Present bias may induce individuals to save too little for old age, when they no longer have labour income. Consider the following model to address this concern formally. Assume that an individual lives for three periods, t = 0, 1, 2 and has income w in both periods 0 and 1 but zero income in period 2. In each period t the individual derives utility ut(xt) = ln(xt) from consumption xt . The individual discounts the future. Lifetime utility Ut from the perspective of period t = 0, respectively t = 1 is given by: U0 = u0 + u1 + 2u2 and U1 = u1 + u2, where 0 < 1 and 0 < < 1. The individual can borrow and save at an interest rate of 0, so that total consumption expenditure must equal the sum of income.

(a) Derive the individual's preferred consumption plan (x 0 , x 1 , x 2 ) from a period t = 0 perspective (i.e. maximising U0). Explain the roles of parameters and .

(b) Derive the individual's preferred consumption pattern (x N 1 , xN 2 ) from a period t = 1 perspective (i.e. maximising U1) for a given value of x0. Explain the difference, if any, between (x N 1 , xN 2 ) and (x 1 , x 2 ).

(c) Suppose that x N 1 > x 1 , but that the individual's preferred consumption x N 0 in period 0 when anticipating that later consumption will be (x N 1 , xN 2 ) is the same as x 0 , i.e. the individual would prefer to save a higher proportion of their period 1 income from a period 0 point of view than from a period 1 point of view. If the individual is aware of their present bias, could a financial product that prevents borrowing, requires an investment k in period 0 and pays back k in period 2 increase period 0 utility U0? Argue verbally or compute.