Now or later

ECO61 Microeconomic AnalysisUdayan Roy

Fall 2008

Inter-temporal budget constraint• Two dates: 0 (present) and 1 (future)• Dollar incomes: M0 and M1

• Food consumed: C0 and C1

• Price of food: P0 and P1 dollars

• Money saved in date 0 = M0 – P0C0 dollars

• Interest rate in date 0: R• Wealth carried over to date 1 = M0 – P0C0 + (M0

– P0C0) R = (M0 – P0C0) (1 + R)

• P1 C1 = M1 + (M0 – P0C0) (1 + R)

Inter-temporal budget constraint

• Let m0 M0/P0 and m1 M1/P1 denote the amounts consumed on dates 0 and 1 when there is no saving (or dis-saving) – That is, mt is the amount consumed on date t if all

of date t’s income—neither more nor less—is spent on date t’s consumption

Inter-temporal budget constraint• Start with the

Inter-temporal budget constraint

• Separate the consumption and income terms

• Divide both sides by P1

• Use the definitions m0 M0/P0 and m1 M1/P1

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Interest rates: nominal and real

• A loan of $1 on date 0 gets you $1 + R on date 1– R is the nominal interest rate

• The loaned amount ($1) could have been used for 1/P0 units of consumption on date 0

• The $1 + R that you get on date 1 would pay for (1 + R)/P1 units of consumption on date 1

• So, a sacrifice of 1/P0 units of consumption on date 0 leads to (1 + R)/P1 units of consumption on date 1

• So, a sacrifice of (1/P0)/(1/P0) = 1 unit of saving on date 0 leads to [(1 + R)/P1 ]/(1/P0) = (1 + R) P0/P1 units of consumption on date 1

Inflation

• Let INFL denote the rate of inflation• P1 = P0 + P0 INFL = P0 (1+ INFL)

• Example: – Let the rate of inflation be 5%– Then, INFL = 0.05– If P0 = $2.00, then P1 = $2.00 (1+ 0.05) = $2.10

Interest rates: nominal and real• Recall that a sacrifice of 1 unit of consumption on date 0 leads to

(1 + R) P0/P1 units of consumption on date 1

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Inter-temporal budget constraint

• Note, from the second line, that if C0 = 0, then C1 = m1 + m0(1 + Rreal)– This is the maximum possible consumption in date 1• Note, from the third line, that if C1 = 0, then C0 = m0 + m1/(1

+ Rreal) • This is the maximum possible consumption in date 1

• If C0 = m0, then C1 = m1, and vice versa.– That is, the consumer always has the option of not saving or dis-

saving

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Inter-temporal budget constraint

• This is just like the usual budget constraintXPX + YPY = M, when

• X is C0, Y is C1

• PX = 1, PY = 1/(1 + Rreal), and• M = m0 + m1/(1 + Rreal)

– M is also called the present discounted value (PDV) of the consumer’s income stream

– The left-hand-side of the budget constraint is the PDV of the consumption stream

– So, the budget constraint is that the PDV of the consumption stream must equal the PDV of the income stream

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Inter-temporal budget line

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Inter-temporal budget line

• When the real interest rate increases, the budget line still passes through the no-saving-no-dissaving (NSNDS) point, but rotates upward and becomes steeper (red line)

• When the real interest rate decreases, the budget line rotates downward and becomes flatter (green line)

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Inter-temporal budget line• When either m0 or

m1 (or both) changes, the no-saving-no-dissaving point changes, and the budget line moves parallel to the old line, still staying on the NSNDS point

• The line moves outward (inward) if M, the PDV of the income stream, increases (decreases)

• If M is unchanged, the budget line stays put even if m0 or m1 (or both) changes

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Preferences for the Timing of Consumption

• Treat identical physical objects as distinct goods if they are available at different points in time

• Consider a consumer who cares about two goods: food this year and food next year

• Indifference curves have customary shape– Slope downward, declining MRS– Bundles on 45o line indicate equal consumption in both years

• If food this year is on horizontal axis, steeper indifference curves show greater impatience in consumption

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Figure 10.4: Preferences for the Timing of Consumption

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Affordable Consumption Bundles

• Consumption bundle is affordable if, through borrowing and lending, the consumer can make all required payments as they come due

• PDV of consumption stream = PDV of income stream

• Slope of the budget line is the negative of the ratio of the goods’ prices:

1

01 linebudget theof SlopeP

PR

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Figure 10.5: Affordable Alternatives

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Consumption Choices

• To determine each consumer’s best choice, apply the no-overlap rule

• Brian’s solution (Figure 10.6(a)):– Chooses point B– Saves some income in first year to boost consumption in

second year• Ryan’s solution (Figure 10.6(b)):

– Chooses point C– Borrows money in first year to consume more food than

current income would allow

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Figure 10.6: Best Choices with Saving and Borrowing

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For both Brian and Ryan, A is the NSNDS point.

