ISI MSQE 2022 SOLUTION SUBJECTIVE ECONOMICS
Consider an agent living for two periods, 1 and 2. The agent maximizes lifetime utility, given by: 1 𝑈 (𝐶1) +1/(1 + 𝜌) 𝑈 (𝐶2), where 𝜌 > 0 captures the time preference, while 𝐶1 and 𝐶2 are the agent’s consumption in period 1 and period 2, respectively. The agent supplies one unit of labor inelastically in period 1, earning a wage 𝑤. A portion of this wage is consumed in period 1 and rest is saved (denoted 𝑠). 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, 𝐶1 + 𝑠 = 𝑤 and 𝐶2 = (1 + 𝑟)𝑠, where 𝑟 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-Ѳ)]-1/(1-Ѳ) WHERE 0<Ѳ<1
(a) Demonstrate, deriving your claim, how optimal savings, 𝑠, would respond to changes in 𝑟 .
(b) Now suppose, initially, 𝑟 = 𝜌. What happens to optimal savings, 𝑠, if 𝑟 and 𝜌 increase by the same amount (so that the condition 𝑟 = 𝜌 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 𝑠 = 𝑤 − 𝐶1, and the lifetime utility function can be rewritten as: U = [𝐶1^(1-Ѳ)]/(1-Ѳ) + 1/(1+𝜌) * [ (1 + 𝑟)𝑠 )^(1-Ѳ) ]/(1-Ѳ) Substituting 𝑠 = 𝑤 − 𝐶1 into the lifetime utility function yields: U = [𝐶1^(1-Ѳ)]/(1-Ѳ) + 1/(1+𝜌) * [ (1 + 𝑟)(𝑤−𝐶1) ]^(1-Ѳ)/(1-Ѳ) Taking the derivative of U with respect to 𝐶1 and setting it equal to zero gives the first-order condition: (1-Ѳ)𝐶1^(-Ѳ) - 1/(1+𝜌) * (1-Ѳ) * [(1+𝑟)(𝑤-𝐶1)]^(-Ѳ) * (1+𝑟) = 0 Simplifying and solving for 𝐶1 yields: 𝐶1 = [ (1+𝑟)/(1+𝑟+𝜌) ] * 𝑤 Therefore, optimal savings are given by: 𝑠 = 𝑤 − 𝐶1 = [ 𝜌/(1+𝑟+𝜌) ] * 𝑤 The optimal savings are decreasing in 𝑟. To see why, note that an increase in 𝑟 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 𝑟 and 𝜌 both increase by the same amount, so that 𝑟 = 𝜌 + 𝛿 for some positive 𝛿, then optimal savings would also increase. To see why, note that the increase in 𝑟 increases the return on savings, making saving more attractive. However, the increase in 𝜌 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 𝑟 is larger than the increase in 𝜌, then the agent would save more. If the increase in 𝑟 is smaller than the increase in 𝜌, then the agent would save less. If the increases are equal, then the optimal savings would increase proportionally.