<p>Since in this case, both utility function and the budget line are perfect substitutes and thus you can't equate the <strong>Marginal Rate of Substitution</strong> to <strong>the price ratio</strong>.</p>

<p>This problem can be solved this way,</p>

<p><span class="math-container">$$\min_{x,y} \quad  3x+3y\\
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textrm{s.t.} \quad  2x+5y=10\\$$</span></p>

<p>or</p>

<p><span class="math-container">$$\min_{x,y} \quad  x+y\\
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \textrm{s.t.} \quad  2x+5y=10\\$$</span></p>

<p>Here,<br>
     <span class="math-container">$$MRS=\frac{MU_{x}}{MU_{y}}=\frac{2}{5}&lt;1=\frac{P_{x}}{P_{y}}$$</span> 
or,  <span class="math-container">$$\frac{MU_{x}}{P_{x}}=2 &lt; 5 = \frac{MU_{y}}{P_{y}}$$</span> </p>

<p>Therefore the expenditure is minimum when the consumer spends his entire income on good y, that is <span class="math-container">$(x,y)=(0,2)$</span> is the bundle that minimizes the expenditure given the utility <span class="math-container">$u(x,y)=2x+5y=10$</span>. This is depicted in the figure below:</p>

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

CC BY-SA 4.0
2020-02-24T05:16:36.500Z
2020-02-24T05:16:36.500Z
Naveen Hari
<p>Here,  Mux/Muy is less than Px/Py and this utility function is perfect substitute type So we are free to choose only x or y. 
Here the optimal solution will be(0, 2)</p>

CC BY-SA 4.0
2020-02-24T05:02:15.847Z
2020-02-24T05:02:15.847Z
PREYASI KUMARI
<p>The utility function shows that x and y are perfect complements with, y giving a MU of 5 and x giving an MU of 2. Both have the same price. Hence in equilibrium the consumer will consume only y. So quantity of x is 0.</p>

CC BY-SA 4.0
2020-02-24T04:08:15.743Z
2020-02-24T04:50:53.720Z
2020-02-24T04:50:53.720Z
Sab Ray
<p>So I have this utility function where I need to find t expenditure minimizing value of x. Normally, I would get the marginal utility of goods x and y, but doing that in this case doesn't leave any variables, so setting that equal to the prices would give me 2/5 = 3/3, which doesn't make sense. How do I solve this? I feel like I know how to do it but I'm just missing a major step or two.</p>

<p><a href="https://i.sstatic.net/TvuOE.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/TvuOE.jpg" alt=""></a></p>

CC BY-SA 4.0
2020-02-24T04:55:26.820Z
Try drawing the indifference curve (for $u(x,y)=10$) and budget line.
Herr K.
<microeconomics>
How to derive this Hicksian Demand?
Layden Monte
# How to derive this Hicksian Demand?

(Asked by: username_0 on 2020-02-24)

So I have this utility function where I need to find t expenditure minimizing value of x. Normally, I would get the marginal utility of goods x and y, but doing that in this case doesn't leave any variables, so setting that equal to the prices would give me 2/5 = 3/3, which doesn't make sense. How do I solve this? I feel like I know how to do it but I'm just missing a major step or two.

[![](https://i.sstatic.net/TvuOE.jpg)](https://i.sstatic.net/TvuOE.jpg)

**Tags:** microeconomics

> **Comments:**
> **username_1**: Try drawing the indifference curve (for $u(x,y)=10$) and budget line.  

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## 3 answers

### Answer by username_2 on 2020-02-24. Score: 1

Since in this case, both utility function and the budget line are perfect substitutes and thus you can't equate the **Marginal Rate of Substitution** to **the price ratio**.

This problem can be solved this way,

$$\min\_{x,y} \quad 3x+3y\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textrm{s.t.} \quad 2x+5y=10\\$$

or

$$\min\_{x,y} \quad x+y\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textrm{s.t.} \quad 2x+5y=10\\$$

Here,  
$$MRS=\frac{MU\_{x}}{MU\_{y}}=\frac{2}{5}<1=\frac{P\_{x}}{P\_{y}}$$
or, $$\frac{MU\_{x}}{P\_{x}}=2 < 5 = \frac{MU\_{y}}{P\_{y}}$$

Therefore the expenditure is minimum when the consumer spends his entire income on good y, that is $(x,y)=(0,2)$ is the bundle that minimizes the expenditure given the utility $u(x,y)=2x+5y=10$. This is depicted in the figure below:

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


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### Answer by username_3 on 2020-02-24. Score: 0

Here, Mux/Muy is less than Px/Py and this utility function is perfect substitute type So we are free to choose only x or y.
Here the optimal solution will be(0, 2)


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### Answer by username_4 on 2020-02-24. Score: 0

The utility function shows that x and y are perfect complements with, y giving a MU of 5 and x giving an MU of 2. Both have the same price. Hence in equilibrium the consumer will consume only y. So quantity of x is 0.


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