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.
- 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^* ?
- 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