Econ303 Homework 9b: Zoe Lives For Two Periods And Has Life

Econ303 Homework 9b 1zoe Lives For Two Periods And Has Life Time Utili

Zoe lives for two periods and her lifetime utility is given by \( U = \sqrt{c_1} + 0.8 \sqrt{c_2} \). Her incomes are \( y_1 = 60,000 \) in the first period and \( y_2 = 25,000 \) in the second period. The real interest rate is \( r = 7\% \). The assignment involves calculating her consumption and savings under various scenarios including changes in discount factors, taxation, and social security policies.

Paper For Above instruction

The analysis of Zoe's consumption and savings choices across different economic policies provides insight into how incentives, preferences, and government interventions influence individual behavior. This paper explores each scenario, systematically deriving Zoe’s optimal consumption in both periods and the corresponding savings, then discusses the implications of policy changes on her utility and the broader economic environment.

1. Optimal Consumption and Savings with Given Income and Discount Rate

To determine Zoe's optimal consumption in each period, we set up her lifetime utility maximization problem subject to her intertemporal budget constraint. The marginal utility per dollar spent in each period, accounting for the discount factor, guides her consumption choices. Her initial problem is:

Maximize \( U = \sqrt{c_1} + 0.8 \sqrt{c_2} \)
subject to:
\( c_1 + \frac{c_2}{1 + r} = y_1 + \frac{y_2}{1 + r} \)

The marginal utilities are \( \frac{1}{2 \sqrt{c_1}} \) and \( \frac{0.8}{2 \sqrt{c_2}} \). Applying the method of Lagrange multipliers and equality of marginal utility ratios adjusted for the discount factor and interest rate, we derive the optimal consumption levels:

\[
\frac{1}{2 \sqrt{c_1}} = \lambda
\]
\[
\frac{0.8}{2 \sqrt{c_2}} = \lambda \times (1 + r)
\]

Dividing the second equation by the first yields:

\[
\frac{0.8}{2 \sqrt{c_2}} \div \frac{1}{2 \sqrt{c_1}} = (1 + r)
\Rightarrow 0.8 \frac{\sqrt{c_1}}{\sqrt{c_2}} = (1 + r)
\Rightarrow \frac{\sqrt{c_1}}{\sqrt{c_2}} = \frac{(1 + r)}{0.8}
\end{pre>
Squaring both sides gives:
\[
\frac{c_1}{c_2} = \left( \frac{1 + r}{0.8} \right)^2
\]

Plugging in \( r = 0.07 \):
\[
\frac{c_1}{c_2} = \left( \frac{1.07}{0.8} \right)^2 \approx (1.3375)^2 \approx 1.79
\]
Using the budget constraint:
\[
c_1 + \frac{c_2}{1.07} = 60,000 + \frac{25,000}{1.07} \approx 60,000 + 23,363.55 = 83,363.55
\]
Expressing \( c_2 \) in terms of \( c_1 \):
\[
c_2 = \frac{c_1}{1.79}
\]
Thus:
\[
c_1 + \frac{c_1/1.79}{1.07} = 83,363.55
\Rightarrow c_1 \left( 1 + \frac{1}{1.79 \times 1.07} \right) = 83,363.55
\]
Calculating denominator:
\[
1.79 \times 1.07 \approx 1.917
\]
then:
\[
c_1 \left( 1 + \frac{1}{1.917} \right) = 83,363.55
\Rightarrow c_1 (1 + 0.521) = 83,363.55
\Rightarrow c_1 \times 1.521 = 83,363.55
\]
\[
c_1 \approx \frac{83,363.55}{1.521} \approx 54,866
\]
and
\[
c_2 = \frac{54,866}{1.79} \approx 30,638
\]
Savings in the first period:
\[
s = y_1 - c_1 = 60,000 - 54,866 \approx 5,134
\]
Results: Zoe consumes approximately \$54,866 in the first period and \$30,638 in the second, saving about \$5,134 in the first period.
2. Effect of Increased Discount Factor (0.9)
Increasing Zoe’s subjective discount factor to 0.9 reflects greater emphasis on future utility, possibly due to medical advancements prolonging life. The utility function parameters remain the same, but the relative preference changes. The key change is in the discounting, which affects her intertemporal choice:
\[
\text{New utility weights: } 1 \text{ in } c_1, \text{ and } 0.9 \text{ in } c_2
\]

Repeating the marginal utility ratio derivation with the new discount factor, the equation becomes:
\[
\frac{0.9}{2 \sqrt{c_2}} = \lambda \times (1 + r) \times \lambda
\]
which leads to:
\[
\frac{\sqrt{c_1}}{\sqrt{c_2}} = \frac{(1 + r)}{0.9}
\]

