From Preference Relation to Utility Function.

From Preference Relation to Utility Function.

A real-valued utility function is then defined to represent the preference relation$\succsim$ and thus simplifies the analysis of many problems in consumer theory.

It has been proved that any binary relation that is complete, transitive, and continuous can be represented by a continuous real-valued utility function.[Barten and Böhm (1982). The classic reference is Debreu (1954).]

$$u(\mathbf{x})\mathbf{e}\sim \mathbf{x}$$

Equation1 provides a method to explains how the utility function works to maintain the preference relation. For any $\mathbf{x}$, which is a bundle of commodities, a number (utility) is assigned to it and the homogenous bundles of commodities $u(\mathbf{x})\mathbf{e}$ then can be compared with each other at a more easy basis.

Then, we can use $u(\mathbf{x})$ in a single axis to represent the preference relation.
$${\mathbf{x}}^1\succsim {\mathbf{x}}^2⟺u({\mathbf{x}}^1)\ge u({\mathbf{x}}^2)$$

Concerning utility functions, they play the same function if they do the same ordering to the bundles of the commodities. People sometimes say the utility function is invariant to positive monotonic transform.

For a utility function, the Marginal Utility of good $i$ is the first-order derivative of $u(\mathbf{x})$ with respect to ${\mathbf{x}}_i$.
$$M{U}_{{\mathbf{x}}_{\mathbf{i}}}=\frac{\partial u(\mathbf{x})}{\partial {\mathbf{x}}_{\mathbf{i}}}$$
Take the example of a bundle of two goods$<x_1,x_2>$, if set $u(\mathbf{x})$ as a constant value $c$, that is $u(x_1,f(x_1))=c$, the line of $x_2=f(x_1)$ in $x_1,x_2$ planes can be drawn and the line is what we call the indifference line.

And the marginal rate of substitution(MRS) is defined as:
$$MR{S}_{12}(x_1^1,x_2^1)\equiv |f^{\prime }(x_1^1)|=-f^{\prime }(x_1^1)$$

As $u(x_1,f(x_1))=c$, its derivative with respect to $x_1$ must be $0$.
$$\frac{\partial u(x_1^1,f(x_1^1))}{\partial x_1^1}+\frac{\partial u(x_1^1,f(x_1^1))}{\partial f(x_1^1)}{f}^{\prime }(x_1^1)=0$$
Combine the last two equations, we define the MRS as:
$$MR{S}_{12}({\mathbf{x}}^1)=\frac{\partial u({\mathbf{x}}^1)/\partial {\mathbf{x}}_1^1}{\partial u({\mathbf{x}}^1)/\partial {\mathbf{x}}_2^1}$$