Answer to Question #165562 in Math for Joseph

Question

The demand for a commodity is given by p = 400 q. The average total cost of producing the commodity is given by 

 

where p is the price in shillings and q is the quantity in kilograms. 

Required

What does  in the ATC equation represent economically?              (1 mark)

Determine the output that leads to maximum profit and the profit at the level of output.   (9 marks)

Alpha industries sells two products, X and Y, in related markets, with demand functions given by:

Px 13 + 2X + Y = 0

Py 13 + X + 2Y = 0

The total cost, in shillings, is given by:

TC = X + Y

Required:

Determine the price and the output for each good which will maximize profits.        

Answer 1

b) The demand for a commodity is given by $p=400-q.$ The average total cost of producing the commodity is given by 

$$ATC=\dfrac{1000}{q}+100-5q+q^2$$

i) $\dfrac{1000}{q}$ in ATC represents the average fixed cost AFC.

ii) The output that leads to maximum profit is:

$MR=MC$

$$MR=(TR)'=(p\cdot q)'$$

$$=((400-q)q)'=400-2q$$

$$MC=(TC)'=(ATC\cdot q)'$$ $$=((\dfrac{1000}{q}+100-5q+q^2)q)'$$

$$=(1000+100q-5q^2+q^3)'=100-10q+3q^2$$

$$400-2q=100-10q+3q^2$$

$$3q^2-8q-300=0$$

$D=(-8)^2-4(3)(-300)=3664$

$$q=\dfrac{8\pm\sqrt{3664}}{6}=\dfrac{4\pm2\sqrt{229}}{3}$$

Since $q>0,$ we take

$$q=\dfrac{4+2\sqrt{229}}{3}\approx11.422$$

The profit at the level of output

$$TP=(p-ATC)q$$

$$=400q-q^2-1000-100q+5q^2-q^3$$

$$=300q+4q^2-1000-q^3$$

$$TP=300(11.422)+4(11.422)^2-1000-(11.422)^3$$

$$=1458.31$$

c) The profit-maximizing condition is $MR=MC$

$$MC_X=(X+Y)'_X=1$$

$$MC_Y=(X+Y)'_Y=1$$

$$MR_X=(X(13-2X-Y))'_X=13-4X-Y$$

$$MR_Y=(Y(13-X-2Y))'_Y=13-X-2Y$$

$$13-4X-Y=1$$

$$13-X-4Y=1$$

$$X=Y$$

$$13-4X-X=1$$

$$X=2.4$$

$$Y=2.4$$

$$P_X=13-2(2.4)-2.4=5.8$$

$$P_X=13-2.4-2(2.4)=5.8$$