Applications of the exact integral. Plan: 1 Integral Definition 2

consumer surplus.
As long as the price stays on the demand function curve, a lower price means a greater quantity sold, and a greater consumer surplus.
In a similar manner, we can focus on the producer side. The area under the supply function, from 0 to the quantity sold, measures the producers’ need for revenue. The area in the rectangle with that same base and height equal to the sale price measures the actual producer revenue. The difference between the two is a quantity we will call producer surplus.
As long as the price stays on the supply function curve, a higher price means a greater quantity sold, and a greater producer surplus. Consider first an example where the supply and demand functions are simple enough that the computations can all be done by hand.
I am trying to sell widgets and have determined the supply and demand functions to be:
supply pricedemand price.supply price(Quantity)=4+Quantitydemand price(Quantity)=106−2∗Quantity.
Find the equilibrium price and quantity. Find the producer and consumer surpluses when the shirts are sold at the equilibrium price. If the producers form a cartel, find the price that maximizes producer surplus.
We now try an example where we need other techniques to evaluate the integrals.
A store trying to sell t-shirts on campus has determined the supply and demand functions to be:
supply pricesupply price(Quantity)=5+ln⁡(Quantity+10)
demand price.demand price(Quantity)=10+100/(Quantity+2).
Find the equilibrium price and quantity. Find the producer and consumer surpluses when the shirts are sold at the equilibrium price.
The sum of the consumer surplus and the producer surplus is referred to as the total social gain. As we looked at consumers' surplus, we assumed that the sales were determined by supply and the price-quantity point was on the supply curve. Similarly, when looking at producers' surplus we assume price is set by demand and the price-quantity point was on the demand curve. If both sides are made up of many individuals acting independently, the price-quantity point is the equilibrium point, which is on both curves. Selling at that point also maximizes the total social gain.
As we can see from the picture, this always lowers the total social gain. However for some reduction of quantity the producers’ surplus is increased. In the equation for producer surplus the price ps is demand functiondemand function(qs) rather than supply function.supply function(qs). If the quantity goes down too far the producer surplus will also go down.
A store trying to sell t-shirts on campus has determined the supply and demand functions to be:
supply pricesupply price(Quantity)=5+ln⁡(Quantity+10)
demand pricedemand price(Quantity)=10+100/(Quantity+2)
The store owner has a monopoly on campus and decides to limit the quantity sold to 200 shirts and charge what the market will bear. Find the price, the producer surplus, and consumer surpluses. Find these numbers if the owner decides to limit sales to 50. How many shirts should the owner sell at what price to maximize producer surplus? If producer surplus is maximized, how much is the total social gain reduced?
Solution.
Similarly, if the consumers form a cartel, they can artificially reduce the demand. Since they will then pay the supply price the total social gain will be decreased, but the consumers’ surplus may be increased. In this case the consumer surplus is the integral of the difference between the demand function and the supply price of the quantity that will be sold.
In the example we just looked at, both the supply and demand curves have a small slope, so the market is quite elastic from both the producers and consumers point of view. Is such a case there is less incentive to form a cartel. In other markets, like gas and oil, where the market is more inelastic, there is more incentive to engage in monopolistic practices.
A question that arises in economics looks at the equity of income or wealth distribution in a country. In standard economic theories either too much or too little equity indicates a lack of opportunity and is a hindrance to growth. However, before being able to address the advantages or disadvantages of a level of inequity we need to be able to quantify the level of equity or inequity. The standard method is to use the