Final answer:
To find the consumers' and producers' surplus at the equilibrium price, determine the equilibrium quantity by equating the demand function to the supply function. The consumer surplus and producer surplus are calculated as the areas of their respective triangular regions on a price-quantity graph. Multiply the result by the actual quantities and the price difference to get the dollar amount.
Step-by-step explanation:
To find the consumers' surplus and producers' surplus at the equilibrium price, we must first determine the equilibrium price itself. To do this, we set the quantity demanded function equal to the quantity supplied function and solve for x. The demand function is p = -0.1x² - x + 30, and the supply function is p = 0.1x² + 2x + 10. Equating the two, we find the equilibrium quantity x.
Once the equilibrium quantity is found, we can substitute it back into either the demand or supply function to obtain the equilibrium price. The consumer surplus is represented by the area above the equilibrium price and below the demand curve. The producer surplus is the area below the equilibrium price and above the supply curve. We calculate the surpluses by finding the area of the respective triangular regions on a price-quantity graph. To get the dollar amount, we need to multiply the units of hundreds by the actual quantities and the price given in the functions.
For example, using the hypothetical scenario from Figure 3.23, if the equilibrium price is $80 and consumers would have been willing to pay $90, the consumer surplus can be calculated as the area of the triangle with a base from 20 to 28 million (in units of hundreds) and a height of $10 (the difference between $90 and $80). The producer surplus would be calculated similarly, using the base from the actual supplied quantity at $45 to the equilibrium quantity and the height as the difference between the equilibrium price and the price the producers would have been willing to accept.