Tax evasion and productivity

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# Tax evasion and productivity
### Hans Martinez
### May 20, 2022

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## Agenda

- Revealed P firms: Acceptance rule
  - pseudo-code: `$log(U) < \gamma'[g(x,e_{jump})-g(x,e_{r-1})]$`. 
  - Actual code twice: 
    - chain `$log (U) < ||g(x,e_{jump})-g(x,e_{r-1}) ||$`
    - Optimization `$log(U) < \gamma'[g(x,e_{jump})-g(x,e_{r-1})]$`
  - `$\gamma$`  
  
- Tax evasion
  - Can we use an IC as location condition?

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# Tax evasion and productivity in Mexico

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### Tax evasion and productivity

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* Firms have incentives to under-report profits to avoid taxes. Productivity estimates are likely to be biased.

* Mexico: Non-detected tax evasion up to $10 billion USD per year .small[(Zumaya et al., 2021)]

* Assume firms only over-report expenses in flexible input, `$\Pi(x)=PY-\rho X$`.

* Consequently, once they over-report and reach zero profits, firms don't have incentives to over-report more, `$\Pi(\tilde x)=0$`.

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* **Question**: Can we recover productivity estimates in the presence of measurement error by using a micro-founded IC as location condition?

* **Data**: INEGI-EMIM, Anonymized tax filings (?) .small[(Zumaya et al., 2021)]

* **Approach**: Hu & Schennach (2008), Hu (2021), Hu & Yao (*J. Econom., forthcoming*)

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`$$\begin{aligned}
  Y&=G(x^*)e^{\omega+\varepsilon} \\
  X&=X^*+\Delta X(\theta) \\
  \Delta X(\theta) &= \left \{ \begin{array}{ll}
    0 & (1-\tau)\Pi(x^*) \ge \Pi(x^*)-\tau\Pi(x)-\kappa Pr(a)-c(\Delta x) \\
    \mathbf (0,\tilde x] & otherwise
  \end{array} \right.\\
  &\text{where }\; \Pi(x)=PY-\rho X \\
  &\text{and}\; \Pi(\tilde x)=0 \\
\end{aligned}$$`

`$\tau$`: tax; `$\kappa$` cheating fine if the firm is caught by the authority; `$Pr(a)$`: probability of getting caught if cheating.

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Can we use a micro-founded IC as location condition? `$M[f]=x^*$`, where `$M[f]\equiv$`

`$$\begin{aligned}
  \sup\left \{ v: \int^{v}_{x^*}  \left \{ \tau [\Pi(x^*)-\Pi(x)]-c(||x-x^*||)\right\} f_{X|X^*}(x|x^*) dx \le\kappa Pr(a)  \right\}
\end{aligned}$$`

Problems: 
  - `$c(||x-x^*||)$` is unknown 
    - `$\implies$` different mass for every approximation
    - function of firm characteristics
    - convex
  - `$Pr(a)$` could be subjective to the firm

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### How to estimate?

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`\begin{equation}
  f(y_{it},x_{it},p_{it})  = \\
  f(y_{it},x_{it},p_{it},\omega_{it},\omega_{it-1},x^*_{it})= \\
  f(y_{it}| x^*_{it},\omega_{it})
  f(p_{it}|x^*_{it},\omega_{it})
  f(\omega_{it}|\omega_{it-1})
  f(x_{it}|x^*_{it})
  f(\omega_{it-1})f(x^*)
\end{equation}`

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`\begin{equation}
  \color{blue}{f(y_{it}| x^*_{it},\omega_{it})}
  \color{green}{f(p_{it}|x^*_{it},\omega_{it})}
  \color{violet}{f(\omega_{it}|\omega_{it-1})}
  \color{red}{f(x_{it}|x^*_{it})}
  \color{orange}{f(\omega_{it-1})f(x^*_{it})}
\end{equation}`

`\begin{equation}
    \color{blue}{ y_{it} = g(x^*_{it})+\omega_{it}+\varepsilon^Y_{it}} \\
  \color{green}{ \ln\left(\frac{\rho_t X^*_{it}}{P_{t}Y_{it}}\right)=\ln\left(\frac{G_x(x^*_{it})X^*_{it}}{G(x^*_{it})}\right) - \varepsilon^Y_{it}} \\
  \color{violet}{\omega_{it}=\rho\omega_{it-1}+\varepsilon^\omega_{it}} \\
  \color{red}{M[f]=x^*} \\
  \color{orange}{\text{Flexible input assumption.}}
\end{equation}`

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In addition:

`$$\begin{aligned}P(cheated)=&\hat P(cheated|Audited)\hat P(Audited)+\\
&P(cheated|Not Audited)\hat P(Not Audited)
\end{aligned}$$`

Note that `$P(cheated|Not Audited)$` and, consequently, `$P(cheated)$` are unobserved.

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- [Hu & Yao (Journal of Econometrics,2022)](https://doi.org/10.1016/j.jeconom.2021.05.007) use sieve maximum likelihood estimator (MLE)

- Mix ELVIS and sieve MLE (?)

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<div class="figure">
<img src="NK2022_files/figure-html/unnamed-chunk-1-1.png" alt="Link to slides: Scan me!" width="100%" />
<p class="caption">Link to slides: Scan me!</p>
</div>

[hansmartinez.com](hansmartinez.com)