# Pigouvian subsidies for non-linear marginal external benefits

Suppose an agent is choosing how much $x$ to consume and $p_x=200$. If the agent's marginal private benefit is $MPB(x)=300/x$ and $x$ takes only integer values, then he will choose to purchase just one unit of $x$. But suppose his consumption of $x$ yields a positive externality such that the marginal external benefit is $MEB(x)=2700/x$ (essentially there are nine other people similar to the agent who benefit as much from the agent's consumption as the agent does). Then the social optimum is where the marginal social benefit equates with the marginal social cost ($p_x$): $$MSB(x)=MPB(x)+MEB(x)=\frac{3000}{x} = 200 \Longrightarrow x^* = 15$$ If I were to ask what the appropriate Pigouvian subsidy was to implement this societally optimal outcome, I can envisage two possible answers:

1. Set the marginal subsidy equal to the marginal external benefit: $MS(x)=\frac{2700}{x}$. Note that, in this case, the government would end up paying the agent $\sum_{x=1}^{15}\frac{2700}{x} \approx 8959$ in subsidies, which is substantially more than the agent's outlay ($200 \cdot 15 = 3000$).

2. Set the marginal subsidy equal to the marginal external benefit evaluated at the optimal output: $MS(x)=MEB(15)=2700/15=180$. Total subsidy is $180 \cdot 15 = 2700$.

3. Set the marginal subsidy as low as possible to get him to purchase each unit up to 15: $MS(1)=0$. $MS(x) = 200 - \frac{300}{x}$ for $2 \leq x \leq 15$. Total subsidy is $\sum_{x=2}^{15} 200 - 300/x \approx 2105$. I suppose here I could even tax the first unit to bring the total net subsidy down further!

Are all of these correct? Is any of them a better answer than the other two?

• I think 2 would be how a Pigouvian subsidy (a per unit subsidy) is usually implemented. – Herr K. Mar 2 '17 at 18:11
• @HerrK. I agree. On the other hand, if $MEB(x)$ were increasing in $x$ (and $MC(x)$ was also increasing enough that the optimal $x$ was still in the interior), then 2) wouldn't implement the optimal solution. But I can't think of a compelling example of that in the real world. – Shane Mar 2 '17 at 19:36
• I'm not sure I follow. A (per unit) Pigouvian subsidy ($s$) just need to satisfy $MPB(x)+s=MC(x)\bigg\vert_{x=x^*}$, at the optimum, where $x^*$ is the socially optimal quantity satisfying $MPB(x^*)+MEB(x^*)=MC(x^*)$ – Herr K. Mar 2 '17 at 22:02
• @HerrK. I think you're right. When I try to come up with an example of such an $s$ failing to implement the optimal outcome, I fail. Not sure how I had that intuition. – Shane Mar 6 '17 at 16:43