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???

I greatly appreciate your help!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.