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Socially $0$ptimal Level$\newline$How does this compare to the socially optimal level of provision? The social optimum is the quantity at which the sum of the individuals' marginal rates of substitution equals the ratio of prices (which is $1$ in this example). Each individual's $M R S$ is the ratio of his marginal utility of fireworks to his marginal utility of private goods, which we obtain by differentiating the previous utility function with respect to fireworks and then again with respect to private goods. So the optimal amount of fireworks is determined by:$\newline$$$\left(100-F_{B}\right) /\left[2 \times\left(F_{B}+F_{J}\right)\right]+\left(100-F_{j}\right) /\left[2 \times\left(F_{B}+F_{j}\right)\right]=1$$$\newline$Using the fact that total fireworks $F=F_{B}+F_{\beta}$ we can rewrite this equation as:$\newline$$$(200-F) / 2 F=1$$$\newline$Solving this, we obtain $F=66.6$. This quantity is much higher than the total provision by the private market, $40$ , due to the free rider problem. The public good is underprovided by the private market.