Given production function ,Y=X^a in maximum profit .
a) The factor demand function
b)The supply function. Show your working clearly
To determine the factor demand function in the production function Y = X^a, we need to consider the profit maximization condition, which states that firms maximize their profit by equating marginal revenue product (MRP) to the factor price.
a) Factor Demand Function:
In this case, the factor demand function represents the amount of factor input (X) that the firm will employ to maximize profit at a given factor price.
To find the factor demand function, we need to calculate the MRP and equate it to the factor price.
Step 1: Calculate the marginal product of the factor (MP):
MPX = dY/dX = a * X^(a-1)
Step 2: Calculate the marginal revenue product (MRP):
MRP = MPX * MR
Here, MR represents the marginal revenue (additional revenue generated from selling one more unit of output), which depends on market conditions and demand elasticity.
Step 3: Set MRP equal to the factor price (PF) to find the factor demand function:
MRP = PF
a * X^(a-1) * MR = PF
The resulting equation represents the factor demand function for factor input X.
b) Supply Function:
To determine the supply function in the production function Y = X^a, we need to consider the cost minimization condition. Firms minimize their cost by equating the marginal cost (MC) to the factor price (PF).
Step 1: Calculate the average cost (AC) of production:
AC = TC / Y
Here, TC represents total cost, which depends on the cost of factor input (X) and the factor price (PF).
Step 2: Calculate the marginal cost (MC):
MC = dTC/dY = dTC/dX * dX/dY = (PF * X) / Y
(This step utilizes the implicit differentiation approach, taking into account that Y = X^a)
Step 3: Set MC equal to the factor price (PF) to find the supply function:
MC = PF
(PF * X) / Y = PF
By rearranging the equation and substituting Y with X^a, we can solve for X. This will provide the supply function, representing the quantity of factor input X that firms are willing to supply at a given factor price PF.
Note: The specific values of MR, PF, and a will depend on the market conditions and demand elasticity, and need to be provided to obtain precise numerical calculations for the factor demand and supply functions in this production function.