<p>Let $x_{ij}$ denote the amount of good $i$ bought by the consumer from store $j$. So, consumer's utility maximization problem is described as follows :
\begin{eqnarray*} \max_{x_{11}, x_{12}, x_{21}, x_{22}} &amp; (x_{11}+x_{12})(x_{21}+x_{22}) \\ \text{s.t.} &amp; \ \ 2\sqrt{x_{11}} + x_{21} + 2x_{12} + \frac{1}{2} x_{22} \leq 6 \\ &amp; x_{11} \geq 0, x_{12} \geq 0, x_{21} \geq 0, x_{22} \geq 0 \end{eqnarray*}
Given that good 2 is cheaper in store 2, we know that the solution to this problem will satisfy $x_{21} = 0$. Consequently, it is clear from the objective that consumer will always choose $x_{22} &gt; 0$. Also consumer's utility is an increasing function, so consumer will spend all his income in optimum.  Taking them into account, we can rewrite the utility maximization problem of the consumer as :
\begin{eqnarray*} \max_{x_{11}, x_{12}, x_{22}} &amp; (x_{11}+x_{12})x_{22} \\ \text{s.t.} &amp; \ \ 2\sqrt{x_{11}} + 2x_{12} + \frac{1}{2} x_{22} = 6 \\ &amp; x_{11} \geq 0, x_{12} \geq 0, x_{22} &gt; 0\end{eqnarray*}
Now we set up the Lagrangian:
\begin{eqnarray*} \mathcal{L}(x_{11}, x_{12}, x_{22}) = (x_{11}+x_{12})x_{22} - \lambda (2\sqrt{x_{11}} + 2x_{12} + \frac{1}{2} x_{22} - 6) + \mu_1 x_{11} + \mu_2 x_{12} \end{eqnarray*}
Corresponding Kuhn-Tucker conditions are :
\begin{eqnarray*} \frac{\partial \mathcal{L}}{\partial x_{11}} &amp; = &amp; x_{22} - \frac{\lambda}{\sqrt{x_{11}}} + \mu_1 = 0 \\ \frac{\partial \mathcal{L}}{\partial x_{12}} &amp; = &amp; x_{22} - 2\lambda + \mu_2 = 0 \\ \frac{\partial \mathcal{L}}{\partial x_{22}} &amp; = &amp; x_{11} + x_{12} - \frac{\lambda}{2}  = 0 \\ \frac{\partial \mathcal{L}}{\partial \lambda} &amp; = &amp; 6 - 2\sqrt{x_{11}} - 2x_{12} - \frac{1}{2} x_{22}  = 0, \ \lambda \geq 0 \\ \frac{\partial \mathcal{L}}{\partial \mu_1} &amp; = &amp; x_{11}  \geq  0, \ \mu_1 \geq 0, \ \mu_1x_{11} = 0 \\ \frac{\partial \mathcal{L}}{\partial \mu_2} &amp; = &amp; x_{12}  \geq  0, \ \mu_2 \geq 0, \ \mu_2x_{12} = 0\end{eqnarray*}</p>

<p>One of the solutions to the above conditions solve the optimization problem. Here is that solution :
$x_{11}^* = 4, x_{12}^* = 0, x_{22}^* = 4, \lambda^* = 8, \mu_1^* = 0,\mu_2^* = 12$.</p>

<p>Therefore, consumer will buy 4 units of commodity 1 and 0 unit of commodity 2 from the first store; and 0 unit of commodity 1 and 4 units of commodity 2 from the other store.</p>

CC BY-SA 4.0
2018-06-13T04:59:04.387Z
Amit, I have learnt perfectly from your solution for this type of questions that I previously asked. I can deal with other parts, but I couldn’t find tha part 2 only. Please can you summarize your solution. I doNot need long answer. I can proceed it. https://economics.stackexchange.com/questions/22442/consumer-theory-with-lump-sum-fee
b11bb
CC BY-SA 4.0
2018-07-26T11:29:27.503Z
Dear Amit I know I disturbed you too much but I have posted two questions that I could not solve. I have an exam again. Please help me. These two questions are  last questions that I ask for you. But I need their solutions too much. Because this exam is my last chance as well. I will be happy if you help me. Please. Many many thanks. I don’t know how to give something like a gift toward your great helps. But this is last help! I really can give you Promise! :)
studentp
CC BY-SA 4.0
2018-07-26T11:30:10.637Z
First question : https://economics.stackexchange.com/questions/23890/question-on-signaling
studentp
CC BY-SA 4.0
2018-07-26T11:30:54.243Z
Second question : https://economics.stackexchange.com/questions/23891/question-on-phd-piliminary-exam-question
studentp
CC BY-SA 4.0
2018-07-26T11:36:50.987Z
Amit, if it is difficult and take much time to write them by latex in this website, I can send them by email and you can send them by your hand-writing, if you want. This is really last help.
studentp
CC BY-SA 4.0
2018-07-26T22:03:00.503Z
My questions are closed. They no longer have any answering part. If you will solve, how can I send you? 
studentp
CC BY-SA 4.0
2018-07-28T09:28:38.867Z
Dear Amit I can type my answer. Please help me to solve?
studentp
CC BY-SA 4.0
2018-07-28T09:48:03.793Z
I have really need how to solve these questions. Otherwise, I never  persist too much. On this Monday, I have an exam.  Please reply me today
studentp
CC BY-SA 4.0
2018-06-08T15:47:57.490Z
2018-06-08T15:58:15.553Z
2018-06-08T15:58:15.553Z
Amit
<p>I have also following budget set </p>

