Assignment 1

Assignment 1

Setup

Consider the following simple model of time allocation. Individual utility is given by:

\[ U(C,L) = (\phi C^{\rho} + (1-\phi) L^{\rho})^{1/\rho} \]

where \(L\) is an aggregate leisure good composed of \(K\) different activities:

\[ L = \prod_{k=1}^{K}l_{k}^{\delta_{k}},\qquad \sum_{k}\delta_{k} = 1 \]

In addition to these leisure activities, the agent may supply labor to the market at a wage rate of \(w\). Letting \(h_{k}\) be hours, the time constraint is:

\[ h_{k} + \sum_{k}l_{k} = 1 \]

The model is static and the individual solves the following problem:

\[ \max_{C,\{l_{k}\}_{k=1}^{K}} U(C,L) \]

subject to the constraint:

\[ C + w \left(\sum_{k} l_{k}\right) \leq w \]

Question 1

Suppose you are interested in using this model to study the effects of a wage subsidy on labor supply.

Notice that the model can be written as

\[ \max_{C,h} U(wh,L^*(1-h)) \]

where

\[ L^*(1-h) = \max_{\{l_{k}\}_{k=1}^{K}} \prod_{k=1}^{K}l_{k}^{\delta_{k}} \]

subject to \(\sum_{k}l_{k} = 1-h\). Given this simplification, what does Marschak’s Maxim (and common sense) suggest about what parameters need to be estimated here?

Question 2

Based on your answer to the above, you simplify the model to the following specification:

\[ h^* = \arg\max (\phi (wh)^{\rho} + (1-\phi) (1-h)^{\rho})^{1/\rho} \]

and you derive the following relationship:

\[ \log\left(\frac{C}{L}\right) = \frac{1}{1-\rho}\log\left(\frac{\phi}{1-\phi}\right) + \frac{1}{1-\rho}\log(w) \]

where \(C=wh^*\) is total labor income and \(L=1-h^*\) is non-market time.

Suppose you have a cross-section of data \((C_{n},L_{n},W_{n})\) where \(C_{n}\) is labor market earnings, \(L_{n}\) is non-market time, and \(W_{n}\) is the wage-rate for person \(n\). This could be taken (for example) from the Outgoing Rotation Group of the CPS monthly survey.

How does this model compare to what you are likely to see in the data?

Question 3

Suppose now you augment the model to acommodate some randomness in how much individuals work by allowing for heterogeneity in preferences (\(\phi\)):

\[ \log\left(\frac{C}{L}\right) = \frac{1}{1-\rho}\log\left(\frac{\phi_{n}}{1-\phi_{n}}\right) + \frac{1}{1-\rho}\log(w) \]

What assumption do you need for an OLS regression of \(\log(C_{n}/L_{n})\) on \(\log(W_{n})\) to consistently recover the elasticity of labor supply, \(1/(1-\rho)\)? Do you consider this credible? Why/why not?