assume that a business firm
produces two commod
ities, X and Y,
with two different inpu
ts, Labour (L) and Capital (K). The total
quantities of L and K available per un
it of time are specified as L = 1600
labour hours; and K = 20
00 units. In addition,
assume that producing 1
unit of commodity X requires 4 units
of labour (L) and 2 units of capital
(K). One unit of commodity Y requires
2 units of L and
5 units of K.
Profits per unit of co
mmodities X and Y are estimated at N10 and N8,
respectively.
To find the profit-maximizing quantities of commodities X and Y that the business firm should produce, we'll use the concept of marginal productivity and marginal cost.
Step 1: Calculate the marginal product of labor (MPL) and marginal product of capital (MPK) for both commodities X and Y.
The MPL is the additional output produced by an additional unit of labor, and the MPK is the additional output produced by an additional unit of capital.
For commodity X:
MPL(X) = Change in output of X / Change in labor input
= 1 unit of X / 4 units of labor
= 1/4
MPK(X) = Change in output of X / Change in capital input
= 1 unit of X / 2 units of capital
= 1/2
For commodity Y:
MPL(Y) = Change in output of Y / Change in labor input
= 1 unit of Y / 2 units of labor
= 1/2
MPK(Y) = Change in output of Y / Change in capital input
= 1 unit of Y / 5 units of capital
= 1/5
Step 2: Calculate the marginal cost (MC) of producing each commodity.
The MC is the additional cost incurred by producing an additional unit of output.
For commodity X:
MC(X) = Cost of labor / MPL(X)
= Cost of labor / (1/4)
= Cost of labor * 4
For commodity Y:
MC(Y) = Cost of labor / MPL(Y)
= Cost of labor / (1/2)
= Cost of labor * 2
Step 3: Determine the profit per unit of each commodity.
Profit(X) = Price of X - MC(X)
= N10 - MC(X)
Profit(Y) = Price of Y - MC(Y)
= N8 - MC(Y)
Step 4: Find the profit-maximizing quantities of X and Y.
To maximize profit, the firm should produce where the profit per unit is maximized.
Set Profit(X) = Profit(Y) and solve for the quantities of X and Y.
N10 - MC(X) = N8 - MC(Y)
N10 - (Cost of labor * 4) = N8 - (Cost of labor * 2)
Solve for Cost of labor, which represents the total cost incurred by the firm.
Cost of labor = (N10 - N8) / 2
= N1
Substitute the value of Cost of labor back into the MC equations to find MC(X) and MC(Y).
MC(X) = N1 * 4
= N4
MC(Y) = N1 * 2
= N2
Finally, substitute the MC values into the profit equations to find the profit per unit of X and Y.
Profit(X) = N10 - N4
= N6
Profit(Y) = N8 - N2
= N6
Therefore, to maximize profit, the firm should produce equal quantities of commodities X and Y, each with a profit per unit of N6.