Solution:

(1) Short Run Production Function:

• In the short run, the technical conditions of production are rigid so that the various inputs used to produce a given outputs are in fixed proportions.

• However, in the short run, it is possible to increase the quantities of one input while keeping the quantities of other inputs constant in order to have more output.

• This aspect of the production function is known as the Law of Variable Proportions.

• The short run production function in the case of two inputs, labour and capital with capital as fixed and labour as the variable input can be expressed as Where K refers to the fixed input.

$$Q = f (L,R)$$

• This production function is depicted in Figure 1 where the slope of the curve shows the marginal production of labor.

  

• A movements along the production function shows the increase in outputs as labour increases, given the amount of capital employed $K_1$ , If the amount of capital increases to $K_2$ , at a point of time, the production function Q = f (L, K1 ) shifts upwards to Q = f (L, K2 ), as shown in the figure.

• On the other hand, if labour is taken as a fixed input and capital as the variable input, the production function takes the form,

$$Q= f(KL)$$

• This production function is depicted in Figure 2 where the slope of the curve represents the marginal product of capital.

(2) Long Run Production Function:

• In the long run all inputs are variable. Production can be increased by changing one or more of the inputs. The firm can changes its plants or scale of production.

• Equations (1) and (2) represent the long-run production function. Given the level of technology, a combination of the quantities of labour and capital produces a specified level of output.

• The long run production function is depicted in Figure 3 where the combination of OK of capital and OL of labour produced 100Q.

  

• With the increase in inputs of capital and labour to and , the output increases to 200Q. The long run production function is shown in terms of an isoquant such as 100 Q.

• In the long run, it is possible for a firm to change all to change all inputs up or down in accordance with its scale. This is known as returns to scale.

• The returns to scale are constant when output increases in the same proportion as the increase in the quantities of inputs.

• The returns to scale are increasing when the increased in output is more than proportional to the increase in inputs. They are decreasing if the increase in output is less than proportional to the increase in inputs.

• Let us illustrate the case of constant returns to scale with the help of our production function.

$$Q = (L, M, N, K T2)$$

• Given , if the quantities of all inputs L,M,N,K are increased n-fold the output Q also increases n-fold. Then the production function becomes,

$$nQ = f(nL, nM, nN, nK )$$

• This is known as linear an homogeneous production function, or a homogeneous function of the first degree. If the homogeneous function is of the kth degree, the production function is,

$$n^k Q = f(nL, nM, nN, nK)$$

• If k is equal to 1, it is a case of constant returns to scale; if it is greater than 1, it is a case of increasing returns of scale; and if it is less than 1, it is a case of decreasing returns to scale.