Consider Two Consumers A And B Both Want Perfect Con

Consider Two Consumers A And B A And B Both Want Perfect Consumption

Consider two consumers, A and B. A and B both want perfect consumption smoothing (c = cf) and both have no current wealth. However, the two consumers have different income streams. Person A’s current income, yA, = 100, and future income, yfA, = 121. Person B’s current income, yB, is 120, and future income, yfB, = 99. The real interest rate is 10%.

(a) Calculate the present value of lifetime resources (PVLR) for consumer A and consumer B, respectively.

(b) Draw consumer A’s budget constraint. How does the budget constraint of consumer A compare to the budget constraint of consumer B? Explain.

(c) Find consumer A’s optimal lifetime consumption plan, (cA, cfA). How does consumer B’s optimal lifetime consumption plan, (cB, cfB), compare to consumer A’s lifetime consumption plan? Explain.

(d) Is consumer A a current saver or a current borrower? Explain. Is consumer B a current saver or a current borrower? Explain.

(e) Draw a graph that illustrates how an increase in the interest rate (above 0.10) will affect the budget constraints of consumer A and consumer B. How does the budget constraint of consumer A compare to the budget constraint of consumer B?

Paper For Above instruction

The problem involves analyzing consumption smoothing behavior for two consumers with differing income streams and understanding how interest rates influence their lifetime consumption choices. It also explores the concepts of present value of lifetime resources, optimal consumption plans, and the implications of changing the interest rate on budget constraints.

Introduction

In modern economics, the intertemporal choices of consumers are critical for understanding savings, consumption, and investment behaviors over time. When consumers seek perfect consumption smoothing, they aim to allocate their resources to achieve a consistent level of consumption across present and future periods, adjusting for income and interest rates. This analysis examines two consumers with distinct income streams—one with higher current income but higher future income and the other with higher current income but lower future income—and explores how they allocate resources over their lifetime. We further investigate how variations in the real interest rate impact their budget constraints and consumption plans, highlighting fundamental principles of intertemporal choice theory.

Calculating Present Value of Lifetime Resources

The present value of lifetime resources (PVLR) plays a crucial role in determining consumers' consumption possibilities. For each consumer, PVLR is computed as the sum of discounted current and future income streams. Given the real interest rate (r) of 10% (or 0.10), we compute the PVLR as follows:

For Consumer A:

• Current income, yA = 100
• Future income, yfA = 121
• PVLR_A = yA + yfA / (1 + r) = 100 + 121 / (1.10) = 100 + 110 = 210

For Consumer B:

• Current income, yB = 120
• Future income, yfB = 99
• PVLR_B = yB + yfB / (1 + r) = 120 + 99 / (1.10) ≈ 120 + 90 = 210

Interestingly, both consumers have the same PVLR of 210, despite differing income streams, which implies similar total lifetime resources available for consumption once discounted.

Budget Constraints and Consumption Smoothing

The budget constraint for each consumer can be depicted graphically with present and future consumption. The key relation is:

c + cf / (1 + r) = PVLR

Plotting Consumer A's budget constraint involves the coordinates (c, cf) satisfying the above relation. The slope of the budget line is -(1 + r), reflecting how future consumption trades off against present consumption at the given interest rate.

Comparison with Consumer B’s budget constraint reveals that despite identical PVLRs, the shape may differ based on the income stream timing. Consumer A's higher future income makes them more inclined to borrow against future income, resulting in different consumption trajectories, but the total discounted resources remain equal.

Optimal Consumption Plans

Consumers seeking perfect consumption smoothing aim for equal consumption across periods: c = cf. Given PVLR and perfect smoothing, the optimal lifetime consumption levels for each consumer are:

c_A = cf_A = PVLR / (1 + 1 / (1 + r)) = 210 / 2 = 105

Similarly, for Consumer B:

c_B = cf_B = 210 / 2 = 105

Therefore, both consumers will choose the same constant consumption level of 105 in each period, leading to a smooth consumption path. This outcome demonstrates the fundamental principle that, with perfect smoothing and identical total resources, consumers allocate equally across periods regardless of income stream timing.

Saver or Borrower?

Since each consumer’s chosen consumption level aligns with their lifetime resources, the question of borrowing or saving depends on their initial income relative to their consumption. Consumer A has current income 100 and plans to consume 105 in the current period, implying they will borrow 5 units. Conversely, Consumer B has current income 120 but consumes only 105—implying they are saving 15 units in the current period.

This illustrates that consumers whose current income exceeds their desired consumption are current savers, while those with lower current income than their consumption target are debtors (borrowers). In this case:

• Consumer A is a current borrower (borrowing 5 units).
• Consumer B is a current saver (saving 15 units).

Impact of Interest Rate Changes on Budget Constraints

An increase in the interest rate above 10% would steepen the slope of the budget constraint (from -1.10), making borrowing more costly and saving more attractive. For Consumers A and B, this translates into tighter constraints for borrowing and increased incentives to save, leading them to adjust their consumption plans accordingly. Consumer A, being a borrower, would face higher costs for future borrowing, reducing their current consumption's attractiveness. Consumer B, being a saver, would benefit from higher returns on savings, possibly increasing their current saving and future consumption.

The comparison indicates that, as interest rates increase, the budget constraints become steeper for both consumers but with different implications based on their initial income streams and consumption plans.

Conclusion

The analysis illustrates the fundamental ties between income streams, discounting, and consumption smoothing. Both consumers, despite their differing income streams, achieve similar total lifetime resources due to the present value calculation. Optimal consumption smoothing leads to equal consumption in both periods, but their roles as savers or borrowers differ based on the alignment of current income and planned consumption. Changes in interest rates significantly influence their budget constraints, highlighting the importance of intertemporal financial decisions in consumption and saving behaviors. Ultimately, understanding these dynamics provides invaluable insights into household financial planning and macroeconomic implications.

References