Question

7. Perfectly competitive firm faces P(Q) = P inverse demand curve and its costs are given by a cost function C(Q), assuming that marginal costs are positive. Firm is also taxed at rate t per unit of output. (a) Write down the firm's profit function. Identify the choice variable, and the parameter if the firm maximizes the profit. (b) Write down the FONC for profit maximization. What does this equa- tion solve for? Can you get it explicitly? Discuss. Under what condi- tion does the equilibrium exist? (c) Write down the SOC for profit maximization. Under what condi- tion(s) the profit is indeed maximized? (d) What is the effect of the tax on the equilibrium quantity ? Support your answer by a graph. (e) What is the effect of the tax on the maximized profit?

Answer 1

& , tax = t percent Demand P/Q=P Cost: clo) 2 (9) Profit f(1) = TR - TC TR - PIO)*Q. Sed Te & ccos. = Ploj" a - ccol Choue variable: Quantity cos Parameters: the given values and in & demand of cost function ofà, slope coefficient in demand curre (b) ina typical P= a-ba. (6) FONC: 7 First desivation In Plo) + Prosta - crosso wita. This equation will give us optimal value of a that will maximize profet. sana Cannot get explicitly because the parameters & the true demand & cost functions are not known. Equilibrium exist when 2 = 0 Quero y Plo) + Pio) a = c'os Pro)&= ('lo)-Plos de crol-POO

cho)> Plo) created & P (o) to is needed for equilibrium to exist (positive o). (asoc: 8 2 (20) 22/103HPCOJQ-cros ?ST = P Q + p"(0) + p'rol - C"co). ORLO SA? HOES Prefit is maxinized when d o 2) P70) + P"(0) + Pro) -C"0) LO- (d) due to tax the marginal cost of output & i increased. This shift the cel mare upwards. frott The effect of tax clos on the equilibrium quantity is that of reducing equilibrium output. afce Се » Веломах аваалц | profit will also reduce after imposition of tax cost c(0) inreases because of taxe. This also has a dampening effect on profit