<p>You can take MRS as the willingness to pay (Take the other good in MRS as money). If the good is a private good that costs 3, your willingness to pay should be 3 because you are buying it for yourself. However, a public good is used by all payers, so we only need the sum of the willingnesses to pay to be equal to 3. (You are simply splitting the bill)</p>

CC BY-SA 4.0
2020-02-19T22:46:41.890Z
I understand thank you. If I can ask, I seem to have confused myself with some of the math. In differentiating the Lagrangian function, how is this the result: $$
\frac{\partial \mathscr{L}}{\partial y_{s}^{A}}=U_{1}^{A} f^{\prime}-U_{2}^{A}+\lambda U_{1}^{B} f^{\prime}
$$? The first term is a utility function composed of two variables: $$
\left(x, y^{A *}-y_{s}^{A}\right)
$$. But for example the first term is $$
U_{1}^{A} f^{\prime}
$$
aisync
CC BY-SA 4.0
2020-02-19T22:49:39.050Z
My best understanding is that it's the derivative of the objective utility function w/ respect to the 1st variable (x), multiplied by the derivative of the production function via chain rule? I don't know where the other terms come from.
aisync
CC BY-SA 4.0
2020-02-05T19:46:58.580Z
2020-02-05T23:58:17.677Z
2020-02-05T23:58:17.677Z
Alex Wang
<ul>
<li>We assume a two person economy, denoted by person A and B. Good y is an ordinary private good, and each person begins with an allocation of this good given by <span class="math-container">$ y^{A*} $</span> &amp; <span class="math-container">$ y^{B*} $</span> respectively. </li>
<li><p>Each person may consume some y directly or devote some portion of it to the production of a single public good, x. The amounts contributed are given by <span class="math-container">$y_{s}^{A}$</span> and <span class="math-container">$y_{s}^{B}$</span>, and the public good has production function defined by
<span class="math-container">$$
x=f\left(y_{s}^{A}+y_{s}^{B}\right)
$$</span></p></li>
<li><p>Resulting utilities for these two people are given:
<span class="math-container">$$
\begin{array}{c}{U^{A}\left(x, y^{A *}-y_{s}^{A}\right)} \\ {U^{B}\left(x, y^{B *}-y_{s}^{B}\right)}\end{array}
$$</span></p></li>
<li>The necessary conditions for efficient resource allocation in this problem consist of choosing the levels of public goods subscriptions (<span class="math-container">$ y^{A*} $</span> &amp; <span class="math-container">$ y^{B*} $</span>) that maximize, say, A’s utility for any given level of B’s utility.
The Lagrangian expression is
<span class="math-container">$$
\mathscr{L}=U^{A}\left(x, y^{A *}-y_{s}^{A}\right)+\lambda\left[U^{B}\left(x, y^{B *}-y_{s}^{B}\right)-K\right]
$$</span>
where K is a constant level of B's utility. The first-order conditions for max are
<span class="math-container">$$
\begin{aligned} \frac{\partial \mathscr{L}}{\partial y_{s}^{A}} &amp;=U_{1}^{A} f^{\prime}-U_{2}^{A}+\lambda U_{1}^{B} f^{\prime}=0 \\ \frac{\partial \mathscr{L}}{\partial y_{s}^{B}} &amp;=U_{1}^{A} f^{\prime}-\lambda U_{2}^{B}+\lambda U_{1}^{B} f^{\prime}=0 \end{aligned}
$$</span></li>
<li>A comparison of these two equations yields
<span class="math-container">$$
\lambda U_{2}^{B}=U_{2}^{A}
$$</span>
and when substituting into the preceding equation the end result is:
<span class="math-container">$$
\frac{U_{1}^{A}}{U_{2}^{A}}+\frac{\lambda U_{1}^{B}}{\lambda U_{2}^{B}}=\frac{1}{f^{\prime}}
$$</span> or more simply,
<span class="math-container">$$
M R S^{A}+M R S^{B}=\frac{1}{f^{\prime}}
$$</span></li>
</ul>

<blockquote>
  <p><strong>For such goods, the MRS in consumption must reﬂect the amount of y that
  all consumers would be willing to give up to get one more x, because
  everyone will obtain the beneﬁts of the extra x output. Hence it is
  the sum of each individual’s MRS that should be equated to dy/dx in
  production (here given by <span class="math-container">$ \frac{1}{f'} $</span>).</strong></p>
</blockquote>

<p>Q: I follow the order of operations mathematically but don't understand the bolded statement. </p>

<microeconomics><mathematical-economics><general-equilibrium><public-goods>
Simple General Equilibrium Model of Public Goods: Intuition?
aisync
# Simple General Equilibrium Model of Public Goods: Intuition?

