# Why is Walras equilibrium inefficient when we are dealing with public goods?

I know that when we have public goods we have that:

$$MRT = MRS_a + MRS_b$$

Though I fail to understand why does this makes Walras equilibrium inefficient.

Thank you very much for your help!

In a competitive market for a private good (y) individuals may consume different quantities but the equilibrium condition requires that: $$\frac{\frac{\delta u^{i}}{\delta y}}{\frac{\delta u^{i}}{\delta x}} = MRS^{i}_{yx} = MRT_{yx} \; \forall \; i$$
In the case of a public good (g) individuals may have different MRS but consume the same amount of the public good. Then for a given amount of a public good, the marginal benefit is the sum of individual marginal rates of substitution. Therefore, the optimal allocation must satisfy: $$\sum_{i} \frac{\frac{\delta u^{i}}{\delta g}}{\frac{\delta u^{i}}{\delta x}} = \sum_{i} MRS^{i}_{gx} = MRT_{gx}$$
• Another way to interpret his response is that in a Walrasian equilibrium each agent maximizes his own utility with disregard of other people's utility (hence each $MRS$ equals the $MRT$, the first one captures agents' preferences while the second one captures the real costs of producing a good. However, if there is a public good, it means that it is a good that can be consumed by all agents simultaneously. For example a Library. Individually it might be too expensive for any one individual to pay its cost (so in a Walras equilibrium it will not be built), since $MRS_i<MRT$ for all $i$. – Regio Apr 22 '19 at 17:15
• [...] but if you add the benefit to all people, then it can be optimal to build the library: $\sum_{i=1}^N MRS_i \geq MRT$. Here I assumed that the decision is just to build or not to build, now think that you can also decide how big of a library to build. Hopefully you can see that in a Walrasian equilibrium people will build a library too small, while the efficient size would be to build a library big enough so that $\sum_{i=1}^N MRS_i=MRT$ – Regio Apr 22 '19 at 17:17