<blockquote>
  <p>Consumer's problem in default scenario:</p>
</blockquote>

<p>\begin{eqnarray*} \max_{x_1, x_2} &amp; \ \frac{x_2}{(1+x_1)^2} \\ \text{s.t.} &amp; \ x_1 + x_2 \leq 10 \\&amp; \ 0 \leq x_2 \leq x_1  \end{eqnarray*}</p>

<p>We can easily show that the solution will satisfy $x_2 = x_1$, and therefore we can rewrite the above problem as:  </p>

<p>\begin{eqnarray*} \max_{x_1} &amp; \ \frac{x_1}{(1+x_1)^2} \\ \text{s.t.} &amp; \ 0 \leq x_1 \leq 5   \end{eqnarray*}</p>

<p>Solving this problem we get $x_1= 1$. Consequently, $x_2 = 1$ and the utility of the consumer in optimum is $\dfrac{1}{4}$.</p>

<blockquote>
  <p>Consumer's problem with a possibility of freely disposing good 1 after paying the fee:</p>
</blockquote>

<p>\begin{eqnarray*} \max_{x_1, x_2} &amp; \ x_2 \\ \text{s.t.} &amp; \ x_1 + x_2 \leq m \\&amp; \ 0 \leq x_2 \leq x_1  \end{eqnarray*}
where $m$ is the net income of the consumer after paying the lump-sum fee.</p>

<p>Solution to this problem is $\left(\dfrac{m}{2}, \dfrac{m}{2}\right)$, and the utility of the consumer in optimum is $\dfrac{m}{2}$. Comparing this with the default scenario yields that consumer is indifferent between the scheme and the default when $m = \frac{1}{2}$. Therefore, the maximum willingness to pay to dispose the good at $0$ price is $10 - \frac{1}{2} = 9.5$.</p>

CC BY-SA 4.0
2018-06-13T09:20:13.210Z
2018-06-13T09:20:13.210Z
Amit
<p><a href="https://i.sstatic.net/V7YHL.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/V7YHL.jpg" alt="enter image description here"></a></p>

<p>Please look at only part ii.</p>

<p>I wrote budget constraint $x_1+x_2=10-f$</p>

<p>When I don’t pay fee, my optimal value values${}=(1,1).$</p>

<p>In the case of free disposal, $b_1=b_2=5$ but I consume only $x_1=0$ and $x_2=5.$</p>

<p>However I couldn’t find the value of $f.$ How can I find?</p>

CC BY-SA 4.0
2018-06-13T08:14:54.510Z
Hint: To be able to pay for free disposal you cannot use all income to buy goods.
Klas Lindbäck
CC BY-SA 4.0
2018-06-13T08:16:38.377Z
Hint 2: With free disposal you need to find the cheapest option that outperforms the optimal value you found when not paying the fee.
Klas Lindbäck
CC BY-SA 4.0
2018-06-13T08:17:25.560Z
Hint 3: The consumer will be willing to spend his remaining income on free disposal.
Klas Lindbäck
<microeconomics>
Consumer theory with lump sum fee
B11b.
# Consumer theory with lump sum fee

(Asked by: username_0 on 2018-06-13)

[![enter image description here](https://i.sstatic.net/V7YHL.jpg)](https://i.sstatic.net/V7YHL.jpg)

Please look at only part ii.

I wrote budget constraint $x\_1+x\_2=10-f$

When I don’t pay fee, my optimal value values${}=(1,1).$

In the case of free disposal, $b\_1=b\_2=5$ but I consume only $x\_1=0$ and $x\_2=5.$

However I couldn’t find the value of $f.$ How can I find?

**Tags:** microeconomics

> **Comments:**
> **username_1**: Hint: To be able to pay for free disposal you cannot use all income to buy goods.  
> **username_1**: Hint 2: With free disposal you need to find the cheapest option that outperforms the optimal value you found when not paying the fee.  
> **username_1**: Hint 3: The consumer will be willing to spend his remaining income on free disposal.  

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## 1 answers

### Answer by username_2 on 2018-06-13. Score: 2 | Accepted answer

> Consumer's problem in default scenario:

\begin{eqnarray\*} \max\_{x\_1, x\_2} & \ \frac{x\_2}{(1+x\_1)^2} \\ \text{s.t.} & \ x\_1 + x\_2 \leq 10 \\& \ 0 \leq x\_2 \leq x\_1 \end{eqnarray\*}

We can easily show that the solution will satisfy $x\_2 = x\_1$, and therefore we can rewrite the above problem as:

\begin{eqnarray\*} \max\_{x\_1} & \ \frac{x\_1}{(1+x\_1)^2} \\ \text{s.t.} & \ 0 \leq x\_1 \leq 5 \end{eqnarray\*}

Solving this problem we get $x\_1= 1$. Consequently, $x\_2 = 1$ and the utility of the consumer in optimum is $\dfrac{1}{4}$.

> Consumer's problem with a possibility of freely disposing good 1 after paying the fee:

\begin{eqnarray\*} \max\_{x\_1, x\_2} & \ x\_2 \\ \text{s.t.} & \ x\_1 + x\_2 \leq m \\& \ 0 \leq x\_2 \leq x\_1 \end{eqnarray\*}
where $m$ is the net income of the consumer after paying the lump-sum fee.

Solution to this problem is $\left(\dfrac{m}{2}, \dfrac{m}{2}\right)$, and the utility of the consumer in optimum is $\dfrac{m}{2}$. Comparing this with the default scenario yields that consumer is indifferent between the scheme and the default when $m = \frac{1}{2}$. Therefore, the maximum willingness to pay to dispose the good at $0$ price is $10 - \frac{1}{2} = 9.5$.


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