Question

1. Consider a variant of the two-period model of consumption-saving behavior. In this version of the model, the consumer has income y in the first period and no income in the second period. Her life-time budget constraint is c+ a - 1+r = y. (a) Draw this budget constraint in a diagram with con horizontal axis and d on vertical axis. What are the slope and vertical intercept of this budget constraint? Label the endowment point in the diagram. (3 points) (b) Suppose the consumer faces the above budget constraint and has regular, convex indifference curves. If the interest rate increases, show in a dia- gram how her equilibrium will change. In the same diagram, decompose the overall effect on d' into income and substitution effects. (3 points) (c) Suppose the consumer faces the above budget constraint and has L- shaped indifference curves. She prefers to have the same consumption in the two periods. If the interest rate increases, show in a diagram how her equilibrium will change. In the same diagram, decompose the overall effect on d' into income and substitution effects. What is the in- tuition behind the income and substitution effects that you have found? (4 points)

Answer 1

According to the given model, the consumer has income y in period 1 and no income in period 2. His consumption in period 1 is c and in period 2 is c'. Rate of interest is r. His budget constraint is given as

c+c'/(1+r)=y.........(1)

With the help of the given informations we will start answering the following questions.

(a) The following diagram shows the budget constraint and the slope, intercepts.

Y(Hrt Slope = -(177) Endow ment (E) Equation of budget line : ot e Y. when, e=0, 6=y. X aseis intercept=y when 8 = 0, é = 961+r). You n = Y(12) slope of the budget line =-(+2) The endowment point is E.(go)

From the diagram we can see

Horizontal intercept is y.

Vertical intercept is y(1+r).

Slope of the budget line is -(1+r).

The endowment point is labelled as E i.e. (y,0) on the horizontal axis.

(b) Now, it is given that the consumer has regular convex indifference curves.

Now if the interest rate r rises to r', then we have to show the effect of the increase in a diagram and also have to decompose the substitution and income effect.

If, r rises then the vertical intercept i.e. y(1+r) will also rise. But the horizontal intercept i.e. y will remain same. The slope of the budget line i.e. -(1+r) will also rise. Hence the budget line would rotate fixing the horizontal intercept. The effect is shown in the following diagram.

ez C* 7 * e * = e * substitution effect. Income effext.

From the diagram above we can see the budget line tilted from AB to AB' due to the increase in rate of interest.

His equilibrum will change from e to e1. Where the consumption bundles would be (c2*,c2'*) as labelled in the diagram.

The substitution effect is decomposed by compensating the consumer in a way that he is now able to buy the previous bundle at the new interest rate. The equilibrium shifts from e to e1 in the diagram as he is on a higher utility curve U2.

The substitution effect is c1*-c*.

Now, theat the new interest rate the consumer is buying new bundle shown in the diagram as e2. The income effect is determined with the shift from e2 to e3 as he is on a higher utility curve U3.

The income effect is c2*-c1*.

(c) Now if the consumer has a L shaped indifference curve. He also prefers to have same consumption in both periods. Now we will look for the effect of rise in interest rate in a new diagram.

- -- Call - - --- no substitution There is effext. c* (* = Income effect.

From the above diagram we can see, due to the rise in interest rate, the budget line tilted fr AB to AB'. His equilibrum will shift from e to e1. The new equilibrum is shown as e1 and the consumption bundle is (c1*,c1'*) i.e. labelled in the diagram.

Now, to decompose the substitution effect, the income is compensated in such a way that he is now able to afford the previous consumption bundle e at the new interest rate. Here we can clearly see that even after the change in budget line the consumer is on the same indifference curve i.e.U1. Hence there is no substitution effect as he will consume the same bundle now.

Hence substitution effect is 0.

Now, the income effect is decomposed by thr shift from e to e1. Here the consumer is on a higher utility curve U1 and is consuming at e1.

Hence income effect is c1*-c*.

Hope the solution is clear to you my friend.