Question

1.      Assume that consumer A consumes two products Q₁ and Q₂  when the first product price increases, the consumer decided to decrease for the second product? Explain

2.      Given that the profit function for a firm p = 150x + 4x² – xy – 9y2 + 100y + 2x + y = 50 with a constraint of 2x + y = 50. Determine the amount of x and y which maximizes profit. A. Using the substitution method  B. Using the language method

Answer 1

1.

the two products are complements. When the price of Q₁ increased, the consumer was forced to decrease the consumption of product Q₂

2.

A. Substitution method

$y=50-2x$

substituting into the profit function we get

$150x+4x^2-50x-2x^2 -22500+1800=36x^2+5000-200x+2x+50-2x=50$

$38x^2-100x-15700=0$

$x=21.68461851$

$y=6.63076298$

B. Lagrange method

$\frac{\delta f}{\delta x} =\frac{\delta g}{\delta x} \lambda , \frac {\delta f}{\delta y} =\frac {\delta g}{\delta y}\lambda$

$\frac{\delta g}{\delta x}=2\lambda, \frac{\delta g}{\delta y} =\lambda$

$g(x,y) =2x+y=50$

$2(2y)+y=50$

$4y+y=50$

$5y=50$

$\frac{5y}{5}=\frac{50}{5}$

$y=10$

$2x-y=50$

since y=10 we substitute with 10

$2x+10=50$

$2x=50-10$

$2x=40$

$\frac{2x}{2}=\frac{40}{2}$

$x=20$