Question

If a demad function of a firm is Qd= 200-0.5p+4M+60Pr where M(income) is 100 and Pr(price of related good) is 4.TC=50-0.3Q+0.01Q^2. What is thw profit maximizing level of output

Answer 1

$Q=200-0.5P+4(100)+60(4)\\Q=840-0.5P\\$

$TR=P\times Q\\\\TR=(840-0.5P)P\\TR=840P-0.5P^2$

$TC=50-0.3Q+0.01Q^2\\TC=50-0.3(840-0.5P)+0.01(840-0.5P)^2\\TC=50-252-0.15P+0.01(705600-840P+0.25P^2)\\TC=50-252-0.15P+7056-8.4P+0.0025P^2\\TC=6854-8.55P+0.0025P^2$

Profit is maximized at a point where MR=MC

$MR=840-P\\MC=-8.55+0.005P$

$MR=MC\\840-P=-8.55+0.005P\\840+8.55=1.005P\\P=\frac{848.55}{1.005}\\P=844.3283582$

$Q=840-0.5P\\Q=840-0.5(844.3283582)\\=840-422.1641791\\=417.8358209\\\approx417\space units$