Chapter 14.4, Problem 38E

Fast Food A fast-food outlet finds that the demand equation for its new side dish, “Sweetdough Tiidbit,” is given by

$p=\frac{128}{{\left(q+1\right)}^2},$ where p is the price (in cents) per serving and q is the number of servings that can be sold per hour at this price. At the same time, the franchise is prepared to sell $q=0.5p-1$ servingsper hour at a price of p cents. Find the equilibrium price $\bar{p}$ and the consumers’ and producers’ surpluses at this price level. What is the total social gain at the equilibrium price? [HINT: See Example 3.]