Question

Milton is a young boy whose parents will give him an allowance of $15 today and an allowance of $15 tomorrow. Suppose Milton's preferences over spending today, x₁, and spending tomorrow, x₂, are represented by the utility function U(x1; x2) = x₁ / x₂^1/2 . Given his utility function Milton's MRS_x1x2 = 2x₂/x_1.

(a) Based on Milton's utility function, does he enjoy consumption more today or tomorrow? Explain. For parts (b) through (d), determine how much money Milton spends, saves and borrows in each period.

(b) Suppose there is no credit market, and also, that Milton's parents take back any money that Milton doesn't spend in each period (that is, he cannot save any money).

(c) Now assume the interest rate is 0, but Milton's parents allow him to save or borrow.

(d) Now suppose that Milton's parents decide to set up a credit market with interest rate r = 0.25 so that he can borrow or lend as much as he wants at this interest rate.

(e) What interest rate r should Milton's parents pick if they want Milton to spend the same amount of money in period 2 as he spends in period 1?

Answer 1

The following problem is a problem of intertemporal choice. This model depicts the consumer's choice of spending his income in current period and in future period. Thus, a consumer faces intertemporal budget constraint.

The first step will be to derive the budget contraint .

The income earned by a consumer is either saved or consumed giveing us the following equation.

S = Y1 - C1 ; where S is savings, Y1 is the income earned in first period and C1 is the consumption in first period.

Here, first period is today and second period is tomorrow.

Now, C2 = (1+r)S + Y2 => C2 = (1+r)(Y1 - C1) + Y2

=> (1+r)C1 + C2 = (1+r)Y1 + Y2 or

C1 +C2/ (1+r) = Y1 + Y2/(1+r) - Intertemporal Budget Constraint

Optimal Choice = Slope of intertemporal budget constraint = Slope of Indifference Curve

where Slope of budget constraint = 1+r

Slope of indifference curve = MRS

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(a)

Milton's allowance in each period i.e. today and tomorrow is $15.

Spending Today -- x1

Spending tomorrow -- x2

Milton's Budget Constraint = x1 + x2/(1+r) = 15 + 15/(1+r)

Utility function =   

MRS = 2 x₂/x₁

Now, Milton would consume at the point where the utility indifference curve will be tangent to the budget constraint or where slope of utility function / IC will be equal to slope of budget constraint.

Assuming r to be 0, Slope of budget constraint = 1

So, 2x1 / x2 = 1 => 2 x1 = x2

Ans. According to Milton's utility function, he enjoys consumption today more than tomorrow.

(b)

When there is no credit market and Milton's money is taken back by his parents when not spent, Milton would not save his money for tomorrow.

In this case, X1 = Y1 i.e. X1 = $15

X2 = Y2 i.e. X2 = $15

Milton would be consuming all of his income of the present period in the present and his income of the future period in future.

(c)

Milton's Budget Constraint = x1 + x2/(1+r) = 15 + 15/(1+r) ---------------------------------- eq 1

Interest rate = 0 ; allowed to save and borrow.

eq1 becomes x1 + x2 = 30 ----------------------------------- eq 2

Optimal consumption at 2 x2/x1 = 1 => x1 = 2 x2 --------------------------------- eq3 [ Slope of budget line = MRS]

Substituting the value of x1 in eq 2

2 x2 + x2 = 30,

X2 = $10

X1 = $20

As found from (a) that Milton gets more utility when he consumes more in period 2 i.e. tomorrow than today. He would save some of his today's income to be able to spend tomorrow.

(d)

Interest Rate = 0.25 ; allowed to borrow or lend.

Milton's Budget Constraint = x1 + x2(1 + 0.25) = 15 + 15(1 + 0.25) ----------------------------------- eq 4

x1 + 1.25 x2 = 33.75 -------------------------- eq 5

Optimal consumption choice will be when 2 x2/x1 = 1.25 => x1 = 1.6 x2 ------------------------- eq 6

Substituting the value of x1 in eq 5

1.6 x2 + 1.25 x2 = 33.75

x2 = $11.84 (Consumption tomorrow)

x1 = $18.94 (Consumption today)