Comparing 2 equilibrium values (competitive and centralized) at general equilibrium: can I compare only the first order conditions?

I have a rather complex model where analytical solutions do not seem achievable (I also tried symbolic solving and 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

@axelle try economics stackexchange:

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First, totally awesome that you asked this question here. Totally unrelated to Julia it seems, but always glad to see economists in the community.

Second, perhaps more related to Julia, can’t you just look at the equilibrium allocations? You say you can’t get analytical solutions, presumably you can get numerical solutions though right? If you are trying to make a proof here, this won’t help you though.

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thanks Albert

Thanks Tyler for answering even though it is not connected to Julia indeed!
Exactly, you are right, i made the simulations in Julia for some values of the parameters but i was hoping for a formal analytical proof.

It is impossible to give a useful reply without seeing the entire model (resource constraints, market clearing conditions, budget constraints, etc.).

From your description (perfect competition everywhere) it is not clear why the welfare theorems should not hold.

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thanks @hendri54 for your answer! I was able to show that the allocations are different (the reason is incomplete markets) but I am not able to say anything (general, with a formal proof - beyond simulations) about the relative positions of equilibrium values. I could give the whole model but i thought my question was more of a general maths questions for which i do not seem to have the tools and concepts :frowning: