<font face="Times New Roman" size="4"><p align="center"><font face="Times New Roman" size="4">Indirect utility</font></p><p align="center"><font face="Times New Roman" size="4">Author: Tieding-Zou</font></p></font><p><font face="Times New Roman" size="4">First, the meaning of indirect utility</font></p><p><font face="Times New Roman" size="4">If there is a special good A , a consumer can feel satisfaction from the consumption of A ,we can call this as utility. Using X as the amount of A ,so the react utility is U(x) .as we knew that each amount consumption can reach a maximum level of utility, this utility is the indirect utility which we want to discuss.And we can use V(p,i) to show indirect utility.among this function ,p is the price of A ,i is the income of consumer.</font></p><p><font face="Times New Roman" size="4"></font></p><p><font face="Times New Roman" size="4">In practice ,this idea is very actually, it based on the </font><a href="http://www.iciba.com/presupposition/"><font face="Times New Roman" color="#ae0405" size="4">presupposition</font></a><font face="Times New Roman" size="4"> of economic man, which tells us that the individuals in market always want to use the minimum inputs to produce the maximum outputs, in a word ,the productivity is very important.</font></p><p><font face="Times New Roman" size="4">Second, the characteristics of indirect utility</font></p><p><font face="Times New Roman" size="4">1.Holding constant the income of consumer ,just let the price of A changes.It is easy to see that , the utility will increase if there is a decreasing in price, on the other side ,the utility will decrease. So we get this conclusion there is a adverse relationship between utility and price . V(p,i) is a reduction function of variable p.as the same as:1) if p↗,then V(p,i)↘;2) if p↘,then V(p,i) ↗.</font></p><p><font face="Times New Roman" size="4">2. With the same price level, and let income changes freely. This change will make the utilities change as price, all of them decrease or increase at the same direction. So there is a same direction between utility and price. V(p,i) is a increasing function of variable i.We can show it as this: 1) if i↗,then V(p,i) ↗;2) if i↘,then V(p,i) ↘</font>.</p><p><font face="Times New Roman" size="4">3. If price and imcome changed for the same direction and the same percentage,the result of this change is consumer’s utilities keep invariant, from this we can know that V(p,i) is a zero correlaed and homogeneous function.We can use a formula to summary this relationships: ￡V(p,i)= V(￡p, ￡i), ￡>0.This is a phenomenon of </font><a href="http://blog.sohu.com/manage/entry.do?m=edit&id=105464724&t=shortcut#"><font face="Times New Roman" color="#ae0405" size="4">constant returns to scale</font></a><font face="Times New Roman" size="4">.</font></p><p><font face="Times New Roman" size="4">4. V(p,i) must be a </font><a href="http://blog.sohu.com/manage/entry.do?m=edit&id=105464724&t=shortcut#"><font face="Times New Roman" color="#ae0405" size="4">quasi-convex function</font></a><font face="Times New Roman" size="4"> to variable p .which means that if there is a coefficient t and two different price levels p1 and p2,and 1>t>0 there must have a </font><a href="http://dict.cnki.net/dict_result.aspx?searchword=%e4%b8%8d%e7%ad%89%e5%bc%8f&tjType=sentence&style=&t=inequation"><font face="Times New Roman" color="#ae0405" size="4">inequation</font></a><font face="Times New Roman" size="4"> V[tp1+(1-t)p2, i]≤max{ V(p1,i), V(p2,i)}.</font></p><p><font face="Times New Roman" size="4">5.If V(p,i) is a Differentiable function , we can get Un=-[△V(p,i)/ △p]÷[△V(p,i)/ △i], this result derived from roy equation.</font></p><p><font face="Times New Roman" size="4"></font></p>