Question 1 Consider individuals who live for two periods. In the first period they earn $500 in income, and in the second period they are retired. Their utility in each period is equal to In(c) where c is their consumption in that period. They equally value utility in Period 1 and Period 2 (B = 1). a) How much money will they consume in each period? b) Suppose they also derive utility from leisure. Will they necessarily consume the same amount in both periods (yeso)? Will they consume more or less if consumption and leisure are substitutes? What if they are complements? Provide some reasons (in words). c) Suppose they no longer value leisure, and no longer equally value utility in Period 1 and Period 2, but instead value utility in the second period at half of the value of utility in the first period (p = 0.5). In other words, now total utility is ln(c1) + 0.5 * In(C2), where C1 is consumption in the first period and C2 is consumption in the second period. How much will they consume in each period? Now suppose there is no discounting (so we are back to = 1), but now individuals might die with 20% probability after the first period. d) Assuming they only care about consumption (not leisure anymore), how much will they consume in each period? e) Suppose the government institutes a welfare program in which everyone with less than $225 in their bank account will receive government money up to the point where their consumption is $225. How much will they consume in each period? Suppose there is no welfare program, but instead an insurance company sells an actuarially fair priced "annuity" for a premium p that pays out an amount z in the second period if they survive. Assuming no savings or borrowing this time, solve for p and z. How much do they consume in each period? Question 1 Consider individuals who live for two periods. In the first period they earn $500 in income, and in the second period they are retired. Their utility in each period is equal to In(c) where c is their consumption in that period. They equally value utility in Period 1 and Period 2 (B = 1). a) How much money will they consume in each period? b) Suppose they also derive utility from leisure. Will they necessarily consume the same amount in both periods (yeso)? Will they consume more or less if consumption and leisure are substitutes? What if they are complements? Provide some reasons (in words). c) Suppose they no longer value leisure, and no longer equally value utility in Period 1 and Period 2, but instead value utility in the second period at half of the value of utility in the first period (p = 0.5). In other words, now total utility is ln(c1) + 0.5 * In(C2), where C1 is consumption in the first period and C2 is consumption in the second period. How much will they consume in each period? Now suppose there is no discounting (so we are back to = 1), but now individuals might die with 20% probability after the first period. d) Assuming they only care about consumption (not leisure anymore), how much will they consume in each period? e) Suppose the government institutes a welfare program in which everyone with less than $225 in their bank account will receive government money up to the point where their consumption is $225. How much will they consume in each period? Suppose there is no welfare program, but instead an insurance company sells an actuarially fair priced "annuity" for a premium p that pays out an amount z in the second period if they survive. Assuming no savings or borrowing this time, solve for p and z. How much do they consume in each period
