Question

please show how to solve

Harvey Habit has a utility function U(c₁, c₂) = min{c₁, c₂}, where c₁ and c₂ are his consumption in periods 1 and 2 respectively. Harvey earns $189 in period 1 and he will earn $63 in period 2. Harvey can borrow or lend at an interest rate of 10%. There is no inflation.

a.	Harvey will save $60.
b.	Harvey will borrow $60.
c.	Harvey will neither borrow nor lend.
d.	Harvey will save $124.
e.	None of the above.

Answer 1

The correct answer is (a) Harvey will save $60.

First lets form inter temporal Budget constraint:

Period 1 : c₁ + s = Y₁ where c₁ = consumption in period 1 , s = saving, which means if s < 0 means he is borrowing and if s > 0 means he is saving, Y₁ = Income in period 1 = 189

=> s = 189 - c₁

Period 2 : c₂ = (189 - c₁) + r((189 - c₁)) + Y₂

where r = interest rate = 10% = 0.10, Y₂ = 63

=> c₂ = (189 - c₁) + r((189 - c₁)) + 36

=> 1.1c₁ + c₂ = 1.1*189 + 63 = 270.9

So Now we have to Maximize : U = min{c₁ , c₂}

subject to : 1.1c₁ + c₂ = 270.9 -----------------(1)

We can see from above that This utility is a Leontief utility or perfect complements utility function. In order to maximize utility is such case then a consumer should consume at a point where kink occurs and budget line is intersecting this kink point.

Here Kink will occur when c₂ = c₁

Thus, putting this in (1) we get:

1.1c₁ + c₁ = 270.9

=> c₁ = 270.9/2.1 = 129

=. s = Y₁ - c₁ = 189 - 129 = 60 > 0. Thus s > 0 => he will save.

So, he will save $60.

Hence, the correct answer is (a) Harvey will save $60.