ECON 302: Intermediate Macroeconomic Theory (Fall 2014)

Discussion Section 2  September 19, 2014

SOME KEY CONCEPTS and REVIEW CHAPTER 3: Credit Market  Intertemporal Decisions • Budget constraint for period (time) t

P ct + bt = P yt + bt−1 (1 + R)  Interpretation of P : note that the unit of the equation is dollar valued. The real value of one dollar is the amount of goods this one dollar can buy is 1/P  that is, the real value of bond is bt/P , the real value of consumption is P ct/P = ct.  Normalization: we sometimes normalize price P = 1, when there is no ination. • A side note for income yt: one may associate yt with `t since yt = f (`t) here  that is, output is produced by labor. • Two-period model: assume household lives for two periods  that is, there are t = 1, 2. Household is endowed with bond or debt b0 at the beginning of time t = 1. The lifetime income is given by (y1, y2). The decision variables are (c1, c2, b1, b2). Hence, we have two budget-constraint equations:

P c1 + b1 = P y1 + b0 (1 + R)

P c2 + b2 = P y2 + b1 (1 + R) Combining the two equations gives the intertemporal (lifetime) budget constraint:

c y b (1 + R) b c + 2 = y + 2 + 0 − 2 1 1 + R 1 1 + R P P (1 + R) | {z } | {z } | {z } | {z } (a) (b) (c) (d) (a) is called real present value of lifetime consumption, (b) is real present value of lifetime income, (c) is real present value of endowment, and (d) is real present value of bequest. (b)+(c)+(d) is real present value of total wealth. • Intertemporal substitution and wealth eect  Substitution eect: an increase in interest rate R leads to a decrease in current consumption (c1 ↓) and an increase in future consumption (c2 ↑). Alternatively, one would also work more to save now (`1 ↑).  Wealth eect: an increase in interest rate R leads to the increase in both current and future consumption (c1 ↑, c2 ↑)  Total eect is ambiguous. In this class, we will assume that substitution eect outweighs wealth eect. • Innite horizon proble  There is no bequest in the innite horizon model, only endowment at the beginning b0. The right-hand-side represents the real present value of total wealth, while the left-hand-side is the real present value of consumption stream:

c2 c3 y2 y3 b0 (1 + R) c1 + + ... = y1 + + + ... + 1 + R (1 + R)2 1 + R (1 + R)2 P • Permanent income hypothesis  The marginal propensity to consume is dened as MPC = ∆ct/∆yt ≤ 1. Thus, the marginal propensity to save is MPS = 1 − MPC.  Permanent income is the amount of constant income P receiving every period such that the real present value of P from every period (P,P,...) is equal to the real present value of uctuating lifetime income (y1, y2,...) stream.  Consumption smoothing refers to the choice of lifetime consumption plan such that consumption

is the same across all time  that is, ct = c, for all t.

1 EXERCISE Question 1 (Intertemporal Choice: Basic Example) Suppose there are two people living for two periords, Ann and Bill. Both persons have identical preferences within each period and across time. They have no endowment at the beginning of their lives and will not bequeath at the end of their lives. Let the interest rate be R = 10%. Suppose the lifetime income of Ann is (y1, y2) = (100, 0) and Bill is (y1, y2) = (0, 110). Whose consumption is larger in the rst period?

Question 2 (Substitution and Wealth Eect in Two-period Model, Problem 3.7)

What are the eects of the following changes on current and future consumption (c1, c2) and work eort (`1, `2)? Explain both substitution and wealth eect. 1. A permanent parallel upward shift in production function in both periods. (Hint: Recall Chapter 2.) 2. An increase in interest rate. 3. A temporary change in the marginal product of labor.

Question 3 (Intertemporal Optimization) 1 Suppose household has a utility function u (c1, c2) = log (c1) + /3 log (c2). The income stream is (y1, y2) = (50, 60). Assume P = 1, b0 = 0, b2 = 0, and R = 20% = 0.2. What is the optimal consumption (c1, c2) and saving (b1 − b0) plan?

Question 4 (Wealth Eects for Innitely-lived Household, Problem 3.10) Consider the household's budget constraint in real term over the innite horizon

c y b (1 + R) c + 2 + ... = y + 2 + ... + 0 1 1 + R 1 1 + R P Evaluate the wealth eect of the following scenarios.

1. An increase in price level P , for a household that has a positive value of b0. 2. An increase in the interest rate R, for a household that has b0 = 0 and ct = yt in all period t = 1, 2,... 3. An increase in the interest rate R, for a household that has b0 = 0 with the consumption plan such that ct < yt for t ≤ T , and ct > yt for t > T .

