4 Consider a three-good consumer optimization problem where the utility function is given:$u_i(x_1,x_2,R)=V_1(x_1)+V_2(x_2) +R$$s.t. I=p_1 x_1 +p_2 x_2 +R$This is also called a quasi-linear utility function. This utility function is corresponding to the situation where- There are two target goods, $x_1$ and $x_2$, and the rest is summarized into R.- The demands are independent of each other in the sense that there is no cross price effect.- $V_1$ and $V_2$ are their benefits function (defined independent of the prices). Thus, $V'_1$ and $V'_2$ are their inversedemand functions.Let's confirm the second and third facts:- First, by substituting the budget constraint for R, we can obtain $u_i=V_1(x_1)+V_2(x_2)+($$-D- By considering FOC for each good, we obtain\(=V'_i(x_i)$ for each i.Thus, we can confirm the two facts.

Answer Budget constraint : I = P121 + P2.02 + R = Implies R=1- P1.21 - 02.02 = Substitute this…

Answered: ng FOC for each

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ng FOC for each

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Question

Consider a three-good consumer optimization problem where the utility function is given:
$u_i(x_1,x_2,R)=V_1(x_1)+V_2(x_2) +R$
$s.t. I=p_1 x_1 +p_2 x_2 +R$
This is also called a quasi-linear utility function. This utility function is corresponding to the situation where
- There are two target goods, $x_1$ and $x_2$, and the rest is summarized into R.
- The demands are independent of each other in the sense that there is no cross price effect.
- $V_1$ and $V_2$ are their benefits function (defined independent of the prices). Thus, $V'_1$ and $V'_2$ are their inverse
demand functions.
Let's confirm the second and third facts:
- First, by substituting the budget constraint for R, we can obtain $u_i=V_1(x_1)+V_2(x_2)+($
$-D
- By considering FOC for each good, we obtain
\(=V'_i(x_i)$ for each i.
Thus, we can confirm the two facts.

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