# Meaning or Interpretation of the denominator in the implicit function theorem

Consider $$p=g(p,y)$$. Rewrite this equation as $$p-g(p,y)=F(p,y)=F(P(y),y)=0$$, where $$P(y)$$ is the implicit function of $$p$$. By the implicit function theorem, we obtain $$\partial P(y)/\partial y=-F_y/F_p$$.

In this case, $$F_y=-\partial g/\partial y$$. This interpretation is straightforward. On the other hand, $$F_p=1-\partial g/\partial p$$. How should I interpret this denominator?

Since $$p=g(p,y)$$ is a recurrent equation of $$p$$, the direct effect of $$y$$ on $$p$$ (the numerator) should be discounted by the denominator. This makes sense.

What I am confused is that, depending on the sign of the denominator $$1−∂𝑔/∂𝑝$$, the effect of the numerator is overturned.

What is the meaning or the intuition of the denominator? What does $$1−∂𝑔/∂𝑝>0$$ or $$1−∂𝑔/∂𝑝<0$$ exactly mean??

For example, $$1−∂𝑔/∂𝑝<0$$ means that a marginal increase in $$p$$ increases $$p$$ itself more than 1, and thus it overturns the direct effect of the numerator???