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Consider an agent living for two periods, 1 and 2. The agent maximizes lifetime utility, given by: 1 ๐‘ˆ (๐ถ1) +1/(1 + ๐œŒ) ๐‘ˆ (๐ถ2), where ๐œŒ > 0 captures the time preference, while ๐ถ1 and ๐ถ2 are the agentโ€™s consumption in period 1 and period 2, respectively. The agent supplies one unit of labor inelastically in period 1, earning a wage ๐‘ค. A portion of this wage is consumed in period 1 and rest is saved (denoted ๐‘ ). In period 2 the agent does not work, but receives interest income on the savings. Principal plus the interest income on savings goes to finance period 2 consumption. Thus, ๐ถ1 + ๐‘  = ๐‘ค and ๐ถ2 = (1 + ๐‘Ÿ)๐‘ , where ๐‘Ÿ is the rate of interest. Assume that the per period utility function can be represented by (and only by) any positive linear transformation of the form U (C ) = [C^(1-ัฒ)]-1/(1-ัฒ) WHERE 0<ัฒ<1

(a) Demonstrate, deriving your claim, how optimal savings, ๐‘ , would respond to changes in ๐‘Ÿ .

(b) Now suppose, initially, ๐‘Ÿ = ๐œŒ. What happens to optimal savings, ๐‘ , if ๐‘Ÿ and ๐œŒ increase by the same amount (so that the condition ๐‘Ÿ = ๐œŒ continues to hold)?

ANSWER (a) To find the optimal savings, we need to maximize the agent's lifetime utility subject to the budget constraint. The budget constraint can be rewritten as ๐‘  = ๐‘ค โˆ’ ๐ถ1, and the lifetime utility function can be rewritten as: U = [๐ถ1^(1-ัฒ)]/(1-ัฒ) + 1/(1+๐œŒ) * [ (1 + ๐‘Ÿ)๐‘  )^(1-ัฒ) ]/(1-ัฒ) Substituting ๐‘  = ๐‘ค โˆ’ ๐ถ1 into the lifetime utility function yields: U = [๐ถ1^(1-ัฒ)]/(1-ัฒ) + 1/(1+๐œŒ) * [ (1 + ๐‘Ÿ)(๐‘คโˆ’๐ถ1) ]^(1-ัฒ)/(1-ัฒ) Taking the derivative of U with respect to ๐ถ1 and setting it equal to zero gives the first-order condition: (1-ัฒ)๐ถ1^(-ัฒ) - 1/(1+๐œŒ) * (1-ัฒ) * [(1+๐‘Ÿ)(๐‘ค-๐ถ1)]^(-ัฒ) * (1+๐‘Ÿ) = 0 Simplifying and solving for ๐ถ1 yields: ๐ถ1 = [ (1+๐‘Ÿ)/(1+๐‘Ÿ+๐œŒ) ] * ๐‘ค Therefore, optimal savings are given by: ๐‘  = ๐‘ค โˆ’ ๐ถ1 = [ ๐œŒ/(1+๐‘Ÿ+๐œŒ) ] * ๐‘ค The optimal savings are decreasing in ๐‘Ÿ. To see why, note that an increase in ๐‘Ÿ makes saving more attractive because the agent earns a higher return on savings. As a result, the agent would reduce current consumption in period 1 and save more for period 2.

(b) If ๐‘Ÿ and ๐œŒ both increase by the same amount, so that ๐‘Ÿ = ๐œŒ + ๐›ฟ for some positive ๐›ฟ, then optimal savings would also increase. To see why, note that the increase in ๐‘Ÿ increases the return on savings, making saving more attractive. However, the increase in ๐œŒ reduces the weight placed on future consumption, making current consumption relatively more attractive. The net effect depends on the relative magnitudes of the two effects. If the increase in ๐‘Ÿ is larger than the increase in ๐œŒ, then the agent would save more. If the increase in ๐‘Ÿ is smaller than the increase in ๐œŒ, then the agent would save less. If the increases are equal, then the optimal savings would increase proportionally.

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