# Suppose the wage is 80 per hour and that the consumer has 100 hours H to work with. Suppose that the MRS is given by c/(l−10) . What will the consumer’s choices of c and l be. Repeat with an upper bound of 10 hours. Repeat both parts with a 10% tax rate for all income levels. Suppose that the tax rate has two brackets so that income from hours above (1/5)H is taxed at 20 percent. How does the solution change? Suppose that consumers must instead pay a lump sum tax that raises the same tax revenues as the one listed above. How will outcomes change?

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Suppose the wage is 80 per hour and that the consumer has 100 hours H to work with. Suppose that the MRS is given by c/(l−10) . What will the consumer’s choices of c and l be. Repeat with an upper bound of 10 hours. Repeat both parts with a 10% tax rate for all income levels. Suppose that the tax rate has two brackets so that income from hours above (1/5)H is taxed at 20 percent. How does the solution change? Suppose that consumers must instead pay a lump sum tax that raises the same tax revenues as the one listed above. How will outcomes change?

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Step 1

Wage (w) = 80 per hour of work

Total hours available (H) = 100

l is the leisure (*l is small L and not numeral 1)

H-l are the labor hours.

Pc = 1 (assumption)

MRS = C / (l-10)

Consumer's budget constraint (BC) is:

Pc*C = w(H- l)

= 80(100 - l)

= 8000 - 80l

Equating the MRS with the price ratio

MRS = w/Pc

C / (l-10) = 80 / 1

C = 80l - 800

Using the value of c from the budget constraint above to find the value of l:

8000 - 80l = 80l - 800

l = 55

Putting the value of l in BC to find C:

C = 3600

Now repeating the entire process with H =10 to find the value of C and l.

New BC: C = 800 - 80l

Again equating MRS with the price ratio to get

l = 10

that is there are no labor hours.

C = 0

Step 2

Now there is a 10% tax rate for all the income levels.

Case I: H = 100

New income = w - 10% of w

= 0.9*w

= 72

BC:  C = 72(100-l)

= 7200 - 72l

Again equating MRS with the BC to find C and l:

C/(l-10) = 72

l = 55

C = 3240

Case II: H =10

New income is 72 only.

BC:  C = 72(10-l)

= 720 - 72l

Again equating MRS with the BC to find C and l:

C/(l-10) = 72

l = 10

C = 0

Step 3

Now suppose that the tax rate has two brackets so that income from hours above (1/5)H is taxed at 20 percent.

(1/5) *H = 20 that means working hours are 20 and leisure is 80. When working hours are greater than 20 tax rate is 20% and l is less than 80. On the other hand, when working hours are less than 20 leisure is greater than 80 and tax rate is 10%.

Case I: H =100

BC:

C = {0.9w(100-l)     ; l >= 80

w(20*0.9) + 0.8w(100- l - 20)     ; l < 80 }

Now when l >= 80 BC is same as in Step II, Case I which gives l as 55 which is not ...

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