<p>I have a rather complex model where analytical solutions do not seem achievable (I also tried symbolic solving in Matlab and Python and could not find any) so that I cannot get an explicit expression for my equilibrim values of two of my choice variables, <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>.</p>
<p>Let <span class="math-container">$U(c_1, c_2)$</span> be the objective function, where <span class="math-container">$U$</span> is some function, concave in <span class="math-container">$c_1$</span> and in <span class="math-container">$c_2$</span>. <span class="math-container">$c_1$</span> and <span class="math-container">$c_2$</span> are functions of choice variables <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>. Given the complexity of my expression, I am unsure whether or not the objective function is concave with respect to X and Y (I was not able to find an explicit sign of the second derivative in Matlab).</p>
<p>Two agents are optimizing. Problem of agent A is of the form: <span class="math-container">$\max_{X} U(c_1(X), c_2(X))$</span> and problem of agent B is of the form: <span class="math-container">$\max_{Y} U(c_1(Y), c_2(Y))$</span>. (B is optimizing the same objective as agent A, due to perfect competition: B is a bank investing deposits of consumers A).</p>
<p>I would like to compare equilibrium values in the competitive and in the centralized equilibrium of those two variables X and Y.</p>
<p>In the competitive equilibrium, agents are price takers whereas in the decentralized one, a benevolent social planner internalizes the price of assets <span class="math-container">$P$</span> which depends on choice variables (it is a constrained efficient equilibrium, a second best) and optimizes on behalf of agents A and B under the same constraints.</p>
<p>Therefore in the constrained efficient equilibrium, the problem takes the form: <span class="math-container">$\max_{X} U(c_1(X, \color{cyan}{P(X)}), c_2(X,\color{cyan}{P(X)}))$</span> and <span class="math-container">$\max_{Y} U(c_1(Y, \color{cyan}{P(Y)}), c_2(Y,\color{cyan}{P(Y)}))$</span>.</p>
<p>As I hav no explicit solutions, I was wondering how I could compare the allocations under the competitive equilibrium (<span class="math-container">$X,Y$</span>) to the one under the constrained efficient equilibrium (<span class="math-container">$X^*,Y^*$</span>). As I did some numerical simulations (in Julia) for a wide range of parameters and under different form of the utility function <span class="math-container">$U$</span>, and based on economic intuitions, I have an idea of the relative positions of <span class="math-container">$X^*,X$</span> and <span class="math-container">$Y^*,Y$</span> but I am still hoping for a formal proof.</p>
<ol>
<li><strong>In a partial equilibrium analysis</strong> (looking at the choice of <span class="math-container">$X$</span> only holding <span class="math-container">$Y$</span> fixed or vice versa), if i find that the derivative with respect to X in the decentralized is higher than in the constrained efficient equilibrium can I conclude that X is lower (for a given value of Y)? Do i need to prove concavity of the objective function with respect to X?</li>
</ol>
<p>i.e. is it true and under which conditions if any that <span class="math-container">$\frac{\partial \mathcal{L} }{\partial X } \leq \frac{\partial \mathcal{L^*} }{\partial X^* } \Rightarrow X \geq X^*$</span> ?</p>
<ol start="2">
<li>How can I translate those results (if 1 can be proven) to <strong>general equilibrium</strong>? Would i need some conditions of the type <span class="math-container">$X(Y)$</span> and <span class="math-container">$Y(X)$</span> increasing, something like that?</li>
</ol>
<p>In general if anyone could point me to the type of mathematical methods i should be looking at?</p>
<p>Many thanks for any help</p>

