Question

25. There are no taxes on the first $500 that Debra earns per week, but on income above $500 per week, she must pay a 60% tax. Debra’s job pays $10 per hour. Her utility function is U(c, r) = rc², where r is hours of leisure and c is dollars worth of consumption. She has 100 hours to divide between work and leisure. How many hours per week will she choose to work?

a. 66.66

b. 50

c. 40

d. 33.33

e. 20

Albert consumes only tangerines and bananas. His only source of income is an initial endowment of 30 units of tangerines and 10 units of bananas. Albert insists on consuming tangerines and bananas in fixed proportions, 1 unit of tangerines per 1 unit of bananas. He initially faces a price of $10 per unit for each fruit. The price of tangerines rose to $30 per unit while the price of bananas stayed unchanged. After the price change, he would

a. increase his consumption of tangerines by exactly 5 units.

b. decrease his consumption or tangerines by at least 5 units.

c. increase his consumption of tangerines by exactly 15 units.

d. decrease his consumption of tangerines by exactly 7 units.

e. decrease his consumption of bananas by at least 1 unit.

show work please

Answer 1

Wage is 10$ per hour. There are no taxes on the first $500 that Debra earns per week. So, for the first 50 hours she will not be taxed.

The budget constraint for r∈ [50,100] sic+ 10r= 10(100).

For r∈ [0, 50] the wage becomes 10(1-0.6) = 4.

So the budget constraint has a kink at r= 50. Starting from the kink at (50,500) and considering that the budget constraint for r ∈[0,50] has a slope of-1/4, we find that its intercept is (0,700).

Equation of the budget constraint for r∈[0,50] isc+ 4r= 700.

The indifference curves for the utility function U(c, r) =rc2 are strictly convex.

As per Cobb-Douglas utility,

The demand function for r is r=m/3w. So for the section of the BC with r be an element of [0,50] we have r = (700/(3*4)) = 100. It is not on the section r ∈[0,50] because r= 100>50. For the part of the BC with r ∈[50,100], we have 1000/(3*10) = 33.¯3 which on the section r ∈[50,100] because r= 33.¯3<50 (so the there is no indifference curve tangent to the second section of the budget line). So the optimal choice must be at the kink i.e. she chooses r= 50. So the answer is that she works 100-50 = 50 hours.

So answer is B