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Econ 387: Assignment 1 David Andolfatto January 2006 Consider an economy populated by a representative agent with preferences for consumption and leisure given by the expected utility function: E[U (c, l)] = h(l) + βE[c], where β > 0 is a parameter, h satisfies h00 < 0 < h0 , and E[.] represents an expectation operator. The agent is endowed with one unit of time, that can be allocated either to work or leisure; i.e., n + l = 1. [1] Explain (in words only) the meaning of the restriction h00 < 0 < h0 . [2] Provide an economic interpretation for the preference parameter β. The return to work (i.e., the real wage) is a random variable, z ∈ {zL , zH }, with zH > zL > 0. The time-allocation decision must be made prior to the realization of z. Given a choice of work eﬀort n and given some realization of z, realized income (and hence consumption) is given by c = zn. Note that since z is a random variable, so is consumption. According to the utility function above, a higher level of expected (or ex ante) consumption generates a higher level of expected utility. Note that actual (or ex post) utility and consumption may diﬀer from their expected (ex ante) levels). To formulate their expectations, individuals need to know how z is distributed. Let π(z, s) denote the probability of z occurring conditional on the signal s. Assume that π(zH , s) is an increasing function of s. Thus, the expected return to work, conditional on s, is given by: E [z | s] = π(zH , s)zH + π(zL , s)zL . [3] Explain why E[z | s] is an increasing function of s. Can you think of any real-world examples in which changes in the expected return to an activity may depend on the arrival of information (or a signal)? We are now in a position to write down the (representative) agent’s choice problem. Since c = zn, we can write this as: max h(1 − n) + βE [z | s] n. n 1 The first-order condition characterizing the optimal (and in this case, equilibrium) level of employment n∗ (s) is given by: h0 (1 − n∗ ) = βE [z | s] . The left-hand-side (LHS) above measures the marginal utility cost of employment (i.e., in the form of foregone leisure); while the RHS measures the marginal utility benefit of employment (i.e., in the form of expected consumption). [5] From what you learned in Econ 331 (math econ), you should be able to derive the comparative static dn∗ /ds. Provide an economic interpretation. Do you think that (exogenous) fluctuations in s may be a plausible source of business cycle behavior? Explain. Is there a a potential role for a government stabilization policy in this world? Explain. [6] Find an expression for dn∗ /dβ and provide an economic interpretation. Do you think that fluctuations in β may be a plausible source of business cycle behavior? Explain. 2