Saving, Borrowing, and the Interest Rate

• When interest rate rises:– Saving becomes more rewarding– Borrowing becomes more costly

• Do people respond by saving more and borrowing less?– Not necessarily!

• To understand, study how changes in the interest rate affect consumers’ budget constraints

• If consumption at each point in time is a normal good and the interest rate rises:– Savers may increase or decrease their savings– Borrowers definitely reduce their borrowing

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Figure 10.7: Effect of a Change in the Interest Rate on Saving

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1. The substitution effect of an increase in the real interest rate encourages less current consumption – that is, more saving.

2. For Brian, the saver, an increase in the real interest rate is like an increase in income. When current consumption is a normal good, this encourages more current consumption – that is, less saving.

3. For a saver, like Brian, an increase in the real interest rate can have an ambiguous total effect on saving.

Figure 10.8: Effect of a Change in the Interest Rate on Borrowing

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1. The substitution effect of an increase in the real interest rate encourages less current consumption – that is, more saving.

2. For Ryan, the borrower, an increase in the real interest rate is like a decrease in income. When current consumption is a normal good, this encourages less current consumption – that is, more saving.

3. For a borrower, like Ryan, an increase in the real interest rate leads to more saving.

PDV of Income Stream

• An individual can attain a higher level of utility only if her inter-temporal budget line is high

• This is possible when the PDV of the consumer’s income stream is high

• Therefore, our focus must be on how the PDV of the consumer’s income stream can be maximized

Investment: An Example

Table 10.4: Snow Stuff’s Proposed Factory

Project Year

Investment Units Revenue Fixed Cost

Variable Cost

Scrap Value

1 300,000 0 0 50,000 0 0

2 100,000 200 80,000 50,000 55,000 0

3 0 400 160,000 50,000 110,000 0

4 0 800 320,000 50,000 220,000 0

5 0 1000 400,000 50,000 275,000 0

6-10 0 1200 480,000 50,000 330,000 0

11 0 0 0 0 0 50,000

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Net Present Value

• Investment refers to up-front costs incurred with the expectation of generating future profits– E.g. when a firm buys capital goods

• Profitability of investment is computed as the difference between the PDV of the revenue stream and the PDV of the cost stream, the net present value (NPV)

• NPV criterion: an investment project is profitable when its NPV is positive; unprofitable when its NPV is negative

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Net Cash Flow

• In practice, we usually compute investment’s net cash flows:– Difference between revenue and cost during a single

year of a project’s life

• Then find NPV by computing PDV of project’s net cash flows:

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R

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R

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R

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11 221

0

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Internal Rate of Return• Every project’s NPV depends on the interest rate• A project’s internal rate of return (IRR) is the rate of interest

at which its NPV is exactly zero• If a project’s cash inflows occur before its cash outflows:

– Project is profitable when interest rate < IRR– Unprofitable when interest rate > IRR

• For a two-period investment, IRR is easy to calculate using NPV equation

• For longer term investments, solve for IRR numerically using spreadsheets or other computer programs

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Investment and the Interest Rate

• When interest rates rise, most potential projects become less profitable– Some become unprofitable– Causes total amount of investment to fall

• Two reasons:– Future dollars become worth less compared to current

dollars, this reduces the value of the investment relative to its cost

– Putting money in the bank becomes more attractive; the opportunity cost of funds is greater. Thus profit is lower

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Choosing Between Investments

• Not all profitable projects should be invested in• Sometimes projects are mutually exclusive• Best choice among mutually exclusive alternatives is

the one with the greatest profit– For investments, this is the one with the highest NPV

• Using criteria other than NPV to compare mutually exclusive projects can lead to poor decision-making– E.g., do not compare IRR or payback period

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Investing in Human Capital

• Human capital consists of marketable skills acquired through investments in education and training

• Use standard investment principles to determine whether to invest in human capital– Compute the NPV of the financial costs and benefits– Include opportunity costs

• Economists often summarize the financial returns to education by calculating an IRR

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The PDV of AttendingBusiness School

Table 10.10: The PDV of Attending Business School

(Hypothetical Data, 8% interest rate)

Age TuitionBooks and Supplies

Increase in

Earnings

Opportunity Cost

Total Net Change

PDV

26 20,000 2,000 0 45,000 -67,000 -67,000

27 20,000 2,000 0 45,000 -67,000 -62,037

28-65 0 0 15,000/yr 0 15,000/yr 164,290

Total 0 436,000 35,253

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