We are now going to assume that the consumer receives from the labor she provides to her employer. Let represent the quantity of labor undertaken by the consumer. For each unit of labor, she is paid wage . We are therefore assuming that = . The consumer does not know the wage she will earn in the future, so + is a random variable from the point of view of the consumer at period , for ≥ 1. We assume that the consumer does not enjoy undertaking labor, and therefore increased labor reduces her utility. We assume the consumer has the following utility function, ∑ =∞ =0 ( 1− 1 − − 1+ 1 + ), where > 0 and ≠ 1, > 0, > 0. (c) [7.5 points] Set up the Lagrangean for the consumer and solve for the first-order conditions for both consumption and labor. Remember that the consumer will be maximizing over the consumption and labor choice each period, i.e. they are choosing both , +1, +2, … and , +1, +2, ….

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Lagrangean and FirstOrder Conditions Step 1 Set up the Lagrangean The consumer wants to maximize their expected utility subject to their budget constraint The budget constraint in each period is Ct wt Lt The expected utility function is EU 1 Ct1 1 UCt1 Lt1 where Ct and Lt are consumption and labor in period t wt is the wage in period t is the discount factor is the parameter for diminishing marginal utility of consumption UCt Lt is the period utility function We can formulate the Lagrangian as follows L 1 Ct1 1 UCt1 Lt1 t wt Lt Ct where t is the Lagrange multiplier for the budget constraint in period t Step 2 FirstOrder Conditions We need to find the firstorder conditions for Ct and Lt to maximize the ... View the full answer