Question

Consider an economy with a flat rate tax system. Each dollar of income over $5000 is taxed at 20%. (Income below $5000 is tax free.) In general, T2(Y-5000) = -1000+ .2Y, where T is taxes and Y is income. Suppose that the population mean income is $20,000 and that the population standard deviation of incomes is $8000. All families have at least $5000 of income. a. find the mean of T b. Find the standard deviation of T. c. if the population contains 20 million families, what is the government's total tax revenue.

Answer 1

a)
`E(T) = 0.2 E(Y) -1000= 0.2*20000 -1000=3000`

b)
`Sd(T) = √(0.2^2 Var(Y))=√(0.2^2 8000^2)= 1600`

c) Assuming 20 million of families and each one with a mean of income of 20000 for each family approximately then total income would be:

`E(T) = 20000000*20000= 40000 millions`

And if we replace into the formula of T we have:

`T = 0.2*400000x10^6 -1000= 790000 millions`

Approximately.

Explanation:

For this case we knwo that Y represenet the random variable "Income" and we have the following properties:

`E(Y) = 20000, Sd(Y) = 8000`

We define a new random variable T "who represent the taxes"

`T = 0.2(Y-5000) = 0.2Y -1000`

Part a

For this case we need to apply properties of expected value and we have this:

`E(T) = E(0.2 Y -1000)`

We can distribute the expected value like this:

`E(T) = E(0.2 Y) -E(1000)`

We can take the 0.2 as a factor since is a constant and the expected value of a constant is the same constant.

`E(T) = 0.2 E(Y) -1000= 0.2*20000 -1000=3000`

Part b

For this case we need to first find the variance of T we need to remember that if a is a constant and X a random variable
`Var(aX) = a^2 Var(X)`

`Var(T) = Var (0.2Y -1000)`

`Var(T)= Var(0.2Y) -Var(1000) + 2 Cov(0.2Y, -1000)`

The covariance between a random variable and a constant is 0 and a constant not have variance so then we have this:

`Var(T) =0.2^2 Var(Y)`

And the deviation would be:

`Sd(T) = √(0.2^2 Var(Y))=√(0.2^2 8000^2)= 1600`

Part c

Assuming 20 million of families and each one with a mean of income of 20000 for each family approximately then total income would be:

`E(T) = 20000000*20000= 40000 millions`

And if we replace into the formula of T we have:

`T = 0.2*400000x10^6 -1000= 790000 millions`

Approximately.