Question

In this assignment, you will need to come up with a production function for a firm. 1. First, you will have to determine what kind of firm you want to be and what product you will (hypothetically) produce. 2. Next, you will have to describe the production function of your firm. Naturally production processes are very complex, so you might have to aggregate inputs in 2 categories, capital and labor. These are things you should keep in mind when choosing your specification: a. Does this production process exhibit increasing, decreasing or constant returns to scale? b. Does your production process require both inputs to be used in a fixed ratio? Can you substitute capital and labor for each other at a fixed ratio? Will the rate at which you can substitute them depend on how much you are already using? In other words, do you think your production process should be described by a Leontief production function, perfect substitutes production function or Cobb-Douglas production function? c. Does your production process have a higher marginal product of labor or marginal product of capital?

Answer 1

Answer to this question number 1:

Let us consider the case of an agricultural firm to explain the possible characteristics of its production function. Let us further assume that this agricultural firm produces only rice.

Answer to the question number 2:

Let us suppose that the production function of the agricultural firm takes the following mathematical form:

R=AK^αL^β

Where R is the Output (that is rice), L is the quantity of labour employed; K is the quantity of capital employed, A is the technological coefficient (or in other words, Total factor productivity). And α, β are positives.

As we know, in the production process of an agricultural product (say rice) needs many inputs such as: labour, land, tractors, seeds, fertilizers etc. in this case, land, tractors, fertilizers etc. could be included under the broad heading of capital.

A]. In our production function, the sum of the exponents of the inputs (that is α+β) measures the returns to scale. As the production function mentioned above exhibits the property of linear homogeneous production function; therefore, when each input of that production function is increased by a constant factor; say, λ; output R also increases to λ^α+β. Therefore the degree of returns to scale in this production function is strictly depends on the fact that whether the sum of the exponents are greater than or equal to or less than one.

If α+β=1, returns to scale are constant.

If α+β>1, returns to scale are increasing.

If α+β<1, returns to scale are decreasing.

B]. As the production of rice needs the use of both inputs simultaneously, therefore, the production will be nil or not possible with only a single input of production. Moreover, the exponents of the inputs; α and β represents the relative share of capital and labour in output. However, this does not mean that both the inputs have to be used in fixed ratios. But yes, these inputs could be substituted for one another in the production process if required.

We know that the marginal rate of technical substitution (MRTS) of a production function depends on the ratios at which the inputs are used or in other words, MRTS of a production function is different at different factor proportions used in production of a good. Thus, we can say that the rate at which factors are substituted depends on the how much the inputs are already being used. Finally, the Cobb-Douglas Production function could be used to describe this production function.

C]. With this production function; it will not be very easy to say whether the capital will have higher marginal product or the labour. This is because the marginal product of any specific input of this production function not only depends on Output (R) but also on its exponents and quantity. Algebraically; in this hypothetical production function, Marginal Product of labour will be β (Q/L) and the Marginal Product of capital will be α (Q/K).

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