Assume that a profit maximizing monopolist faces an inverse demand function give
ID: 1252204 • Letter: A
Question
Assume that a profit maximizing monopolist faces an inverse demand function given by p(), and a total cost function given by c(y). Suppose the government wishes to combat the undesirable allocational effect of a monopoly through the use of a subsidy.d) Show that if the per unit subsidy is chosen so that s = -yp'(y) where y is the output produced by an archetype competitive industry, then the monopolist will produce the same output that the competitive industry would produce.
No clue how to do this...help is greatly appreciated! Thanks.
What I do have is that, and what I THINK is correct is the first order necessary condition which is p'(y) - c'(y) + s = 0. Is this correct? If so, would I plug in the s given above? If so, how would I prove the outputs are equal???
Explanation / Answer
Assume that a profit maximizing monopolist faces an inverse demand function given by p(y), and a total cost function given by c(y). Suppose the government wishes to combat the undesirable allocational effect of a monopoly through the use of a subsidy.
d) Show that if the per unit subsidy is chosen so that s = -yp'(y) where y is the output produced by an archetype competitive industry, then the monopolist will produce the same output that the competitive industry would produce.
supply = -yp'(y) is the same thing as
supply = -(price)(slope of price)
Thus, is supply is determined by price and the slope of the price (the rate of change for the price), the price of the monopolist wull be the same as the price of the competitive industry - and so, the monopolist will sell the same output as the competitive industry.
No clue how to do this...help is greatly appreciated! Thanks.
What I do have is that, and what I THINK is correct is the first order necessary condition which is p'(y) - c'(y) + s = 0. Is this correct? If so, would I plug in the s given above? If so, how would I prove the outputs are equal??? No, you just break the equation up into its parts and realize that the supply function is now the same as the competitors', because a perfect competitor sells only as much as the market price will allow the people to demand. This is that same function.
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