Calculating the ratio:
\[
\frac{(1.07)}{0.9} \approx 1.1889
\]
Squaring:
\[
\frac{c_1}{c_2} \approx 1.1889^2 \approx 1.413
\]
Using the budget constraint as before, we solve for \( c_1 \):
\[
c_1 + \frac{c_1/1.413}{1.07} = 83,363.55
\Rightarrow c_1 \left( 1 + \frac{1}{1.413 \times 1.07} \right) \approx 83,363.55
\]
Calculating denominator:
\[
1.413 \times 1.07 \approx 1.512
\]
Thus:
\[
c_1 (1 + 0.661) = 83,363.55
\Rightarrow c_1 \times 1.661 \approx 83,363.55
\Rightarrow c_1 \approx \frac{83,363.55}{1.661} \approx 50,218
\]
Similarly,
\[
c_2 = \frac{50,218}{1.413} \approx 35,530
\]
Savings calculation:
\[
s = y_1 - c_1 = 60,000 - 50,218 \approx 9,782
\]
Comparison: With a higher discount factor, Zoe consumes less now (\$50,218 vs. \$54,866) and consumes more in the future (\$35,530 vs. \$30,638). Her savings increase, reflecting a higher valuation of future utility and an increase of approximately \$4,648.
3. Impact of Capital Income Tax (40%)
Implementing a flat tax on capital income modifies Zoe’s budget constraint to:
\[
c_1 + \frac{c_2}{1 + (1 - \tau_k) r} = y_1 + \frac{y_2}{1 + (1 - \tau_k) r}
\]
with \( \tau_k = 0.4 \), so the after-tax interest rate is:
\[
r_{after} = (1 - 0.4) \times 0.07 = 0.042
\]
Calculations:
\[
\text{Budget constraint}:
c_1 + \frac{c_2}{1 + 0.042} = 60,000 + \frac{25,000}{1 + 0.042} \approx 60,000 + 23,987 = 83,987
\]
Using the previous budget constraint structure and similar derivations for marginal utility ratios:
\[
\frac{\sqrt{c_1}}{\sqrt{c_2}} = \frac{1 + r_{after}}{0.8} = \frac{1.042}{0.8} \approx 1.3025
\]
Squaring:
\[
\frac{c_1}{c_2} \approx 1.696
\]
From the budget constraint:
\[
c_1 + \frac{c_1/1.696}{1.042} = 83,987
\Rightarrow c_1 (1 + \frac{1}{1.696 \times 1.042}) \approx 83,987
\]
Calculating denominator:
\[
1.696 \times 1.042 \approx 1.768
\]
So:
\[
c_1 (1 + 0.566) = 83,987
\Rightarrow c_1 \times 1.566 \approx 83,987
\Rightarrow c_1 \approx 53,735
\]
Correspondingly,
\[
c_2 = \frac{53,735}{1.696} \approx 31,716
\]
Savings:
\[
s = 60,000 - 53,735 \approx 6,265
\]
Government revenue:
\[
\text{Tax revenue} = \tau_k \times r \times \frac{c_2}{1 + r_{after}} \approx 0.4 \times 0.07 \times (\text{capital income in second period})
\]
Since the tax is on capital income, total tax revenue can be approximated from the taxed interest on savings and income, which roughly amounts to:
\[
\text{Revenue} \approx 0.4 \times 0.07 \times \text{total capital income}
\]
which indicates the government collects significant revenue, primarily from savings and investment activity (the precise amount depends on Zoe's actual savings which we approximated as \$6,265). For simplicity, the total tax revenue is approximately:
\[
0.4 \times 0.07 \times 6,265 \times \text{some factor}
\]
but exact calculation requires detailed modeling of her savings and investment behavior. Overall, the tax reduces her savings and consumption slightly compared to the baseline scenario.
4. Impact of Consumption Tax (2%)
Imposing a flat consumption tax modifies the budget constraint to:
\[
(1 + \tau_c) c_1 + (1 + \tau_c) c_2 \times \frac{1}{1 + r} = y_1 + \frac{y_2}{1 + r}
\]
with \( \tau_c = 0.02 \). Re-arranged as:
\[
(1 + 0.02)(c_1 + c_2/(1 + r)) = 83,363.55
\Rightarrow 1.02 \times \text{intertemporal budget} = 83,363.55
\]
Thus, the effective budget constraint becomes:
\[
c_1 + \frac{c_2}{1 + r} = \frac{83,363.55}{1.02} \approx 81,731
\]
Repeating similar derivations as in part 1, the marginal utility ratio becomes:
\[
\frac{\sqrt{c_1}}{\sqrt{c_2}} = \frac{1 + r}{0.8} \approx 1.3375
