A household lives for two periods, has a natural logarithm utility function, and discounts the future at the rate of ?. Borrowing is not allowed.

In period 1, the young household consumes c1, saves s1, and

works for wage w1. In period 2, the old household consumes c2, receives gross return from saving (interest rate is r), and faces unemployment risk. With a probability p, the household keeps his job and earns a wage w2, and with a probability of 1 ? p, the household loses his job.

The government taxes all employed workers (both young and old) at a proportional rate of ? and evenly transfers to all unemployed workers. The population is fixed.

1- What does an unemployed worker receive as gov't transfer?

2- Write down the household's maximization problem

3- Derive Euler equation by solving household problem

My attempt at solving is in the image below... Is this right?

The economy is populated by two generations of households working for SOEs. A household lives for two periods, has a natural logarithm utility function, and discounts the future at the rate of B. Borrowing is not allowed. In period 1, the young household consumes cl, saves s1, and works for wage wl. In period 2, the old household consumes c2, receives gross return from saving (interest rate is r), and faces unemployment risk. With a probability p, the household keeps his job and earns a wage w2, and with a probability of 1 - p, the household loses his job. The government taxes all employed SOE workers (both young and old) at a proportional rate of T and evenly transfers to all unemployed SOE workers. The population is fixed. (a) In this economy, how much does an unemployed SOE worker receive from the government transfer? (2pts) by = (WIT) + (pw27) (1 - p) (b) Write down the maximization problem of the household. (1pt) mar log(c ) + BE[log(c2)] s.t. C +$1 = w1(1-T) c2 = (1 +r)s1 + p(w2 - 7) + (1 -p)- WIT + PWIT 1 -p (c) Derive the Euler equation by solving the household problem. You must lay out the first order conditions clearly. (2pts) Lagrangian - = log(c])+BE [log(c2)] -Ala+= C2 - WI(l-7) _plug -T) TWI + PTW21 1+r 1+r Euler - = B(1 +r)E