Proceedings from the ECFIN Workshop "The budgetary implications of structural reforms" - Brussels, 2 December 2005

unemployed worker finds a new job with probability p_t^u while a currently employed worker
looses his job and becomes unemployed with probability p_t^e . Both employed and unemployed
workers can expect the same pension. The budget constraint of both types is given by

A_t^wj =(1+r_t)A_t^w_-^j₁ +w_t^w -C_t^wj                                                      (9)

where the superscript w = {e, u} indicates the employment status of worker j. The FOC of worker
ej is given by

(1-λt^w)((1-pt^e)Ct^e+^j1+pt^eCt^u+^j1)+λt^wΛt+1Ct^r+^j1=((1+rt)Ωtβ)^σCt^ej              (10)

The FOC of unemployed worker uj is given by

(1-λt^w)((1- pt^u)Ct^u+^j1+ pt^uCt^e+^j1)+λt^wΛt+1Ct^r+^j1= ((1+rt)Ωtβ)^σCt^uj             (11)

σ

Where Λ_t = ε_t^1-σ is the marginal rate of substitution of consumption across work and retirement
and Ω_t is a factor that adjusts the rate of time preference for the fact that the worker can be in a
different state next period. It is given by

1

ω t = (1 - λw ) + λwεp^σ                                           (12)

The worker determines the level of consumption in period t such that the ratio of the marginal
utility of consumption tomorrow vs. today is equal to the difference between the real interest rate
and the rate of time preference. The expected marginal utility of consumption in the next period is
a weighted average of consumption in the two possible states. Notice, because of risk neutrality
only the mean of consumption in the two states matters. A closed form decision rule for workers
consumption can be derived and is given by

C_t^wj = π_t [A_t^wj +H_t^wj +S_t^wj]                                                     (13)

where the marginal propensity to consume out of wealth is given by

∏t = 1 -(((1 + r )Ω t )^σ-1 β^σ )^π.                                            (14)

πt+1

The term H ^w is the present discounted income of an employed or unemployed worker. Because
workers can switch employment status randomly in the next period, H ^e and H ^u are best
represented by the following arbitrage conditions

(1 -λ^w)

He = -(----((1 - pe ) Hej₁ + peHuj ₁ )+ we                              (15)

t (1 ₊ r )_Ω t     t+1      t t+1       t

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