<p>$$B=\{x=(x_1,x_2)\in R^2_+ \mid 2\sqrt{x_1}+x_2\le y\}$$</p>

<p>where y is income. </p>

<p>Assume that there are two stories. The agent can shop in both of them. The first store has a quantity discount on good1 as described above. And a fixed unit price of good 2. $p_2=1$. The other store charges fixed unit prices, $p_1=2$ and $p_2=1/2$ respectively. Assume that income y is 6. How can I write the feasible budget constraint for utility maximization $U(x_1, x_2)=x_1x_2$. And since there is two stores, how can I write Lagrangian function for calculation of the optimization problem.</p>

<p>I know I have to write my trials. But I cannot write properly any budget  constraint and Lagrangian therefore, I cannot post my solution. </p>

<p>Any help is appreciated. Thank you.   </p>

<p>Edit </p>

<p><a href="https://i.sstatic.net/kXX2o.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/kXX2o.jpg" alt="enter image description here"></a></p>

CC BY-SA 4.0
2018-05-09T14:14:10.893Z
your description of the problem is insufficient. is the consumer able to go to just one store or to both?
saguru
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2018-05-09T19:33:55.597Z
@saguru the question say nothing about this. But I guess the consumer can go to both.
studentp
CC BY-SA 4.0
2018-05-10T08:22:13.553Z
so the consumer is able to shop good 1 at one store and then go to the other for good 2? Then for good 2 it is obvious. Could you please add the quantity discount information too?
saguru
CC BY-SA 4.0
2018-05-10T08:23:27.060Z
and you should probably reconsult your micro books to understand the connection between the budget set and the buget constraint. as you are given the budget set, the budget constraint is given too
saguru
CC BY-SA 4.0
2018-05-10T08:52:51.093Z
@saguru this is the only one part of my question. I post whole question. Thanks for helps:)
studentp
CC BY-SA 4.0
2018-05-11T10:43:38.003Z
ok, I guess going to one store excludes you from going to the other store. but then it should be simple, for which bundles would you prefer to go to store 1? for which would you prefer to go to store 2?
saguru
CC BY-SA 4.0
2018-05-12T01:06:46.410Z
Dear @saguru will I write two budget sets? For store 1, the budget set is $2\sqrt{x_1}+x_2\le 6$. And for the store 2, $2\sqrt{x_1}+(1/2)x_2\le 6$. But I think that the question asks only one feasible set in the last part (iv)? What do you think about my answers?
studentp
<microeconomics><mathematical-economics><utility><optimization><self-study>
Question about budget constraint and utility maximization
studentp
# Question about budget constraint and utility maximization

(Asked by: username_0 on 2018-05-09)

I have also following budget set

$$B=\{x=(x\_1,x\_2)\in R^2\_+ \mid 2\sqrt{x\_1}+x\_2\le y\}$$

where y is income.

Assume that there are two stories. The agent can shop in both of them. The first store has a quantity discount on good1 as described above. And a fixed unit price of good 2. $p\_2=1$. The other store charges fixed unit prices, $p\_1=2$ and $p\_2=1/2$ respectively. Assume that income y is 6. How can I write the feasible budget constraint for utility maximization $U(x\_1, x\_2)=x\_1x\_2$. And since there is two stores, how can I write Lagrangian function for calculation of the optimization problem.

I know I have to write my trials. But I cannot write properly any budget constraint and Lagrangian therefore, I cannot post my solution.

Any help is appreciated. Thank you.

Edit

[![enter image description here](https://i.sstatic.net/kXX2o.jpg)](https://i.sstatic.net/kXX2o.jpg)

**Tags:** microeconomics, mathematical-economics, utility, optimization, self-study

> **Comments:**
> **username_1**: your description of the problem is insufficient. is the consumer able to go to just one store or to both?  
> **username_0**: @username_1 the question say nothing about this. But I guess the consumer can go to both.  
> **username_1**: so the consumer is able to shop good 1 at one store and then go to the other for good 2? Then for good 2 it is obvious. Could you please add the quantity discount information too?  
> **username_1**: and you should probably reconsult your micro books to understand the connection between the budget set and the buget constraint. as you are given the budget set, the budget constraint is given too  
> **username_0**: @username_1 this is the only one part of my question. I post whole question. Thanks for helps:)  
> **username_1**: ok, I guess going to one store excludes you from going to the other store. but then it should be simple, for which bundles would you prefer to go to store 1? for which would you prefer to go to store 2?  
> **username_0**: Dear @username_1 will I write two budget sets? For store 1, the budget set is $2\sqrt{x_1}+x_2\le 6$. And for the store 2, $2\sqrt{x_1}+(1/2)x_2\le 6$. But I think that the question asks only one feasible set in the last part (iv)? What do you think about my answers?  

