and β are exponents that determine the output elasticities. The given production function does not match this form exactly but is similar with the addition of a third degree term (-K^3).
- Leontief Production Function: Leontief production functions have the form: y = min(αK, βL), where α and β are constants that represent the fixed proportions in which inputs are combined in production. In the given production function, there is no fixed proportionality between capital and output, so it cannot be classified as a Leontief production function.
1.4 To find the points where the MP of capital is equal to the minimum or maximum point on the average production curve, you would first need to calculate the derivative of the average product of capital (APK) with respect to capital (K):
d(APK) / d(K) = 0
This equation represents the condition where the MPK is equal to the marginal value of APK. By solving this equation, you can find the values of K and corresponding y where this condition is met. The resulting points would give you the minimum or maximum points where the MP of capital is equal to the APK.