# Comparative statics with a system of equations

I am self-learning macroeconomics and have got stuck due to a mathematical difficulty. I have a set of equations like this-

$$f(y, r, p, \bar A, \bar M)=0$$ $$g(y, r, p, \bar A, \bar M)=0$$ $$h(y, r, p, \bar A, \bar M)=0$$

$$\bar A$$ and $$\bar M$$ are parameters and the rest of the variables are endogenous. For comparative statics, I would want to find, for example the expression for $$\frac{\partial y}{\partial \bar A}$$ in terms of the functions $$f, g$$ and $$h$$ and their various partial derivatives, such that all the three constraints are always/ identically satisfied. But I am completely stuck, as I do not know how to take into account all the equations simultaneously.

The reason I am asking the question in the Mathematics SE instead of Economics SE is because I want to learn more about the general theory behind such comparative statics analysis. That is, suppose there are $$n$$ identities

$$f_1(x, a)=0$$ $$f_2(x, a)=0$$ $$....$$ $$f_n(x,a)=0$$

$$x$$ is the vector of endogenous variables and $$a$$ is the vector of parameters. I want to know how to find out the derivative of $$x$$ with respect to $$a$$ given that the $$n$$ identities are always satisfied. How should I go about it in general? What conditions must be satisfied for an answer to exist? Also, what are some good sources to read more about such analysis?

## 1 Answer

This will be more intuitive but less precise, but it may get you started. Start with implicit differentiation $$f(x,y)=k$$ and start with the total differential $$f_xdx+f_ydy=0$$ $$\frac{dy}{dx}=-\frac{f_x}{f_y}.$$ But this works with vectors and matrices as well, giving us something that looks like $$\frac{dy}{dx}= f_x f_y^{-1}$$ See the wikipedia https://en.wikipedia.org/wiki/Implicit_function_theorem for more precision, or Spivak's Calculus on Manifolds for more on implicit differentiation with matrices. It wouldn't hurt to look at at math econ book, such as Intrilligator's Mathematical Optimization and Economic Theory or Takayama's Mathematical Economics.