Partial derivatives

Functions of two variables

  1. Why should we bother considering functions of two variables?

  2. For example, we have the following economic contexts:

    • A firm can produce more than one product.
    • A firm can use multiple inputs to create a product.
    • An individual can consume different types of goods.

  3. How do we extend the ideas of the course in this situation?

Functions of two variables

  1. A function \(f\) of two variables \(x\) and \(y\) with domain \(D\) is a rule that assigns a specified number \(f(x,y)\) to each \((x,y)\) in \(D\).

  2. Examples based on linear functions of one variable:

    • \(f(x,y)=1\)
    • \(f(x,y)=x\)
    • \(f(x,y)=y\)

  1. Examples based on quadratic functions of one variable:

    • \(f(x,y)=x^2\)
    • \(f(x,y)=y^2\)
    • \(f(x,y)=x^2+y^2\)
    • \(f(x,y)=x^2-y^2\)

  1. More complicated examples:

    • \(f(x,y)=x^2+2xy+y^2\)
    • \(f(x,y)=\dfrac{x^2+y^3}{y-x+2}\)
    • \(f(x,y)=\dfrac{1}{e^{x+y}-3}\)
    • \(f(x,y)=2\ln(x-3)+2\ln(y+1)\)

  2. A commonly used example in economics:

    • \(f(x,y)=2x^{1/2}y^{1/2}\)

Level curves

  1. A common visualization of functions of two variables is a level curve.

  2. The idea is to cut through \(z=f(x,y)\) using horizontal planes parallel to the \(xy\)-plane.

  3. The level curve for \(f\) at height \(c\) consists of the points satisfying the equation \(f(x,y)=c\).

  4. In other words, we want the set of all \((x,y)\) in the domain \(D\) such that \(f(x,y)=c\) where \(c\) is given.

  1. It is a good time to return to our examples earlier and visualize the level curves.

  2. In economic applications, we use level curves to proceed with economic analysis.

    • Firms choosing how much inputs to use for production.
    • Consumers choosing how much of goods and services should they buy.

  1. A production function \(Q=F(L,K)\) using \(L\) units of labor and \(K\) units of capital transforms these inputs into output \(Q\). Fix values of \(Q\) then draw the combinations of \((L,K)\) which will produce the same level of output.

  2. A consumer’s utility function \(U=F(x,y)\) depends on the consumption of \(x\) units of a first good and \(y\) units of a second good. Fix values of \(U\) then draw the combinations of \((x,y)\) which will produce the same level of output.

  3. The theme of substitution becomes important in these discussions and is best seen in terms of level curves.

Partial derivatives

  1. Why not extend the idea of a Newton quotient?
  2. Consider a Newton quotient where \[\frac{f(x+h, y)-f(x,y)}{h}\] and another Newton quotient where \[\frac{f(x, y+k)-f(x,y)}{k}\]

  1. So, we can think of the partial derivative of \(f\) with respect to \(x\) as \[f_x^\prime(x,y)=\lim_{h\to 0}\frac{f(x+h, y)-f(x,y)}{h}\]

  2. In other words, hold \(y\) constant and allow \(x\) to change.

  3. This idea is actually compatible with how economic analysis is done – so compatible that economic analysis has a name for it: ceteris paribus.

  1. To get the partial derivative of \(f\) with respect to \(y\), we let \(k\to 0\) this time for the other Newton quotient: \[f_y^\prime(x,y)=\lim_{k\to 0}\frac{f(x, y+k)-f(x,y)}{k}\]

  2. But how do you actually calculate these derivatives?

    • Turns out that we don’t even have to use the definition to find partial derivatives
    • This should not be surprising.

  1. You literally have to treat \(y\) as if it were constant if you want the partial derivative of \(f\) with respect to \(x\). Similarly, treat \(x\) as if it were constant if you want the partial derivative of \(f\) with respect to \(y\).

  2. For example, if \(f(x,y)=x\), then \(f_x^\prime(x,y)=1\) and \(f_y^\prime(x,y)=0\).

  3. For example, if \(f(x,y)=x^2+2xy+y^2\), then \(f_x^\prime(x,y)=2x+2y\) and \(f_y^\prime(x,y)=2x+2y\).

Notation

Let \(z=f(x,y)\). Then the partial derivative with respect to \(x\) can be denoted as \[\begin{eqnarray*} & f_x^\prime(x,y), f_x^\prime, f_x \\ & f_1^\prime(x,y), f_1^\prime, f_1 \\ & z_x^\prime, z_x \\ & \frac{\partial f}{\partial x}, \frac{\partial z}{\partial x}\end{eqnarray*}\]

Working on some examples

Find the partial derivatives with respect to \(x\) and \(y\) for the examples earlier.