Editing Expenditure function (section)

== Expenditure and indirect utility ==
The expenditure function is the inverse of the [[indirect utility]] function when the prices are kept constant. I.e, for every price vector <math>p</math> and income level <math>I</math>:<ref>{{Cite Varian Microeconomic Analysis 3}}</ref>{{rp|106}}
:<math>e(p, v(p,I)) \equiv I</math>
There is a duality relationship between the expenditure function and the utility function. If given a specific regular quasi-concave utility function, the corresponding price is homogeneous, and the utility is monotonically increasing expenditure function, conversely, the given price is homogeneous, and the utility is monotonically increasing expenditure function will generate the regular quasi-concave utility function. In addition to the property that prices are once homogeneous and utility is monotonically increasing, the expenditure function usually assumes

# Is a non-negative function, i.e., <math> E(P \cdot u)>O; </math>
# For P, it is non-decreasing, i.e., <math> E(p^1 u)> E(p^2 u),u> Op^l>p^2> O_N </math>;
# E(Pu) is a concave function. That is, <math> e(np^l+(1-n)p^2)u )>\lambda E(p^1u)(1-n)E(p^2u)y>0 </math> <math> O<\lambda<1p^l\geq O_Np^2 \geq O_N </math>

Expenditure function is an important theoretical method to study consumer behavior. Expenditure function is very similar to cost function in production theory. Dual to the utility maximization problem is the cost minimization problem <ref>{{Cite book|url=https://www.worldcat.org/oclc/34287945|title=Jing ji xue da ci dian|date=1994|publisher=Tuan jie chu ban she|others=Xiaomin Liang, 梁小民.|isbn=7-80061-954-0|edition=Di 1 ban|location=Beijing Shi|oclc=34287945}}</ref><ref>{{Cite web|title=CONSUMER CHOICE AND DUALITY|date=23 February 2024 |url=http://www2.econ.iastate.edu/classes/econ501/Hallam/documents/DualConsumer.pdf}}</ref>