CC BY-SA 4.0
2021-01-28T15:52:18.497Z
Your description of the problem is too vague. It would help if you can write down the problem in mathematical terms. For example, $\max_{c}f(c)$ subject to $c=g(x,y)$, where $f$ is concave (and $g$ or the graph of $g$?).
Herr K.
CC BY-SA 4.0
2021-01-29T10:19:43.840Z
thanks @HerrK. i edited my post to formalize the problem a bit more
axelle
CC BY-SA 4.0
2021-02-01T16:36:53.013Z
I'm afraid I still don't have full clarity about the problem you're facing. But regarding your first question: Shouldn't $\frac{\partial\mathcal L}{\partial X}=\frac{\partial\mathcal L^*}{\partial X^*}=0$, if solutions are interior? If either solution is at the corner, then comparing the corner value(s) should give you something to start.
Herr K.
CC BY-SA 4.0
2021-02-01T16:48:36.347Z
If you can establish that $\mathcal L$ is strictly concave, and $X=\arg\max_{x}\mathcal L(x)$, i.e. $$\frac{\partial\mathcal L(x)}{\partial x}\bigg\vert_{x=X}=0,$$ then it is valid to conclude that $$\frac{\partial\mathcal L(x)}{\partial x}\bigg\vert_{x=X^*}\le0\quad\Rightarrow\quad X^*\ge X,$$ where $X^*=\arg\max_{x}\mathcal L^*(x)$.
Herr K.
CC BY-SA 4.0
2021-02-03T13:10:16.863Z
thanks a lot @HerrK ! And that would be true only at partial equilibrium i.e. holding the other choice variable $Y$ at the same given value, as we have only  $\frac{\partial \mathcal{L} }{\partial X }\biggr\rvert_{\color{cyan}{Y=\overline{Y}}}$, right? Is there any way to prove that this is true at general equilibrium for some type of relationship between $X$ and $Y$ (monotonic relationship? Or some stronger conditions, $Y$ strictly increasing in $X$?)
axelle
CC BY-SA 4.0
2021-02-03T14:37:12.420Z
Yes, what I said was based on the assumption that $Y$ is held fixed. In what you call a general equilibrium, I suppose you'd be solving $\max_{x,y}\mathcal L(x,y)$. The FOCs will define, if only implicitly, solutions $X(y)$ and $Y(x)$ to the problem. The same for the starred version of the problem. If you can show/assume that these functions are monotonic (either increasing or decreasing), then you should be able to pin down the relative positions of $X(y)$ and $X^*(y)$ for particular values of $y$.
Herr K.
CC BY-SA 4.0
2021-02-09T08:36:46.623Z
@HerrK.Many thanks for your very useful answers!!
axelle
<mathematical-economics><optimization><general-equilibrium><welfare-economics>
Comparing 2 equilibrium values (competitive vs centralized): can I compare only 1st derivative of objective function?
axelle
# Comparing 2 equilibrium values (competitive vs centralized): can I compare only 1st derivative of objective function?

(Asked by: username_0 on 2021-01-28)

I have a rather complex model where analytical solutions do not seem achievable (I also tried symbolic solving in Matlab and Python and could not find any) so that I cannot get an explicit expression for my equilibrim values of two of my choice variables, $X$ and $Y$.

Let $U(c\_1, c\_2)$ be the objective function, where $U$ is some function, concave in $c\_1$ and in $c\_2$. $c\_1$ and $c\_2$ are functions of choice variables $X$ and $Y$. Given the complexity of my expression, I am unsure whether or not the objective function is concave with respect to X and Y (I was not able to find an explicit sign of the second derivative in Matlab).

Two agents are optimizing. Problem of agent A is of the form: $\max\_{X} U(c\_1(X), c\_2(X))$ and problem of agent B is of the form: $\max\_{Y} U(c\_1(Y), c\_2(Y))$. (B is optimizing the same objective as agent A, due to perfect competition: B is a bank investing deposits of consumers A).

I would like to compare equilibrium values in the competitive and in the centralized equilibrium of those two variables X and Y.

In the competitive equilibrium, agents are price takers whereas in the decentralized one, a benevolent social planner internalizes the price of assets $P$ which depends on choice variables (it is a constrained efficient equilibrium, a second best) and optimizes on behalf of agents A and B under the same constraints.