(Asked by: username_0 on 2020-02-05)

* We assume a two person economy, denoted by person A and B. Good y is an ordinary private good, and each person begins with an allocation of this good given by $ y^{A\*} $ & $ y^{B\*} $ respectively.
* Each person may consume some y directly or devote some portion of it to the production of a single public good, x. The amounts contributed are given by $y\_{s}^{A}$ and $y\_{s}^{B}$, and the public good has production function defined by
  $$
  x=f\left(y\_{s}^{A}+y\_{s}^{B}\right)
  $$
* Resulting utilities for these two people are given:
  $$
  \begin{array}{c}{U^{A}\left(x, y^{A \*}-y\_{s}^{A}\right)} \\ {U^{B}\left(x, y^{B \*}-y\_{s}^{B}\right)}\end{array}
  $$
* The necessary conditions for efficient resource allocation in this problem consist of choosing the levels of public goods subscriptions ($ y^{A\*} $ & $ y^{B\*} $) that maximize, say, A’s utility for any given level of B’s utility.
  The Lagrangian expression is
  $$
  \mathscr{L}=U^{A}\left(x, y^{A \*}-y\_{s}^{A}\right)+\lambda\left[U^{B}\left(x, y^{B \*}-y\_{s}^{B}\right)-K\right]
  $$
  where K is a constant level of B's utility. The first-order conditions for max are
  $$
  \begin{aligned} \frac{\partial \mathscr{L}}{\partial y\_{s}^{A}} &=U\_{1}^{A} f^{\prime}-U\_{2}^{A}+\lambda U\_{1}^{B} f^{\prime}=0 \\ \frac{\partial \mathscr{L}}{\partial y\_{s}^{B}} &=U\_{1}^{A} f^{\prime}-\lambda U\_{2}^{B}+\lambda U\_{1}^{B} f^{\prime}=0 \end{aligned}
  $$
* A comparison of these two equations yields
  $$
  \lambda U\_{2}^{B}=U\_{2}^{A}
  $$
  and when substituting into the preceding equation the end result is:
  $$
  \frac{U\_{1}^{A}}{U\_{2}^{A}}+\frac{\lambda U\_{1}^{B}}{\lambda U\_{2}^{B}}=\frac{1}{f^{\prime}}
  $$ or more simply,
  $$
  M R S^{A}+M R S^{B}=\frac{1}{f^{\prime}}
  $$

> **For such goods, the MRS in consumption must reﬂect the amount of y that
> all consumers would be willing to give up to get one more x, because
> everyone will obtain the beneﬁts of the extra x output. Hence it is
> the sum of each individual’s MRS that should be equated to dy/dx in
> production (here given by $ \frac{1}{f'} $).**

Q: I follow the order of operations mathematically but don't understand the bolded statement.

**Tags:** microeconomics, mathematical-economics, general-equilibrium, public-goods


---

## 1 answers

### Answer by username_1 on 2020-02-05. Score: 2 | Accepted answer

You can take MRS as the willingness to pay (Take the other good in MRS as money). If the good is a private good that costs 3, your willingness to pay should be 3 because you are buying it for yourself. However, a public good is used by all payers, so we only need the sum of the willingnesses to pay to be equal to 3. (You are simply splitting the bill)

> **Comments:**
> **username_0**: I understand thank you. If I can ask, I seem to have confused myself with some of the math. In differentiating the Lagrangian function, how is this the result: $$ \frac{\partial \mathscr{L}}{\partial y_{s}^{A}}=U_{1}^{A} f^{\prime}-U_{2}^{A}+\lambda U_{1}^{B} f^{\prime} $$? The first term is a utility function composed of two variables: $$ \left(x, y^{A *}-y_{s}^{A}\right) $$. But for example the first term is $$ U_{1}^{A} f^{\prime} $$  
> **username_0**: My best understanding is that it's the derivative of the objective utility function w/ respect to the 1st variable (x), multiplied by the derivative of the production function via chain rule? I don't know where the other terms come from.  

---

