Consumers’ and Producers’ Surplus
a. Find the consumers’ surplus at a price level of p = $15 for the price demand equation p = D(x) = (7500 - 30x)/(300-x). Graph the price demand equation and the price level equation p = $15. What region represents the consumers’ surplus?
b. Find the equilibrium price p and equilibrium demand x for the pair of price-demand equation in a. and for the price supply equation of p = S(x) = 10x / (300 –x). Then find the consumers’ surplus and the producers’ surplus at that equilibrium price.
c. The government places a sales tax of 6% on items purchased by the consumer. What does that do to the equilibrium demand of the pair of price-demand and price supply equations in b.? What change is there in the Consumer's and Producer's Surplus? If the government increased the sales tax above 6% would the equilibrium demand go up or down? Is there a limit to the effect on demand in increasing the tax?
d. Consider again the price-demand equation p = D(x) = (7500-30x)/(300-x) and the price-supply equation p = S(x) = 10x / (300 –x). At what value of x is the sum of the Consumer's and Producer's surplus the maximum? Give a reason for your answer. Hint: identify the graphed regions of the consumer and producer surplus for different values of x. At what value of x does it appear that the sum is a maximum? Can you verify your answer using the first and/or second derivative tests?