Question

Consider a firm producing two goods, good A and good B, by using a fixed amount of labor. The production possibilities set of the firm is given by Y = {(a,b) ∈ R2 + | a2 +4b2 ≤ 16}. Assume the price of A and the price of B are equal to $1. Solve the revenue maximizing level of outputs for the firm.

Answer 1

The firm is producing two goods, good A and good B, by using a fixed amount of labor.

The production possibilities set of the firm is given by

Y= {(a,b) ∈ R2 + | a2 +4b2 ≤ 16}

Here, a = Output of good A and b = Output of good B.

Hence, this is a concave PPF or Production Possibility Frontier. The PPF line is

........(1)

Now, the prices of a and b are both $1. Hence,

Pa = $1 and Pb = $1.

Hence, Total Revenue is

TR = a.Pa + b.Pb

or, TR = a.1 + b.1

or, TR = a + b.........(2)

Hence, the maximization problem is the following,

Max {a + b} subject to

Hence, the Lagrange's equation for the problem is

L = (a + b) + k.{}

where, k>0

Now, the first order conditions or FOCs are,

dL/da = 0

or, 1 - k.(2.a) = 0

or, k.(2.a) = 1.........(3)

dL/db = 0

or, 1 - k.(8.b) = 0

or, k.(8.b) = 1...........(4)

And,

dL/dk = 0

or, = 0

or, .........(5)

Now, deviding equation (1) with equation (2) we get

{2.k.a}/{8.k.b} = 1

or, a/4.b = 1

or, a = 4.b..........(6)

Now putting a = 4b in equation (5) we get

or,

or,

or, = 4/5

or, b* = 2/√5

And, putting b* in equation (6) we get

a* = 4.b* = 4×(2/√5)

or, a* = 8/√5

a* and b* are the revenue maximizing outputs of good A and good B respectively for the firm.

Hence, revenue maximizing output of good A is

a* = 8/√5.

And, revenue maximizing output of good B is

b* = 2/√5.

Hope the solution is clear to you my friend.