Homework Soursethe demand equation for a monopolist is p2003x and the cost
the demand equation for a monopolist is p=200-3x and the cost function is C(x)=75 + (80 + T)x - x^2, where x is between 0 and 40 and T is a govt\' imposed tax. What is the value of x that maximizes the monopolists profit as a function of T? What is the tax revenues received by the govt\' as a function of T? what value of T will maximize the tax revenue received by the govt\'?
Solution
the demand equation for a monopolist is p=200-3x and the cost function is C(x)=75 + (80 + T)x - x^2, where x is between 0 and 40 and T is a govt\' imposed tax. What is the value of x that maximizes the monopolists profit as a function of T?
profit=r=pC(x)=(200-3x)(75+(80+T)x-x^2) where x is between 0 and 40
r=15000-225x+200(80+T)x-3(80+T)x^2-200x^2+3x^3
r\' ==-225+16000x+200T-6x(80+T)-400x+9x^2=0
14875+200T+15180x-880x-6xT-9x^2=0
x=(-b±(b^2-4ac))/2a=(-6T-880±((-6T-880)^2-4(-18)(15875+200T)))/2(15875+200T) where x is between 0 and 40
What is the tax revenues received by the govt\' as a function of T? T is tax.
what value of T will maximize the tax revenue received by the govt\'?
