Question

A monopolist w11ishes to maximize total revenue. She produces two outputs, (x1, x2) and faces the following demands for her products,

X1 = 20 – 2p1, and

X2 = 20 – 4p2

Where p1 and p2 are, respectively, the prices of the two goods.

To produce one unit of x1 the monopolist must use one unit of land and one unit of capital. And, to produce one unit of x2 requires two units of land and one unit of capital. The firm has available 10 units of land and 6 units of capital.

Specify the firm’s short-run maximization problem.

Set up the Kuhn-Tucker conditions for maximization (you do not need to solve).

Assume that the solution is x*1 = 5 1/3 (i.e. 16/3) and x*2 = 2/3. Explain which constraints are binding and whether the Lagrange multipliers are positive or zero and what they mean.

Answer 1

Given:

$X_1 = 20 – 2p_1 \\ p_1 = 10 - 0.5X_1\\ X_2 = 20 – 4p_2\\ p_2 = 5 - 0.25X_2$

To produce x₁ the monopolist must use = one unit of land and one unit of capital.

To produce one unit of x₂ requires = two units of land and one unit of capital. 

Land (L) = 10 units

Let cost of land be C_L 

Capital (K) = 6 units

Let cost of land be C_K

Part a

Total Revenue of good $X_1 (TR1) = P_1 X_1$

$TR1 = 10X_1 - 0.5X_1^2$

Total Revenue of good $X_2(TR2) = P_2 X_2$

$TR2 = 5X_2 - 0.5X_2^2$

$Total\space Revenue (TR) = TR1+ TR2\\ TR = 10X_1 - 0.5X_1^2 + 5X_2 - 0.25X_2^2$

When the land and labor is divided amongst good 1 and good 2.

Good 1 uses 2 units of land and 2 units of capital

Good 2 uses 8 units of land and 4 units of capital

Thus, 2 units of x₁ is produced and 4 units of x₂ is produced.

In this way the inputs are fully exhausted.

$Total \space cost\space for\space x_1 (TC1) = 2C_L + 2C_K\\ Total \space cost\space for\space x_2 (TC2) = 8C_L + 2C_K\\ Total \space Cost (TC) = TC1+ TC2\\ TC =2C_L + 2C_K + 8C_L + 2C_K\\ TC = 10C_L + 4C_K$

The firm’s short-run maximization problem: 

$TR = 10\times 2 - 0.5\times 22 + 5\times 4 - 0.25\times 4^2 = 34\\ Profit = TR - TC\\ Profit = 34 - 10C_L - 4C_K$

part b.

$Max\space TR = 10X_1 - 0.5X_1^2 + 5X_2 - 0.25X_2^2\\ Subject \space to:\\ L≤10\\K≤6$

The Kuhn-Tucker conditions for maximization:

$KT = 10X_1− 0.5X_1^2 + 5X_2 − 0.25X_2^2−λ_1 (L−10)−λ_2 (K−6)$

part c.

$x^*_1 = \frac{16}{3}\\ and\\ x^*_2 = \frac{2}{3}.$

Constraints L and K both are binding as there is a perfect mix of them and they can be used together in the quantities given. Units of L is 10 and units of capital is 6. 

The Lagrange multipliers are positive and here it is used to find local maxima in this case.