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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 $\frac{\partial \mathcal{L} }{\partial X } \leq \frac{\partial \mathcal{L^*} }{\partial X^* } \Rightarrow X \geq X^*$ ?

  1. 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

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    $\begingroup$ 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 $g$?). $\endgroup$ – Herr K. Jan 28 at 15:52
  • $\begingroup$ thanks @HerrK. i edited my post to formalize the problem a bit more $\endgroup$ – axelle Jan 29 at 10:19
  • $\begingroup$ 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. $\endgroup$ – Herr K. Feb 1 at 16:36
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    $\begingroup$ 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 $X^*=\arg\max_{x}\mathcal L^*(x)$. $\endgroup$ – Herr K. Feb 1 at 16:48
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    $\begingroup$ 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 $y$. $\endgroup$ – Herr K. Feb 3 at 14:37

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