Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff95edb07-fce6-4fe2-9528-1ac1bc343004%2F6524f9fb-6e06-42d5-acac-8288e6bb5122%2Fp3thz5w_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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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