Profit-maximizing price for kangaroo steaks

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The company EleFanti produces kangaroo steaks. The production cost is 20\,\$$ each. At a price of 30,$$zoos buy$500$; if the price rises to$50,$$they buy only$300$. Find the price that maximizes EleFanti’s profit and compute that maximum profit.

Solution

Linear demand curve $q=mp+b$. From the two points $(30,500)$ and $(50,300)$: $m=\frac{300-500}{50-30}=-10,\qquad 500=-10\cdot 30+b\Rightarrow b=800\quad\Rightarrow\quad q=800-10p.$ Profit = (price − cost) × quantity: $\pi(p)=(p-20)(800-10p)=-10p^2+1000p-16000.$ A downward parabola; the maximum is at the vertex: $p^\ast=-\frac{1000}{2\cdot(-10)}=50\,\$.$ Quantity sold $q=800-10\cdot 50=300$; profit $\pi_{\max}=(50-20)\cdot 300=9000\,\$.$

$\boxed{p^\ast=50\,\$,\quad \pi_{\max}=9000\,\$\ (q=300)}$