Section 1
Functions of several variables
One general form to present a function of two variables is \(z = f(x,y)\), or simply to write \(f(x,y)\). Here, \(x\) and \(y\) are the independent variables, for each pair of which in the domain there exists one and only one value of the function \(f(x,y)\).
Many economic activities are influenced by more than one independent variable. For example, when we write \(Q_{d} = f(P)\), that is, the quantity demanded is a function of price, we rely on two Greek words, Ceteris Paribus, other factors remaining constant. In fact, there are many other important variables which may affect the quantity demanded, for example, income, prices of related goods, taste, season, and religious beliefs, etc.
As many important economic activities are influenced by more than one independent variable, it is important to study the calculus of multivariable functions.
Example 1
Some examples of functions of several variables: $$f(x,y) = 2x + y + 2$$ $$f(x,y,z) = 2x + y - z + 7$$ $$f(x,y) = x^{0.5}y^{0.5}$$ $$f(x_{1}, x_{2}) = 10x_{1}^{0.6}x_{2}^{0.4}$$
Example 2
Some examples of economic activities which are functions of several variables:
- Demand function $$Q_{d}(P,I) = 10 - 2P + 0.2I$$ Here, \(P\) is the price of the good and I is the income of the consumer.
- Utility function $$u(x,y) = x^{0.5}y^{0.5}$$ Here, \(x\) and \(y\) are two commodities.
- Production function $$Q(K,L) = 10K^{0.6}L^{0.4}$$ Here, \(K\) and \(L\) are factors of production.
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