\]
and squaring:
\[
\frac{c_1}{c_2} \approx 1.79
\]
The constraint on consumption becomes:
\[
c_1 + \frac{c_1/1.79}{1.07} = 81,731
\Rightarrow c_1 (1 + \frac{1}{1.79 \times 1.07}) \approx 81,731
\]
Calculating the denominator:
\[
1.79 \times 1.07 \approx 1.917
\]
So:
\[
c_1 (1 + 0.522) = 81,731
\Rightarrow c_1 \times 1.522 \approx 81,731
\Rightarrow c_1 \approx 53,803
\]
Corresponding \( c_2 \):
\[
c_2 = \frac{53,803}{1.79} \approx 30,056
\]
Savings:
\[
s = 60,000 - 53,803 \approx 6,197
\]
Government revenue from consumption tax:
\[
\text{Revenue} = \tau_c \times (c_1 + c_2) \approx 0.02 \times (53,803 + 30,056) \approx 1,754
\]
which is relatively modest compared to capital income tax revenue. Furthermore, the consumption tax tends to reduce immediate consumption, encouraging deferred consumption and savings, reflected in the slightly higher savings level compared with part 1. Comparing parts 3 and 4, consumption taxes preferentially target spending rather than savings, potentially leading to more overall savings, but less taxable income revenue compared to capital taxes.
5. Social Security with 15% Tax and Fully Funded System
Implementing a social security tax of 15% on young income affects Zoe’s intertemporal budget constraint, as her future benefits derive from her current contributions plus interest. Her new budget constraint is:
\[
c_1 + \frac{c_2}{1 + r} = (1 - 0.15) y_1 + y_2 + (1 + r) \times 0.15 y_1
\]
Plugging in values:
\[
= 0.85 \times 60,000 + 25,000 + 1.07 \times 0.15 \times 60,000
\]
Calculations:
\[
0.85 \times 60,000 = 51,000
\]
\[
1.07 \times 0.15 \times 60,000 \approx 1.07 \times 9,000 \approx 9,630
\]
Total:
\[
51,000 + 25,000 + 9,630 = 85,630
\]
The intertemporal budget constraint therefore becomes:
\[
c_1 + \frac{c_2}{1.07} = 85,630
\]
From utility maximization similar to previous parts, the marginal utility ratio:
\[
\frac{\sqrt{c_1}}{\sqrt{c_2}} = \frac{1 + r}{0.8} \approx 1.3375
\]
and squaring yields:
\[
\frac{c_1}{c_2} \approx 1.79
\]
Using the budget constraint:
\[
c_1 + \frac{c_1/1.79}{1.07} = 85,630
\Rightarrow c_1 (1 + \frac{1}{1.79 \times 1.07}) = 85,630
\]
Calculating denominator:
\[
1.79 \times 1.07 \approx 1.917
\]
Thus:
\[
c_1 (1 + 0.522) \approx 85,630
\Rightarrow c_1 \times 1.522 \approx 85,630
\Rightarrow c_1 \approx 56,319
\]
Corresponding \( c_2 \):
\[
c_2 = \frac{56,319}{1.79} \approx 31,468
\]
Savings:
\[
s = y_1 - c_1 = 60,000 - 56,319 \approx 3,681
\]
Comparison: The social security system reduces Zoe’s immediate consumption (from about \$54,866 to \$56,319), but ensures some consumption in the second period, with her savings marginally decreasing compared to the no-tax case. Her consumption in the second period increases due to the safety net, and overall her lifetime utility may improve in terms of risk mitigation despite lower savings.
6. Social Security with 4% Return on Contributions
When the rate of return on social security contributions is 4%, different from the market interest rate, Zoe’s budget constraint adjusts accordingly. Her future benefits are based on her current contribution plus interest at 4%, not 7%. The budget constraint becomes:
\[
c_1 + \frac{c_2}{1 + r} = (1 - 0.15) y_1 + y_2 + (1 + 0.04) \times 0.15 y_1
\]
Calculations:
\[
0.85 \times 60,000 = 51,000
\]
\[
1.04 \times 0.15 \times 60,000 = 1.04 \times 9,000 \approx 9,360
\]
Total:
\[
51,000 + 25,000 + 9,360 = 85,360
\]
The budget constraint:
\[
c_1 + \frac{c_2}{1.07} = 85,360
\]
The marginal utility ratio:
\[
\frac{\sqrt{c_1}}{\sqrt{c_2}} = \frac{1 + r