Question

The division manager of an appliance manufacturing firm must decide on the daily production level 9 for an electric mixer. The plant manager where these mixers are produced has reported that daily total fixed costs are $750, while average variable costs (in dollars) are approximately v = 75 - .049 +.0000897 where q is the production level per day. The marketing director reports that the mixers can be sold to wholesale distributors for a price of p = 85-.039 (in dollars per unit), and that the market can absorb all of the plant's output at this price. Given all this information, how should the division manager select q to maximize the profit from the production and sale of these mixers? (Be sure to check your second-order condition.)

Answer 1

Answer

Profit(Pr) = TR - TC

where TC = Total Cost = Total Fixed cost + Total variable cost\

Also, Total Variable cost = AVC*Quantity = vq and TR = Total Revenue = Price*Quantity = pq

=> Profit(Pr) = pq -(750 + vq)

=> Pr = (85 - 0.03q)q - (750 + (75 - 0.04q + 0.00008q²)q)

First order condition :

d(Pr)/dq = 0 => 85 - 0.06q - (75 - 0.08q + 0.00024q²) = 0

=> -0.00024q² + 0.02q + 10 = 0

Solving this we get :

q = 250

Hence In order to maximize profit a manager should select q = 250 units.

SECOND ORDER CONDITION :

Profit will get maximized if at this critical value(q = 250) second derivative of Pr will q is negative i.e. lesser than 0

Pr = (85 - 0.03q)q - (750 + (75 - 0.04q + 0.00008q²)q)

=> d(Pr)/dq = 85 - 0.06q - (75 - 0.08q + 0.00024q²)

$=>\frac{\mathrm{d} \frac{\mathrm{d} Pr}{\mathrm{d} q}}{\mathrm{d} q} = 0.02 - 0.00024q^2$

$=>\frac{\mathrm{d} \frac{\mathrm{d} Pr}{\mathrm{d} q}}{\mathrm{d} q} = 0.02 - 0.00024*250^2 = -14.98 < 0$

Hence it satisfies second order test and hence At q = 250, Its profit will gets maximized.