Rationality (Economics)

Microeconomics Assumptions §

def. Rational Preference. Consider two basket of goods $A,B$. A rational preference is one that is all of the following:

1. Complete: either $A\succ B,B\succ A,\text{or }A\sim B$
   1. Comparable: You have to choose either of the above three
2. Transitive: $\text{if }A\succ B\text{ and }B\succ C\text{, then }A\succ C$
3. Monotonic: $x_1⪰x_2$

The following are optional conditions

• Convexity: Averages are bette than extremes. (Preference of variety)

• The rational preference assumption and the convexity assumption will together be enough to define an Utility Function.

• Two tastes are same if they have the same utility function, or the utility functions are Monotonic Transformations of one another.

Notation §

• $A\sim B$: A is equally preferable with B
• $A\succ B$: A is strictly preferred to B
• $A\succsim B$: A is equally or more preferable to B
• $A\sim B$: A is equally preferred to B (indifferent)

def. Homothetic Tastes. Two equivalent definitions

1. Tastes are homothetic if the utility function (or its Monotonic Transformation) is HD of degree $k>0$
2. Tastes are homothetic if the MRS depends only on $\frac{x_2}{x_1}$

$$\begin{array}{rl}& u(x_1,x_2)\text{ is homogenous of degree }k\\ & ⟹u(t{x}_1,t{x}_2)=t^{k}u(x_1,x_2)\end{array}$$

Quasilinear Taste §