Eco 303 1 Of 3 Stony Brook University Fall 2016 Alejandro Me

Eco 303 1 Of 3stony Brook University Fall 2016alejandro Melo Ponceass

Eco 303 1 Of 3stony Brook University Fall 2016alejandro Melo Ponceass

ECO 303 1 of 3 Stony Brook University Fall 2016 Alejandro Melo Ponce ASSIGNMENT: MIDTERM I PREPARATION Due: Optional, but you should use it to prepare for the midterm. Instructions: This is an optional assignment whose purpose is to prepare you for the midterm. It consists of six problems. I strongly recommend that you attempt to prepare all questions. On Tuesday’s Midterm I will pick four questions at random from this assignment which you will need to answer.

1. Michele, who has a relatively high income I, has altruistic feelings toward Sofia, who lives in such poverty that she essentially has no income. Suppose Michele’s preferences are represented by the utility function UM(cM, cS) = c1-σM cσS, where cM and cS are Michele and Sofia’s consumption levels, respectively, with a standard Cobb-Douglas utility structure. Michele can spend her income either on her own or Sofia’s consumption (charitable donations), with prices pM = pS = 1. (a) Interpret the exponent σ as a measure of Michele’s altruism by explaining the significance of extreme values σ = 0 and σ = 1. What value would make her a perfect altruist (regarding others the same as oneself)? (b) Solve for Michele’s optimal consumption choices and analyze how these choices change with σ. (c) Consider the introduction of an income tax rate τ; net income becomes (1 - τ)I. Solve for Michele’s optimal choices under this tax. (d) Now include charitable deductions where spending on charity is not taxed, effectively changing pS to 1/(1 - τ). Derive the optimal choices considering both the tax and deductions, and compare the incentive effects for more versus less altruistic individuals.

Sample Paper For Above instruction

In exploring the relationship between altruism and consumption choices within a Cobb-Douglas utility framework, Michele's case illustrates how preferences and fiscal policies influence individual behavior. Initially, Michele's utility function U(M, S) = c_M^{1-σ} c_S^{σ} encapsulates her altruism, with σ ∈ [0,1]. When σ approaches 0, Michele's utility is almost entirely dependent on her own consumption, indicating low altruism. Conversely, as σ approaches 1, her utility is heavily weighted toward Sofia's consumption, reflecting high altruism. A sigma of 0.5 signifies a balanced concern for self and others, representing moderate altruism.

Optimal choices are derived using the Cobb-Douglas utility properties. Michele allocates her income based on her preferences; the optimal consumption levels satisfy the ratio c_M / c_S = ( (1-σ) / σ ). Given that prices p_M and p_S are both 1, her budget constraint is I = c_M + c_S. Solving these simultaneously yields the optimal consumption functions:

c_M^ = (1 - σ) I, and c_S^ = σ I.

The degree of altruism (σ) directly influences consumption: higher σ increases Sofia's consumption share. When a tax rate τ is introduced, Michele's net income reduces to (1 - τ)I, leading to adjusted consumption calculations:

c_M^ = (1 - σ)(1 - τ) I, and c_S^ = σ(1 - τ) I.

Charitable deductions further modify the effective price for Sofia's consumption to p_S' = 1 / (1 - τ), which incentivizes increased charitable giving, especially for more altruistic individuals (higher σ), since their utility weighted toward Sofia's consumption makes the deduction more attractive. For less altruistic individuals, the incentive effect of deductions is comparatively smaller because their utility is less sensitive to Sofia's consumption level.

Overall, these fiscal policies demonstrate how individual preferences interact with economic incentives, shaping consumption choices, and illustrating the importance of considering altruism in policy design. The analysis underscores that boosting deductions can significantly influence highly altruistic individuals' charitable behavior, while having a more muted effect on those with lower altruism levels.

References

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