Question 5 (Permanent Income, Adapted from Midterm 2003) A consumer lives for three years. He can save or borrow to smooth his consumption over his lifetime. Assuming that his goal in life is to smooth his consumption over time. His income in Year 1, 2 and 3 are $27000, $50000, and $25000, respectively. The interest rate between the rst and second year is 25%, while between the second and third year is 100%. Assume he receives no endowment and plans to leave no bequest. 1. What is his permanent income? 2. What is his annual consumption? 3. What is the amount that he saves or borrow, if any, in the rst year?

Question 6 (Money Demand, Adapted from Midterm 2003 and Problem 4.9, 4.10) Consider a worker with an annual income of $12000. Suppose he receives wage payments (of equal value) once a month. Consumption is constant at $12000 per year. Assume that the worker holds no bonds; that is, he holds all nancial assets in the form of money. 1. Assume that he shops continuously. (a) What is his average money balance? Draw the graph denoting the money balance over time. (b) If he receives wage twice a month instead of once a month, what is his average money balance? 2. Revert to the case that he receives his wage once a month. Suppose instead that he shops 5 times each month, also, he immediately shops when he has received his wage. (a) What is his average money balance? Draw the graph denoting the money balance over time. (b) If he shops 2 times each month instead of 5 times each month, what is his average money balance? (c) If it is more expensive to go shopping, e.g. UW-Madison charges higher bus fare, what will happen to his shopping frequency? What will happen to his average money balance?

2 BRIEF SOLUTION Question 1 Since the present values of lifetime wealth for both Ann and Bill are the same, as well as the preferences; then the consumption plan must be the same for both.

Question 2 1. Note that when production function shifts upward parallely, there is only wealth eect such that ` ↓ while output increases. Furthermore, such shifts in production function incurs no substitution eect.

Since this happens in both periods, the wealth eects lead to c1 ↑, c2 ↓, `1 ↓ and `2 ↓, which is also the total eect.

2. In class, we have concluded that, when R increases, the substitution eect leads to c1 ↓ and c2 ↑, while wealth eect leads to c1 ↑ and c2 ↓. With the assumption that substitution eect prevails, then c1 ↓ and c2 ↑. To analyze labor supply choice, consider that people is willing to work more now to save; thus, `1 ↑, while `2 ↓ since the present value of y2 = f (`2) falls. 3. Suppose MPL1 increases while MPL2 remains the same. From Chapter 2, we have concluded that substitution eect leads to `1 ↑ while income eect leads to `1 ↓, so we have `1 ↑ in the total eect assuming substitution eect prevails, output and consumption in period 1 increase, so c1 ↑  these are intratemporal eects. Furthermore, labor supply today worths more than labor supply tomorrow;

thus, `2 ↓. With the increase in lifetime wealth, future consumption all increases, so c2 ↑  all these are called intertemporal eects.

Question 3 c1 = 75, c2 = 30, y1 − c1 = b1 − b0 = −25

Question 4

1. If b0 > 0  that is, he receives positive endowment, the increase in P leads to the decrease in the real value of endowment. Thus, a negative wealth eect.

2. Since this household consumes only what he earns, then yt − ct = 0. Also, he has nothing to begin with as b0 = 0. So, he does not save; therefore, any change in R leads to no eect. 3. If b0 = 0, while yt − ct 6= 0, we revert to the simple case of the wealth eect in the two-period economy. Note also that, if ct < yt, household is saving, vice versa, household is borrowing when ct > yt. The income eect from the increase in R leads to the increase in consumption in every period.

Question 5 1. Solve for P from P P 50000 25000 P + + = 27000 + + 1 + 0.25 (1 + 0.25) (1 + 1) 1 + 0.25 (1 + 0.25) (1 + 1)  1 1  P 1 + + = 27000 + 40000 + 10000 1.25 2.5 2.2P = 77000 P = 35000

2. Permanent income hypothesis suggests c1 = c2 = c3 = P = 35000 3. Since y1 = 27000, but c1 = 35000, then he saves y1 − c1 = b1 − b0 = −8000  that is, he borrows $8000 in period 1.

Question 6 1. The graph is left as an exercise. (a) $500, (b) $250 2. The graph is left as an exercise. (a) $400, (b) $250, (c) If it is more expensive to go shopping, he will go less frequently. This causes the average cash balance to decrease.

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