Agents are assumed to live for three periods in this model. In periods 1 and 2 they work, in period 3 they retire. Wages are given by \(w_{1},w_{2}\) in periods 1 and 2. All agents are identical conditional on age. There exists a perfect capital market with constant interest rate \(r\), individuals are standard exponential discounters with discount factor \(\beta\) and the price of consumption acts as the numeraire in each period, i.e. it is normalized to one. Let’s call assets at the start of period 1 \(A_{1}\), and we assume that after period 3 all individuals die, and they must have non-negative assets at that point. There is no bequest motive, so everything needs to be consumed by the end of period 3. We assume the following period perferences in an otherwise time-separable lifetime utility function (very standard): \[
U(c_{t},l_{t})=\alpha\ln c_{t}+(1-\alpha)\ln l_{t}
\] and point out that \(L-h_{t}=l_{t}\), i.e. \(h\) is hours worked and \(L\) is total time endowment.
- Write down the consumers lifecycle maximization problem at age 1.
- Call \(\lambda\) the Lagrange Multiplier on the budget constraint and solve this problem. Provide an expression for \(\lambda\). Show and provide intuition for \(\frac{\partial\lambda}{\partial A_{1}}<0,\frac{\partial\lambda}{\partial w_{t}}<0\).
- Find both the Marshallian and Frischian Labor Supply functions \(h_{t}^{*}(w_{1},w_{2},A_{1})\) and \(h_{t}^{F}(w_{t},\lambda)\).
- Take the following parameter values and evaluate your optimal policy functions for consumption, leisure and assets: \[\alpha=0.3,\beta=0.9,L=8700,A_{1}=1000,r=0.05,w_{1}=5,w_{2}=10\]
- Your friend estimates the regression equation \[\Delta\ln h_{2}=\sigma\Delta\ln w_{2}+u_{2}\] using OLS and he claims to be estimating the Frisch elasticity of labor supply. What’s the value of the estimate \(\hat{\sigma}\)? What’s the estimate’s standard error? (Hint: no statistics software needed to answer this question.)
- Evalute the Frisch elasticity under the numerical values from question 4. How would those results change if \(A_{1}=20000\)? Why? For the rest of the problem, use \(A_{1}=1000\). Then calculate the Hicksian elasticity of labor supply in period 1 (i.e. keep discounted lifetime utility constant).
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