Question

A firm produces two substitute goods . Their demand curves and total cost function are given below : 2 2 P1 = 130 - 401-02 P2 = 160-201-5Q2 TC = 201 +201 O2 + 402 ( a ) Determine the profit maximizing outputs of both goods . ( Restrict your answers to 2 decimal points ) ( b ) Find the price elasticity of demand . ( c ) Evaluate the second order " Hessian " required for profit to be maximized .

Answer 1

profit is maximized at the point MR=Mc

MC wrt Q₁$=2+2Q_2\\$

$TR_1=P_1Q_1=(130-4Q_1-Q_2)Q_1\\=130Q_1-4Q_1^2-Q_1Q_2\\MR_1=130-8Q_1-Q_2$

MR=MC

$130-8Q_1-Q_2=2+2Q_2\\130-2=2Q_2+8Q_1+Q_2\\Q_1=\frac{128-3Q_2}{8}$

$TR_2=P_2Q_2=(160-2Q_1-5Q_2)Q_2\\=160Q_2-2Q_1Q_2-5Q_2^2\\MR_2=160-2Q_1-10Q_2$

MR=MC

$160-2Q_1-10Q_2=2Q_1+4\\160-4-4Q_1=10Q_2\\Q_2=\frac{156-4Q_1}{10}$