1. ## Multi Variable Functions

Apologies if this is in the wrong section.

I've always been stumped by multi variable functions as seen in economics:

Use of Functions and Variables in Economics

If you have one variable, you get a linear function. If that variable has a power, then it's non linear.

I don't understand why having multiple variables for example all to the power of one... yields a line?

Doesn't the addition of more variables turn the graph into a multi dimensional one? Is this even related to multivariate calculus?

2. ## Re: Multi Variable Functions

No it's not a line with multiple variables all power 1, if it's three variables it would be a plane, if it's more than three variables it's a hyper-plane. But geometrically speaking, the "flatness" of the surface is what makes it linear.

3. ## Re: Multi Variable Functions

$u = f(w,\ x,\ y,\ z)$

simply means that the numeric value of u depends on the numeric value of w, x, y, and z. There is some formula or rule involving w, x, y, and z that will unambiguously tell you what the numeric value of u is if you know what are the numeric values of w, x, y, and z, the so-called "independent" variables. Example, the demand for bananas will depend on the price of bananas, blueberries, peaches, and hourly wages. (Of course, the more realistic the economic model, the more variables must be taken into account, and those variables are inter-dependent rather than strictly independent.)

A common practice in economics is to analyze under the assumption that only one independent variable can vary (the assumption of ceteris paribus.) This greatly simplifies economic analysis at the cost of making it hopelessly naive. If you reduce a multivariable linear function under the assumption that all but one of the independent variables cannot change, the resulting graph is a straight line in a plane.

One of the simplest functions imaginable is a linear function. The basic idea in a linear function is that a change in the independent variable causes a strictly proportional change in the dependent variable. But that is clearly untrue of many economic functions. Your happiness today may be increased by eating a chocolate bar, but your happiness will not be increased at all by being required to eat 100 chocolate bars in a day. Consequently, economists sensibly refuse to treat all economic functions as linear. Polynomial functions have a very nice mathematical property: you can create one that exactly fits any finite set of data.

4. ## Re: Multi Variable Functions

Thanks guys, food for thought.