Topic 3. Welfare, aggregation

1.CV an EV

CV: compensation variation, EV: equivalence variation.

Consider a change in price vector from $p^0$$p^0$ to $p^1$$p^1$ :

• $CV(p^0,p^1,y^0)$$C V\left(p^0, p^1, y^0\right)$ : holding nominal income constant at $y^0,CV$$y^0, C V$ is the minimum amount of money required to keep the consumer as well of as $\mathrm{s}$$\mathrm{s}$ was in the initial state $(p^0,y^0)$$\left(p^0, y^0\right)$ as she is in the new state $(p^1,y^0+CV)$$\left(p^1, y^0+C V\right)$. In terms of the expenditure function:

where $U^0=V(p^0,y^0)$$U^0=V\left(p^0, y^0\right)$ ​ In terms of the indirect utility function:

• $EV(p^0,p^1,y^0)$$E V\left(p^0, p^1, y^0\right)$ : uses utility after the price change as the basis for comparison:

In this figure, green line represents the original price level budget constraints.

• CV and EV are opposite in sign.

  If price of good 1 drops, then CV measures how much expenditure that we have to take from agent that makes him same utility as before. $\Rightarrow$$\Rightarrow$ Negative sign.

  If price of good 1 drops, then EV measures how much expenditure that we have to give to the agent to maintain the new utility level if we do not have the price change. $\Rightarrow$$\Rightarrow$ Positive sign.

• CV EV and CS

  

  price fall from $p_1$$p_1$ to $p_2$$p_2$

  $$\begin{array}{}\text{(4)}& \begin{array}{l}CV=x\\ EV=w+x+z\\ \mathrm{\Delta }S=x+z\end{array}\end{array}$$

  price rise from $p_2$$p_2$ to $p_1$$p_1$

  $$\begin{array}{}\text{(5)}& \begin{array}{l}CV=-(w+x+z)\\ EV=-x\\ \mathrm{△}S=-(x+z)\end{array}\end{array}$$

• Path Dependence

  Consider first a case when only one price changes from $p_1^0$$p_1^0$ to $p_1^1$$p_1^1$, all other prices are held constant: (a) Integrating between the two prices gives:

  $$\begin{array}{}\text{(6)}& \begin{array}{rl}CV(p^0,p^1,y^0)& =e(p^1,U^0)-e(p^0,U^0)\\ & ={\int }_{p_1^0}^{p_1^1}{h}_1(p,U^0)d{p}_1={\int }_{p_1^0}^{p_1^1}\frac{\partial e(p,U^0)}{\partial p_1}d{p}_1\end{array}\end{array}$$

  (b) Alternatively:

  $$\begin{array}{}\text{(7)}& \begin{array}{rl}EV(p^0,p^1,y^0)& =e(p^1,U^1)-e(p^0,U^1)\\ & ={\int }_{p_1^0}^{p_1^1}{h}_1(p,U^1)d{p}_1={\int }_{p_1^0}^{p_1^1}\frac{\partial e(p,U^1)}{\partial p_1}d{p}_1\end{array}\end{array}$$

• How do we interpret the consumer surplus that derived from the Marshallian demand?

  CS is an uncompensated measurement.

  $$\begin{array}{}\text{(8)}& \mathrm{CS}(p^0,p^1,y^0)={\int }_{p_1^0}^{p_1^1}{x}_i(p,y^0)d{p}_1=-{\int }_{p_1^0}^{p_1^1}\frac{\frac{\partial V(p,y^0)}{\partial p_1}}{\frac{\partial V(p,y^0)}{\partial y}}d{p}_1\end{array}$$

  To keep the individual at the same indifference curve, income level must be constantly adjusted along the path of price change. Since in CS, the income is fixed, this produce the difference between compensated and uncompensated welfare measures.

• More than one price change

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2. Homothetic function

• Let $y=f(x_1,\dots ,x_n)$$y=f\left(x_1, \ldots, x_n\right)$ be homogenous of degree $k$$k$, and let $z=F(y)$$z=F(y)$, where $F^{\prime }(y)>0$$F^{\prime}(y)>0$ (then $F$$F$ is a monotonic transformation of $y$$y$ ), the function $z(x_1,x_2,\dots ,x_n)$$z\left(x_1, x_2, \ldots, x_n\right)$ is homothetic.

• Alternative definition: as monotonic transforms of linear homogenous functions. But this is the same because any homogenous function of degree " $k$$k$ " can be converted to a linear homogeneous function $(k=1)$$(k=1)$ by taking the $k^{t}h$$k^t h$ root of $f(x_1,x_2,\dots ,x_n)$$f\left(x_1, x_2, \ldots, x_n\right)$, i.e. ${\left[f(x_1,\dots ,x_n)\right]}^{1/k}$$\left[f\left(x_1, \ldots, x_n\right)\right]^{1 / k}$. Then one can transform this function with some function $F$$F$.

• Homothetic functions have MRS homogeneous of degree 0 :

  $$\begin{array}{}\text{(9)}& \frac{\frac{\partial U}{\partial x_i}(tx)}{\frac{\partial U}{\partial x_j}(tx)}=\frac{\frac{\partial U}{\partial x_i}(x)}{\frac{\partial U}{\partial x_j}(x)}\text{ for all }t>0\end{array}$$

• When the ratio of goods consumed is independent of level of income, all income elasticity have to be equal 1. (所有商品的income elasticity 都是1)

  Proof

  $$\begin{array}{}\text{(10)}& \begin{array}{l}\frac{\partial }{\partial M}\left(\frac{x_i}{x_j}\right)=0\\ \frac{\frac{\partial x_i}{\partial M}{x}_j-x_i\frac{\partial x_j}{\partial M}}{x_j^2}=0\\ \frac{\partial x_i}{\partial M}{x}_j=\frac{\partial x_j}{\partial M}{x}_i\text{ multiply by M }\\ \frac{\partial x_i}{\partial M}\frac{M}{x_j}=\frac{\partial x_j}{\partial M}\frac{M}{x_i}\\ {\epsilon }_i={\epsilon }_j.\end{array}\end{array}$$

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3. Quasilinear utility function and Gorman form