Question

Consider an individual that must decide how much to consume in a two period model. Let us suppose that her preferences for present consumption (c1) and future consumption (c2) can be characterized by the following utility function: u(c1, c2) = c1^0.5 × c2 Further assume that her income in both the present period (M1) and the future period (M2) is equal to 105, the price index in the present period (p1) is 1, the price index in the future period (p2) is 1.05, and the nominal interest rate (i) is 0.05. (a) Illustrate this individual’s intertemporal budget constraint with c1 on the horizontal axis. What is the slope of this budget constraint and her present and future consumption 1 at her zero savings point? Add the zero savings point to your diagram. (b) Solve for this individual’s optimal values of c1 and c2 respectively. What is her total utility at this equilibrium? Illustrate this equilibrium in your diagram from part (a). (c) Now suppose that the nominal interest rate decreases to 0.04. Let all of the other parameters in the model remain unchanged. Calculate the total effect of this increase in i on c1. (d) Decompose the total change in c1 consumption in part (c) into an income effect and a substitution effect. You will find it helpful to use a new diagram to first illustrate the income and substitution effect.

Answer 1

Her Utility function is given by

U(c₁, c₂ ) = c₁^1/2 c₂^1/2

Price of C1 is P1 and Price of C2 = P2

Now dU = (U/c₁ )dc₁ + (U/c₂)dc₂  

= (1/2)c₂^1/2 c₁^-1/2 dc₁ + (1/2)c₁^1/2 c₂^-1/2

For fixed Utility dU = 0

Therefore, (1/2)c₂^1/2 c₁^-1/2 dc₁ + (1/2)c₁^1/2 c₂^-1/2 =0

=> (c₂ / c₁ )^1/2 = (c₁ /c₂ )^1/2

=> C₂² = C₁²  

=> (C₂ - C₁ )(C₂ + C₁ ) = 0

Therefore, C₂ = C₁ or C₂ = -C₁

Now cost of her education = M_{1 ;where M1 is her income and is some factor lying beween 0 and 1}

Money Left out for future co nsumption = M₁ -M₁  = (1-)M₁

The Budget Constraint is M₁ = P₁ C₁ + P₂ C₂ where C₁ = M₁/P₁ , and C₂ = (1-)M₁ /P₂

Her Future income for given lambda becomes

M₂ = (1+ ^1/2 )M₁ --------------------------------(i)

This means that a dollar spent on University education has income generating impact, as income is rising by square root lambda times M₁ . -------------------------------Answer (a)

Answer (b)

She has saved (1-)M₁ to invest at nominal interest rate i

In future her income becomes (1+^1/2)M₁ -----------------------(ii)

Therefore, Rise in incomeis (1+^1/2)M₁ -(1-)M₁

= M₁ ( -^1/2 )

This is positive. This implies that investment rate i has positive relation with .

Answer (c)

WE can differentiate the Utility function w.r.t C1 and C2, and equate with the p1 and P2, and thereafter proceed to give the budget constraint.