Question

Alex needs some caffeine to function well (and the more the better). Suppose that a cup of coffee contains twice as much caffeine as a cup of tea, so that Alex can perfectly substitute one cup of coffee for two cups of tea, and his utility function is of the form U(C, T)=2C+T where C denotes cups of coffee, and T denotes that of tea.

a. Suppose that the price of a cup of tea is fixed at pt =$1, but the price of coffee, pc , can vary. Describe Alex’s choice as a function of the price of coffee. Namely, identify the region of pc , where Alex drinks only tea and the region, where Alex drinks only coffee. Illustrate your answer with a graph.

b. Suppose that pc =$2, and Alex has a budget of $10. What will be his optimal consumption bundle? Illustrate your answer with a graph.

c. Now Alex’s cafeteria runs a special promotion: if he buys 3 cups of coffee (at pc =$2) he can have the forth cup of coffee for free. Draw his new budget constraint and show his new choice.

Answer 1

a. Let Alex's income be M. Then the budget constraint Alex faces is T+ pc*C = M.

Utility function is U = 2C + T where the marginal rate of substitution is 2.

Now it depends on the relationship of price ratio with MRS. The price ratio is Pc/1 or Pc

If Pc is greater than 2, then Alex will only consume tea.

If Pc is equal to 2, Alex is indifferent between tea and coffe

If Pc is less than 2, Alex will only consume coffee.

b. If Pc is 2, and income is 10

Budget constraint will be : T + 2C = 10

At MRS = pc/pt which is the case here, Alex will consume equal amounts of coffee and tea, hence T = C

Putting this in budget constraint

C+ 2C = 10

3C = 10

C = 3.33

c. Alex is getting 4 cups of coffee at the price of 3 i.e he is paying 6 to get 4 coffee cups. So the new price will be 1.5. New budget constraint will be 1.5C+T = 10

Now since the price ratio is 1.5 which is less than the MRS, Alex will consume only coffee. He will spend the entire income on coffee i.e. he will consume 10/1.5 = 6.66 cups of coffee