Nash equilibrium - mistake in proof of paper?

Question

I have a question regarding the proof of Proposition 1 in Besley and Ghatak (2007) in Appendix A of their paper. It is a quite highly cited paper but I believe there is a mistake in the proof of their Proposition 1. The model is described in Section 2. of their paper. It is a very simple model that seeks to derive the Nash equilibrium of the market. In particular, they show that $\theta_c^*$ must satisfy $f'(n \theta_c^*) = \alpha$ in equilibrium. In their proof in the appendix they state:

However, I think there is a mistake because the package $(p', \widehat{\theta'})$ does not make each caring consumer strictly better off.

The new package $(p', \widehat{\theta'})$ gives utility:

$$b-p'+f(\widehat{\theta'} + (n-1)\widehat{\theta}) $$

The old package $(p, \widehat{\theta})$ gives utility:

$$b-p+f(n\widehat{\theta}) $$

The new package is strictly preferred to the old package iff

$$b-p'+f(\widehat{\theta'} + (n-1)\widehat{\theta}) > b-p+f(n\widehat{\theta})$$

$$ f(n\widehat{\theta} + \Delta \widehat{\theta}) - f(n\widehat{\theta}) > \Delta p$$

If we make the substitution as stated in the paper: $\Delta p = f'(n\widehat{\theta}) \Delta \widehat{\theta}$ and assuming we want a positive $\Delta \widehat{\theta}$, then the above becomes:

$$\frac{f(n\widehat{\theta} + \Delta \widehat{\theta}) - f(n\widehat{\theta})}{\Delta \widehat{\theta}} > f'(n \widehat{\theta}) $$

If we pick a negative $\Delta \widehat{\theta}$, then the above condition becomes

$$\frac{f(n\widehat{\theta} + \Delta \widehat{\theta}) - f(n\widehat{\theta})}{\Delta \widehat{\theta}} < f'(n \widehat{\theta}) $$

But $f$ is a increasing and strictly concave function, so the above inequalities are always false. So the new package is never preferred to the old package. Everything else in the proof is fine apart from this step and this is the most crucial step of the proof as it characterizes what $\widehat{\theta}$ should be in equilibrium, if this is wrong then many other results in the paper are also wrong.

So, is there a mistake in this part of the proof of the paper? And if so, is there a way to correct it so that the results still hold?

Answer 1

The objective is to show that, as long as $f'(n\hat\theta)\ne \alpha$, a firm can always engineer a package $(p',\hat\theta')=(p+\Delta p,\hat\theta+\Delta \hat{\theta})$ such that (i) a caring consumer strictly prefers this package, and (ii) the firm makes positive profits.

1) Your maths are correct: the inequalities are always false when f is increasing and strictly concave. In that case, the proof in the paper is unclear. The package $(p',\hat\theta')=(p+f'(n\hat\theta)\Delta \hat{\theta},\hat\theta+\Delta \hat{\theta})$ does not satisfy (i).

2) You can show the proposition with the following reasoning. Assume $f'(n\hat\theta)>\alpha$. Take any $h$ within the interval $(\alpha,f'(n\hat\theta))$. For $\Delta \hat{\theta}$ small enough and positive, a firm that offers the package $(p',\hat\theta')=(p+h\Delta \hat{\theta},\hat\theta+\Delta \hat{\theta})$ satisfies (i) and (ii). With a first-order appoximation, we show (i): $$b-p'+f(\hat\theta'+(n-1)\hat\theta)>b-p+f(n \hat\theta),$$ and (ii): $$\Delta p - \alpha \Delta \theta>0.$$

Assume $f'(n\hat\theta)<\alpha$, and take $h$ within the interval $(f'(n\hat\theta),\alpha)$. For $\Delta \hat{\theta}$ small enough and negative, a firm that offers the package $(p',\hat\theta')=(p+h\Delta \hat{\theta},\hat\theta+\Delta \hat{\theta})$ satisfies (i) and (ii).