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

Lorenz curve and the Gini index.
The Lorenz curve is defined by a function ,L(x), with ,0≤x≤1, that measures the proportion of something is held by the bottom x proportion of the population. Thus, if ,L(0.2)=.1, for the Lorenz function for income in a country, then the bottom 20% of the population earns 10% of the income in the country. Since, under usual definitions, a person cannot have negative income, the Lorenz functions are nonnegative and increasing. Since the Lorenz functions are measured from the bottom, we also have L(x)≤x for all .x.
We can make a few more observations. The population as a whole has the entire income of the population. An empty set of the population has none of the population's income. Any bottom segment will have nonnegative income. In formulas these observations become ,L(1)=1, ,L(0)=0, and ,L(x)≥0, for all ,x, respectively.
If we had perfect equity, our Lorenz function would be .L(x)=x. Any Lorenz curve we find for a real population will be below this curve. The Gini index (or Gini coefficient) measures the percentage that a real Lorenz curve is below the ideal curve.
Te Lorenz curve for income in a certain country is given by .L(x)=.8x3+.2x. What proportion of the income is earned by the bottom half of the population? Find the Gini index.
In practice, the Gini index is an application where a numeric approximation of an integral is the method most likely to be used. We are unlikely to get a formula for income distribution. Instead we are likely to find data points. Since there is no good model for how the income will be distributed, we can simply connect the points with line segments and find the area using the area formula for a trapezoid.

1. 
   Find the consumer surplus, producer surplus, and total social gain at market equilibrium.
   
2. 
   If the producers can form a cartel and restrict the available quantity to 5, selling at the demand price for 5 (for a price of 185), what are the consumer surplus, producer surplus, and total social gain?
   
3. 
   Find the price where a producer cartel will maximize the producer surplus. Find the producer surplus at that price.