Elasticity of Substitution

tastes.)

def. Elasticity of Substitution. The EoS of good $x_1$ of $x_2$ asks “how much % $x_2$ are you willing to give up to get one more % of $x_1$“?

$${\sigma }_{2,1}:=\frac{\text{\% change in x1}}{\text{\% change in x2}}=|\frac{\%\Delta \frac{x_2}{x_1}}{\%\Delta {\text{MRS}}_{1,2}}|\equiv \frac{d\ln \frac{x_2}{x_1}}{d\ln |{\text{MRS}}_{1,2}|}$$

⇒ MRS is the Marginal Rate of Substitution (MRS) of good Higher elasticity of substitution (EoS) means the following things:

• Utility Function has less curvature
• $x_1$ and $x_2$ are better substitutes

Taste and elasticity

• Homothetic tastes when MRS depends on $\frac{x_2}{x_1}$ alone:
  • i.e. $MRS(t{x}_1,t{x}_2)=t^{k}MRS(x_1,x_2)$

Elasticity & Indifference Curve Shapes §

Cobb-Douglas EOS §

1. $u(x_1,x_2)=x_1^{\alpha }{x}_2^{1-\alpha }$
2. $\text{MRS}=-\frac{\alpha }{1-\alpha }\cdot \frac{x_2}{x_1}$

⇒ Then we observe:

$$\begin{array}{rl}& \frac{x_2}{x_1}={\text{MRS}}_{1,2}\frac{1-\alpha }{\alpha }\\ & ⟹\frac{d}{d\ln |\text{MRS}|}\ln \frac{x_1}{x_2}=\frac{d}{d\ln |\text{MRS}|}\left(\ln \text{MRS}+\ln \frac{\alpha }{1-\alpha }\right)\\ & ⟹\frac{d\ln \frac{x_1}{x_2}}{d\ln |\text{MRS}|}=1={\sigma }_{2,1}\end{array}$$

⇒ We also can see that it is homogeneous and thus homothetic: