Assignment 1

Setup 1: A Simple Labor Supply Model

Consider a dynamic labor supply model (with no uncertainty) where each agent \(n\) chooses a sequence of consumption and hours, \(\{c_{t},h_{t}\}_{t=1}^{\infty}\), to solve: \[ \max \sum_{t=0}^\infty \beta^{t} \left(\frac{c_{t}^{1-\sigma}}{1-\sigma} - \frac{\alpha_{n}^{-1}}{1 + 1/\psi}h_{t}^{1+1/\psi}\right)\] subject to the intertemporal budget constraint: \[ \sum_{t}q_{t}c_{t} \leq A_{n,0} + \sum_{t}q_{t}W_{n,t}h_{t},\qquad q_{t} = (1+r)^{-t}.\] Let \(H_{n,t}\) and \(C_{n,t}\) be the realizations of labor supply for agent \(n\) at time \(t\). Labor supply in this model obeys: \[H_{n,t}^{1/\psi} = (\alpha_{n}W_{n,t})C^{-\sigma}_{n,t}.\] To simplify below, assume that \(\beta=(1+r)^{-1}\), so that the optimal solution features perfectly smoothed consumption, \(C^*_{n}\). Making appropriate substitutions gives \(C^*_{n}\) as the solution to: \[ \left(\sum_{t}q_{t}\right)C^*_{n} = \sum_{t}\left(q_{t}W_{n,t}^{1+\psi}\right)\alpha_{n}^{\psi}(C_{n}^*)^{-\psi\sigma} + A_{n,0}.\]

Setup 2: A Diff-in-Diff Analysis

You are talking to a friend (let’s call them Mr Straw Man) who has data on labor supply for two groups, \(A\), and \(B\), who are not directly comparable. Assume they have two periods of data (\(s=1,2\)), and that group \(A\) is subjected to a proportional wage subsidy, \(1+\tau\), in period 2 (\(s=2\)). In this case the effective wage is \(\tilde{W}_{n,t} = (1+\tau)W_{n,t}\). Your friend claims that they can estimate the elasticity of labor supply by first estimating the response to the tax using difference-in-differences, which has a population limit:

\[ \alpha = \mathbb{E}[\log(H)|A,s=2] - \mathbb{E}[\log(H)|B,s=2] - (\mathbb{E}[\log(H)|A,s=1] - \mathbb{E}[\log(H)|B,s=1])\] and then constructing the elasticity as: \[ \tilde{\psi} = \frac{\alpha}{\log(1+\tau)}.\]

Your friend also claims that this elasticity is sufficient to forecast the effects of any other labor supply subsidy in the future.

Questions

In answering the questions below, you can assume that the distribution of wages and preferences adhere to the following relationship for each group \(g\in\{A,B\}\): \[\mathbb{E}[\log(W_{n,t})|g] = \mu_{W,g} + \gamma_{t}\] \[\mathbb{E}[\log(\alpha_{n})|g] = \mu_{\alpha,g}\]

  1. Show that this model satisfies the “parallel trends assumption”, namely that if the policy had never been introduced, we would get \[ \mathbb{E}[\log(H)|A,s=2] - \mathbb{E}[\log(H)|B,s=2] - (\mathbb{E}[\log(H)|A,s=1] - \mathbb{E}[\log(H)|B,s=1]) = 0 \]
  2. Show that when the policy is introduced unexpectedly, in period 2, the population limit \(\tilde{\psi}\) can be written as: \[ \tilde{\psi} = \psi\left(1 - \sigma\frac{\Delta\mathbb{E}\left[\log(C^*)|A\right]}{\log(1+\tau)}\right)\] where \[\Delta\mathbb{E}\left[\log(C^*)|A\right]\] is the growth in log consumption induced by the policy.
  3. Derive a similar expression for \(\tilde{\psi}\) for the case where the policy is announced in the first period (\(s=1\)) and implemented in the second period (\(s=2\)).
  4. Use the model to argue that the size of the effect depends on whether the subsidy is perceived as temporary or permanent.
  5. Suppose that the exact same policy (same anticipation, same duration, same \(\tau\)) were to be implemented on group \(B\). Under what assumptions on the distribution of assets could we expect the same labor supply response in the model? Does this seem reasonable?
  6. Use these derivations to argue that:
    1. The estimand \(\tilde{\psi}\) does not map directly to an underlying structural parameter of interest; and
    2. The parameter \(\tilde{\psi}\) cannot be used to forecast the effect of wage subsidies that are either (a) larger than \(\tau\); or (b) implemented over different horizons; or (c) implemented on group \(B\).