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## 1 answers

### Answer by username_2 on 2018-06-08. Score: 1 | Accepted answer

Let $x\_{ij}$ denote the amount of good $i$ bought by the consumer from store $j$. So, consumer's utility maximization problem is described as follows :
\begin{eqnarray\*} \max\_{x\_{11}, x\_{12}, x\_{21}, x\_{22}} & (x\_{11}+x\_{12})(x\_{21}+x\_{22}) \\ \text{s.t.} & \ \ 2\sqrt{x\_{11}} + x\_{21} + 2x\_{12} + \frac{1}{2} x\_{22} \leq 6 \\ & x\_{11} \geq 0, x\_{12} \geq 0, x\_{21} \geq 0, x\_{22} \geq 0 \end{eqnarray\*}
Given that good 2 is cheaper in store 2, we know that the solution to this problem will satisfy $x\_{21} = 0$. Consequently, it is clear from the objective that consumer will always choose $x\_{22} > 0$. Also consumer's utility is an increasing function, so consumer will spend all his income in optimum. Taking them into account, we can rewrite the utility maximization problem of the consumer as :
\begin{eqnarray\*} \max\_{x\_{11}, x\_{12}, x\_{22}} & (x\_{11}+x\_{12})x\_{22} \\ \text{s.t.} & \ \ 2\sqrt{x\_{11}} + 2x\_{12} + \frac{1}{2} x\_{22} = 6 \\ & x\_{11} \geq 0, x\_{12} \geq 0, x\_{22} > 0\end{eqnarray\*}
Now we set up the Lagrangian:
\begin{eqnarray\*} \mathcal{L}(x\_{11}, x\_{12}, x\_{22}) = (x\_{11}+x\_{12})x\_{22} - \lambda (2\sqrt{x\_{11}} + 2x\_{12} + \frac{1}{2} x\_{22} - 6) + \mu\_1 x\_{11} + \mu\_2 x\_{12} \end{eqnarray\*}
Corresponding Kuhn-Tucker conditions are :
\begin{eqnarray\*} \frac{\partial \mathcal{L}}{\partial x\_{11}} & = & x\_{22} - \frac{\lambda}{\sqrt{x\_{11}}} + \mu\_1 = 0 \\ \frac{\partial \mathcal{L}}{\partial x\_{12}} & = & x\_{22} - 2\lambda + \mu\_2 = 0 \\ \frac{\partial \mathcal{L}}{\partial x\_{22}} & = & x\_{11} + x\_{12} - \frac{\lambda}{2} = 0 \\ \frac{\partial \mathcal{L}}{\partial \lambda} & = & 6 - 2\sqrt{x\_{11}} - 2x\_{12} - \frac{1}{2} x\_{22} = 0, \ \lambda \geq 0 \\ \frac{\partial \mathcal{L}}{\partial \mu\_1} & = & x\_{11} \geq 0, \ \mu\_1 \geq 0, \ \mu\_1x\_{11} = 0 \\ \frac{\partial \mathcal{L}}{\partial \mu\_2} & = & x\_{12} \geq 0, \ \mu\_2 \geq 0, \ \mu\_2x\_{12} = 0\end{eqnarray\*}

One of the solutions to the above conditions solve the optimization problem. Here is that solution :
$x\_{11}^\* = 4, x\_{12}^\* = 0, x\_{22}^\* = 4, \lambda^\* = 8, \mu\_1^\* = 0,\mu\_2^\* = 12$.

Therefore, consumer will buy 4 units of commodity 1 and 0 unit of commodity 2 from the first store; and 0 unit of commodity 1 and 4 units of commodity 2 from the other store.

> **Comments:**
> **username_3**: username_2, I have learnt perfectly from your solution for this type of questions that I previously asked. I can deal with other parts, but I couldn’t find tha part 2 only. Please can you summarize your solution. I doNot need long answer. I can proceed it. https://economics.stackexchange.com/questions/22442/consumer-theory-with-lump-sum-fee  
> **username_0**: Dear username_2 I know I disturbed you too much but I have posted two questions that I could not solve. I have an exam again. Please help me. These two questions are  last questions that I ask for you. But I need their solutions too much. Because this exam is my last chance as well. I will be happy if you help me. Please. Many many thanks. I don’t know how to give something like a gift toward your great helps. But this is last help! I really can give you Promise! :)  
> **username_0**: First question : https://economics.stackexchange.com/questions/23890/question-on-signaling  
> **username_0**: Second question : https://economics.stackexchange.com/questions/23891/question-on-phd-piliminary-exam-question  
> **username_0**: username_2, if it is difficult and take much time to write them by latex in this website, I can send them by email and you can send them by your hand-writing, if you want. This is really last help.  
> **username_0**: My questions are closed. They no longer have any answering part. If you will solve, how can I send you?   
> **username_0**: Dear username_2 I can type my answer. Please help me to solve?  
> **username_0**: I have really need how to solve these questions. Otherwise, I never  persist too much. On this Monday, I have an exam.  Please reply me today  

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