<p>In a competitive market for a private good (<em>y</em>) individuals may consume different quantities but the equilibrium condition requires that: <span class="math-container">$$ \frac{\frac{\delta u^{i}}{\delta y}}{\frac{\delta u^{i}}{\delta x}} = MRS^{i}_{yx} = MRT_{yx} \; \forall \; i $$</span></p>

<p>In the case of a public good (<em>g</em>) 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 <strong>sum</strong> of individual marginal rates of substitution. Therefore, the optimal allocation must satisfy: <span class="math-container">$$ \sum_{i} \frac{\frac{\delta u^{i}}{\delta g}}{\frac{\delta u^{i}}{\delta x}} = \sum_{i} MRS^{i}_{gx} = MRT_{gx} $$</span></p>

<p>You can easily see that both conditions are not the same. In the case of an economy with only private goods, the benefit to
society of the last unit of a private good provided (expressed as the
willingness to forgo units of another good) is equal to the benefit of the
one person in society who receives this last unit. If there are some
people who receive a higher benefit from the last unit than others, we
don’t have Pareto efficiency. Hence, the marginal benefit of a private
good must be the same across all people for the allocation to be
efficient. <a href="http://jaimedv.com/eco/3c2-publi/intermediate-public-economics.pdf" rel="nofollow noreferrer">Hindriks and Myles</a> explain this thoroughly if you want to read further on the Samuelson condition versus competitive equilibrium (see Chapter 8 in particular). </p>

<p>Edit: There is no benefit that producers get here. Think of the MRT as the opportunity cost of producing an extra unit of a good and of the MRS as the marginal benefit of another unit of a good. The marginal cost of a unit of public good is one unit of private good (i.e. the MRT). Therefore the rule says that an efficient allocation is achieved when the total marginal benefit of another unit of the public good, which is the sum of the individual benefits, is equal to the marginal cost of another unit. If the good were private then the marginal benefit of an extra unit can only be attained by a single individual, so there is no sum. </p>

CC BY-SA 4.0
2019-04-22T11:56:36.830Z
Thank you very much for your answer, so if I get it right one of the main problems is that the benefit that the producer gets isn't the same as the benefit that the consumers get, is that correct? (though that the case for Pareto efficiency, what about the Walras efficiency?)
Fozoro
CC BY-SA 4.0
2019-04-22T12:28:33.957Z
I edited my answer given your comment, I hope this helps.
Ali
CC BY-SA 4.0
2019-04-22T17:15:54.810Z
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
CC BY-SA 4.0
2019-04-22T17:17:27.763Z
[...] 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
CC BY-SA 4.0
2019-04-22T11:24:04.493Z
2019-04-22T12:29:15.370Z
2019-04-22T12:29:15.370Z
Ali
<p>I know that when we have public goods we have that:</p>

<p><span class="math-container">$$MRT = MRS_a + MRS_b$$</span></p>

<p>Though I fail to understand why does this makes Walras equilibrium inefficient.</p>

<p>Thank you very much for your help!</p>

<microeconomics><competitive-equilibrium><public-goods>
Why is Walras equilibrium inefficient when we are dealing with public goods?
Fozoro
# Why is Walras equilibrium inefficient when we are dealing with public goods?

(Asked by: username_0 on 2019-04-22)

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!

**Tags:** microeconomics, competitive-equilibrium, public-goods

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## 1 answers

### Answer by username_1 on 2019-04-22. Score: 3 | Accepted answer

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: $$ \sum\_{i} \frac{\frac{\delta u^{i}}{\delta g}}{\frac{\delta u^{i}}{\delta x}} = \sum\_{i} MRS^{i}\_{gx} = MRT\_{gx} $$

You can easily see that both conditions are not the same. In the case of an economy with only private goods, the benefit to
society of the last unit of a private good provided (expressed as the
willingness to forgo units of another good) is equal to the benefit of the
one person in society who receives this last unit. If there are some
people who receive a higher benefit from the last unit than others, we
don’t have Pareto efficiency. Hence, the marginal benefit of a private
good must be the same across all people for the allocation to be
efficient. [Hindriks and Myles](http://jaimedv.com/eco/3c2-publi/intermediate-public-economics.pdf) explain this thoroughly if you want to read further on the Samuelson condition versus competitive equilibrium (see Chapter 8 in particular).

Edit: There is no benefit that producers get here. Think of the MRT as the opportunity cost of producing an extra unit of a good and of the MRS as the marginal benefit of another unit of a good. The marginal cost of a unit of public good is one unit of private good (i.e. the MRT). Therefore the rule says that an efficient allocation is achieved when the total marginal benefit of another unit of the public good, which is the sum of the individual benefits, is equal to the marginal cost of another unit. If the good were private then the marginal benefit of an extra unit can only be attained by a single individual, so there is no sum.

> **Comments:**
> **username_0**: Thank you very much for your answer, so if I get it right one of the main problems is that the benefit that the producer gets isn't the same as the benefit that the consumers get, is that correct? (though that the case for Pareto efficiency, what about the Walras efficiency?)  
> **username_1**: I edited my answer given your comment, I hope this helps.  
> **username_2**: 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 $i$.  
> **username_2**: [...] 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 $\sum_{i=1}^N MRS_i=MRT$  

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