Question

A short run cost function for an entrepreneur is q³- 8q² + 30q + 60. Determine the price at which the entrepreneur cases production in an ideal market. Also, derive the supply function.

Answer 1

$$C(q)=q^3-8q^2+30q+60$$

$$MC=3q^2-16q+30$$ $$MR=MC=>p=3q^2-16q+30$$ $$p=3(q^2-2(\dfrac{8}{3})q+\dfrac{64}{9})-\dfrac{64}{3}+30$$ $$\dfrac{p}{3}=(q-\dfrac{8}{3})^2+\dfrac{26}{9}$$ $$(q-\dfrac{8}{3})^2=\dfrac{p}{3}-\dfrac{26}{9}$$ $$q-\dfrac{8}{3}=\sqrt{\dfrac{p}{3}-\dfrac{26}{9}}$$

Supply function

$$q=\sqrt{\dfrac{p}{3}-\dfrac{26}{9}}+\dfrac{8}{3}$$