Implicit Functions

Implicit Functions

Explicit function: \[y=f\left(x_{1}, x_{2}, \cdots, x_{n}\right)\]

Implicit function: \[F\left(y, x_{1}, x_{2}, \cdots, x_{n}\right)=0\]

Example

Implicit function: \[F(x,y)=y-3x^2 =0\]

Corresponding explicit function: \[y = f(x) = 3x^2\]

However, not all implicit functions have a corresponding explicit function. E.g. \(F(x,y) = x^2 + y^2 -9=0\)

Implicit Function Theorem

Given, \[F\left(x, y \right)=0\]

If the following conditions are met:

• \(F_y\) and \(F_x\) are continuous, and

• At some point \((a,b)\), \(F_y\) is non-zero

Then in a neighborhood around \((a,b)\), an implicit function exists. Moreover, this function is continuous and has continuous partial derivatives.

Derivatives of Implicit Functions

Totally differentiating \(F\), we have \(dF=0\), or \[F_{y} d y+F_{1} d x_{1}+\cdots+F_{n} d x_{n}=0\]

Suppose that only \(y\) and \(x_{1}\) are allowed to vary: \[\frac{\partial y}{\partial x_{1}}=-\frac{F_{1}}{F_{y}}\]

In the simple case where the given equation is \(F(y, x)=0\), the rule gives \[\frac{d y}{d x}=-\frac{F_{x}}{F_{y}}\]

Example: Market Equilibrium

Demand: \(Q^d = 100 - 2P + 0.02Y\), where \(Y\) is income

Supply: \(Q^s = 20 + 3P - c\), where \(c\) is an input cost

In equilibrium \(Q^d = Q^s\), that is, \[F(P, Y, c) = 80 - 5P + 0.02Y + c = 0\]

This defines the equilibrium price \(P\) as an implicit function of \(Y\) and \(c\). Find \(\partial P/\partial Y\) and \(\partial P/\partial c\).

Check your answer by solving for \(P\) explicitly.

Example: When You Cannot Solve for \(y\)

Given the following function, let’s find \(\partial y/\partial x\) and \(\partial y/\partial z\). \[F(x,y,z) = x^3 z^2+y^3+4xyz = 0\]

Another Example

Estimate the following model for demand for fast food: \[\text{orders} = \beta_0 + \beta_1 \, \text{price} + \beta_2 \, \text{quality} + \varepsilon\]

What is the interpretation of \(\beta_1\)?

Another Example (cont.)

What if instead we estimate: \[\ln(\text{orders}) = \beta_0 + \beta_1 \ln(\text{price}) + \beta_2 \, \text{quality} + \varepsilon\]

What is the interpretation of \(\beta_1\)?

Another Related Example

Production function: \[Y = A L^{\alpha} K^\beta\]

To estimate the elasticities from data: \[\ln Y = \ln A + \alpha \ln L+ \beta \ln K + \varepsilon\]

Elasticity in Logs

Recall that elasticity is a ratio of percentage changes: \[\varepsilon = \frac{dQ/Q}{dP/P}\]

Since \(d \ln Q = \dfrac{dQ}{Q}\) and \(d \ln P = \dfrac{dP}{P}\), \[\varepsilon = \frac{d \ln Q}{d \ln P}\]

So if \(\ln Q = a - \alpha \ln P\), then \(\varepsilon = -\alpha\)

Find the Derivative by Taking the Log

\[\text{Demand:} \quad Q(p) = \frac{c}{p^{\alpha}}\]

Homework Questions

• Exercise 8.5: 1, 2 (d), 3 (a)

Textbook reference: 8.5

Midterm 2 is on Wednesday, October 21: a sample exam and a help sheet are posted on the Exams page