Question81 A widely used utility function in the economics literature is the constant rate of risk aversion utility function. It is given by: [math] Assume that an agent lives for three periods (t=0,1,2) and discounts future utility at rate β=0.8 (per period), and the degree of risk aversion [math] The agent is born with asset level a0 =5, and his/her labour market income is y0 =10, and y1 =15, for periods 0 and 1 respectively, the agent retires in the last period (no labour income in period 2). The interest rate in this economy is r=5%. Please answer the following questions based on the information displayed here. Choose the best option available. The optimal consumption at t=1 is approximately (2-decimal places) c1 = 11.12 The optimal consumption at t=1 is approximately (2-decimal places) c1 = 16.67 The optimal consumption at t=1 is approximately (2-decimal places) c1 = 9.34 The optimal consumption at t=1 is approximately (2-decimal places) c1 = 16.49 The optimal consumption at t=1 is approximately (2-decimal places) c1 = 10.19 ResetMaximum marks: 2 Flag question undefined

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Question restatement: The problem provides a constant relative risk aversion (CRRA) utility function, three periods (t = 0,1,2), discount factor β = 0.8, risk aversion parameter (assumed to be the CRRA coefficient) with given values, initial assets a0 = 5, labor income y0 = 10 in period 0 and y1 = 15 in period 1, no labor income in period 2, and an interest rate r = 5%. We are asked to identify which of the listed options is the best approximation for the optimal consumption in period t = 1, c1, to two decimals.

Option A: The optimal consumption at t=1 is approximately (2-decimal places) c1 = 11.12
- This option proposes a period-1 consumption around 11.12. To assess its plausibility, one would normally compute the intertemporal Euler equation under CRRA utility with risk aversion parameter and the budget constraints across periods, accounting for the interest rate, discounting, and future income. If 11.12 were correct, it would imply a certain feasible splitting of resources between periods 0, 1, and 2, given a0 = 5 and the labor incomes; any inconsistency with the intertemporal conditions or budget balance would falsify this value.

Option B: The optimal consumption at t=1 is approximately (2-decimal places) c1 = 16.67
- A c1 value as high as 16.67 would require substantial smoothing of consumption across periods, given the relatively modest initial assets and incomes, and the fact that part of period 2 has no income. Under CRRA with a positive risk aversion coefficient, one would expect a balance that often keeps c1 below or around the mid-point of available resources unless there is very high impatience or favorable financing conditions. If 16.67 overshoots the intertemporal budget given a0=5 and future incomes, this option would be inconsistent.

Option C: The optimal consumption at t=1 is approximately (2-decimal places) c1 = 9.34
- A c1 near 9.34 sits in a plausible range between a0 + y0 and the present value of future resources, considering that some income occurs in period 1 and period 2 has no income but can be funded via savings from period 0 and 1. Under CRRA with the given β and r, the Euler equation would typically yield a c1 that reflects a balance between current and future utility, with the ability to borrow or lend at rate r. If 9.34 satisfies the intertemporal budget constraint and the marginal rate of substitution aligns with (1+r) and β, this could be a feasible solution.

Option D: The optimal consumption at t=1 is approximately (2-decimal places) c1 = 16.49
- Similar to option B, 16.49 is relatively high given the asset base and incomes. This would require substantial consumption in period 1 while preserving enough resources to cover period 2, especially since there is no labor income in period 2. The CRRA Euler condition would need to justify why substantial c1 is optimal despite the need to smooth across a later period with limited income. If the implied present value of lifetime resources allows such a high c1 while still fulfilling the budget and the Euler equation, this could be feasible; otherwise, it would be unlikely.

Option E: The optimal consumption at t=1 is approximately (2-decimal places) c1 = 10.19
- This value sits in a mid-range that often arises in standard intertemporal consumption problems with CRRA utility, given the three-period horizon, β = 0.8, r = 5%, a0 = 5, y0 = 10, y1 = 15, and no income in period 2. The period-1 budget constraint combines assets, current income, and the present value of future resources, while the Euler equation ties the marginal utility of consumption across adjacent periods with the 5% return and the time preference β. If the computed shadow prices and the intertemporal trade-off point yield c1 around 10.19, this option would align well with typical results in such settings.

Putting these together, the correct option would be the one that best satisfies both the intertemporal budget constraint across periods 0–2 and the CRRA Euler condition, given the data provided. The other choices can be ruled out if their values cannot simultaneously satisfy the available assets, incomes, interest rate, and discounting structure, or if they conflict with the implied marginal rate of substitution implied by CRRA utility and β.

Note: Without performing the explicit numerical optimization (solving the Euler equations and budget constraints with the given CRRA parameters, β, r, and incomes), we cannot definitively confirm which single option is the numerical optimum. The reasoning above explains how one would evaluate each option against the model’s conditions and why some values are more or less plausible given standard intertemporal optimization logic.

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