# Comparing 2 equilibrium values (competitive vs centralized): can I compare only 1st derivative of objective function?

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

• 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. Jan 28 at 15:52
• thanks @HerrK. i edited my post to formalize the problem a bit more – axelle Jan 29 at 10:19
• 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. Feb 1 at 16:36
• 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. Feb 1 at 16:48
• 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. Feb 3 at 14:37