Quasi-Linear

EC487_Advanced_Microeconomics

We use an example in the Exam2024LT to introduce.

The normal form:

$u(x_1,x_2,\dots ,x_n)=v(x_1,x_2,\dots ,x_{n-1})+x_n$

where

• $v$ is a utility function (it may not be linear)
• $x_n$ here is about transferable utility (in exchange economy). And it is linear.

suppose in a utility function

$u^j(x^j,{\ell }_A^j,{\ell }_B^j,{\ell }_A^j,{\ell }_B^j,{\iota }^j)=\underset{v(\cdot )}{\underbrace{(x^j{)}^{\alpha }({\ell }_A^j+{\ell }_B^j{)}^{1-\alpha }-\frac{({\ell }_A^j+{\ell }_B^j{)}^2}{2}}}+{\iota }^j$

其中 ${\iota }^j$ 是跨国劳动的净收入，可以理解为”货币收入”。

In the question, ${\iota }^j=l_j^{-j}{w}_j^{-j}-l_{-j}^j{w}^j$.

It’s obvious that the change in the $\iota$ won’t lead to change in your desire of $x^j$ (land here).

Also for the Marginal Utility,

$$\frac{u^j}{{\iota }^j}=1$$

Thus there’s no diminishing Marginal Utility happens here.