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Question 2 Consider two consumers (consumer A and consumer B), which live for two periods. Both have income Y1 = 60 in the first period and Y2 = 40 in the second period. They can borrow and save at the interest rate r = 10%. Their utility functions are respectively: Consumer A’s utility function: U A (C1 , C 2 ) ln C1 ln C 2
(1 r )
C2 [ (1 r )]2 C1
Since they have the same income profile:
C1
C2 Y 40 Y1 2 60 1 r 1 r 1 r
Consumer A: C2 1 1.1 1 C1 C2 C C1 1.1
C2 Y1 = 210, Y2 = 0
New optimal choice
C1 = C2 = 100
C1 Clearly from the picture, the higher interest rate restricts Jill’s possible choices. Therefore, she will be worse off.
b. Suppose the interest rate increasetion in the first period? Is Jack better off or worse off than before the interest rate increase? Show your result in graphs. Jack is initially consuming his own income (he does not borrow or save):
C1 = C2 = 100 A Y1 = 100 C1
Why is the new optimal choice to the left of the income point in the new budget constraint? Consider a point A to the right of the income point (as shown above). Such point was available in the old budget constraint, but was not chosen. This implies that Jack prefers to consume his income instead of point A. In the new budget line, both points are again available. This implies that point A cannot be chosen. Moreover, Jack will not choose his income again: at that point, the slope of the new budget constraint is different than the slope of the indifference curve. Therefore, Jack will pick a point to the left of his income. This implies that consumption today will fall and consumption tomorrow will rise. Jack will become a saver. Jack will be better off because he can get at least the same utility as before if he consumes his income (that is still available). But he decides not to do so, meaning that the new choice improves his welfare. c. What will happen to Jill’s consumption in the first period when the interest rate increases? Is Jill better off or worse off than before the interest rate increase? Show your result in graphs. Jill’s initial choice is describe in the following graph:
2
C2 (1+r)Y1+ Y2 = 210 Y1 = 0, Y2 = 210
C1 = C2 = 100
Y1+ Y2/(1+r) = 190.91
C1
Jill is a borrower. A higher interest rate makes her poorer, meaning that the income effect contributes to lower both C1 and C2. But this also means that the relative price of consumption today is higher; according to the substitution effect, C1 goes down and C2 goes up. Therefore, C1 surely decreases (income and substitution effects reinforce each other), but the impact of C2 is ambiguous. C2 will go up (down) if the substitution (income) effect is dominant. In the following graph, the substitution effect is stronger.
C
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Savings: S Y1 C1 60 50.48 9.52
C2 (1+r)Y1+ Y2 = 104 C1 = C2 = 50.48 Y1 = 60, Y2 = 40 Y1+ Y2/(1+r) = 96.36
Consumer B:
C1
C2 1 1.1 1 C1 C2 C C1 1.1 Since both consumers have the same income profile, the solution will also be the same. b. Suppose now that the interest rate is increased to 20%. Before calculating the new optimal choices, what do you expect to happen to C1 and C2? An increase in r has two effects: - Income effect: since both consumers are savers, the higher interest rate makes them richer. Given that C1 and C2 are normal goods, this would lead to an increase in consumption in both periods. - Substitution effect: higher r means that the consumption today is more expensive relative to consumption tomorrow. Consumers will substitute away from C1, towards C2. Consumption today falls; consumption tomorrow rises. Total effect: C2 goes up (both effects reinforce each other) for sure. C1 will increase (decrease) if the income (substitution) effect is stronger. c. Now calculate the new optimal choices for both consumers. In a graph, illustrate the change. What does this tell you about the magnitude of income and substitution effects? Now let r = 20%
Guanghua School of Management, Peking University Intermediate Macroeconomics Answer Key to Homework Assignment 4
Se Yan Spring 2012
Question 1 Jack and Jill both obey the two-period model of consumption. Jack earns $100 in the first period and $100 in the second period. Jill earns nothing in the first period and $210 in the second period. Both can of them can borrow or lend at the interest rate r. a. You observe both Jack and Jill consuming $100 in the first and second period. What is the interest rate r? Using the intertemporal budget constraint for Jill: C1Jill C2Jill Y Jill 100 210 110 Y1Jill 2 100 100 1 r 1 r 1 r 1 r 1 r 1 r 110 / 100 r 10%