Therefore in the constrained efficient equilibrium, the problem takes the form: $\max\_{X} U(c\_1(X, \color{cyan}{P(X)}), c\_2(X,\color{cyan}{P(X)}))$ and $\max\_{Y} U(c\_1(Y, \color{cyan}{P(Y)}), c\_2(Y,\color{cyan}{P(Y)}))$.

As I hav no explicit solutions, I was wondering how I could compare the allocations under the competitive equilibrium ($X,Y$) to the one under the constrained efficient equilibrium ($X^\*,Y^\*$). As I did some numerical simulations (in Julia) for a wide range of parameters and under different form of the utility function $U$, and based on economic intuitions, I have an idea of the relative positions of $X^\*,X$ and $Y^\*,Y$ but I am still hoping for a formal proof.

1. **In a partial equilibrium analysis** (looking at the choice of $X$ only holding $Y$ fixed or vice versa), if i find that the derivative with respect to X in the decentralized is higher than in the constrained efficient equilibrium can I conclude that X is lower (for a given value of Y)? Do i need to prove concavity of the objective function with respect to X?

i.e. is it true and under which conditions if any that $\frac{\partial \mathcal{L} }{\partial X } \leq \frac{\partial \mathcal{L^\*} }{\partial X^\* } \Rightarrow X \geq X^\*$ ?

2. How can I translate those results (if 1 can be proven) to **general equilibrium**? Would i need some conditions of the type $X(Y)$ and $Y(X)$ increasing, something like that?

In general if anyone could point me to the type of mathematical methods i should be looking at?

Many thanks for any help

**Tags:** mathematical-economics, optimization, general-equilibrium, welfare-economics

> **Comments:**
> **username_1**: Your description of the problem is too vague. It would help if you can write down the problem in mathematical terms. For example, $\max_{c}f(c)$ subject to $c=g(x,y)$, where $f$ is concave (and $g$ or the graph of $g$?).  
> **username_0**: thanks @HerrK. i edited my post to formalize the problem a bit more  
> **username_1**: I'm afraid I still don't have full clarity about the problem you're facing. But regarding your first question: Shouldn't $\frac{\partial\mathcal L}{\partial X}=\frac{\partial\mathcal L^*}{\partial X^*}=0$, if solutions are interior? If either solution is at the corner, then comparing the corner value(s) should give you something to start.  
> **username_1**: If you can establish that $\mathcal L$ is strictly concave, and $X=\arg\max_{x}\mathcal L(x)$, i.e. $$\frac{\partial\mathcal L(x)}{\partial x}\bigg\vert_{x=X}=0,$$ then it is valid to conclude that $$\frac{\partial\mathcal L(x)}{\partial x}\bigg\vert_{x=X^*}\le0\quad\Rightarrow\quad X^*\ge X,$$ where $X^*=\arg\max_{x}\mathcal L^*(x)$.  
> **username_0**: thanks a lot @HerrK ! And that would be true only at partial equilibrium i.e. holding the other choice variable $Y$ at the same given value, as we have only  $\frac{\partial \mathcal{L} }{\partial X }\biggr\rvert_{\color{cyan}{Y=\overline{Y}}}$, right? Is there any way to prove that this is true at general equilibrium for some type of relationship between $X$ and $Y$ (monotonic relationship? Or some stronger conditions, $Y$ strictly increasing in $X$?)  
> **username_1**: Yes, what I said was based on the assumption that $Y$ is held fixed. In what you call a general equilibrium, I suppose you'd be solving $\max_{x,y}\mathcal L(x,y)$. The FOCs will define, if only implicitly, solutions $X(y)$ and $Y(x)$ to the problem. The same for the starred version of the problem. If you can show/assume that these functions are monotonic (either increasing or decreasing), then you should be able to pin down the relative positions of $X(y)$ and $X^*(y)$ for particular values of $y$.  
> **username_0**: @username_1Many thanks for your very useful answers!!  

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