<p>To provide another answer with less equations:</p>
<p>Consider first that the inner utility function <span class="math-container">$u_1 = \sqrt{x+y}$</span> and <span class="math-container">$u_2 = z + w$</span> are perfect substitutes implying that consumer only buy the cheapest of <span class="math-container">$x$</span> and <span class="math-container">$y$</span> and similarly only the cheapest of <span class="math-container">$z$</span> and <span class="math-container">$w$</span>. Because prices are given we know that <span class="math-container">$y$</span> is cheaper than <span class="math-container">$x$</span> and <span class="math-container">$w$</span> is cheaper than <span class="math-container">$z$</span>. Using this we can simply set <span class="math-container">$x=0$</span> and <span class="math-container">$z=0$</span> and get the utility maximization problem</p>
<p>Maximize
<span class="math-container">$$U(y,w) = \min\{\sqrt y,w\}$$</span></p>
<p>which implies that <span class="math-container">$\sqrt y = w$</span> for any solution.</p>
<p>Therefore there are two equations in two unknowns</p>
<p><span class="math-container">$$p_y y + p_w w = m \Rightarrow 2y + w = 1$$</span></p>
<p>and
<span class="math-container">$$\sqrt y = w$$</span></p>
<p>Hence it follows that</p>
<p><span class="math-container">$$2y + \sqrt y = 1$$</span></p>
<p>which is satisfied for <span class="math-container">$y=1/4$</span> and since <span class="math-container">$\sqrt y = w$</span> it follows that <span class="math-container">$w = \sqrt(1/4) = 1/2$</span>. Adding that <span class="math-container">$x=0$</span> and <span class="math-container">$z=0$</span> we have all demands.</p>

CC BY-SA 4.0
2021-05-19T14:53:43.003Z
2021-05-19T14:53:43.003Z
Jesper Hybel
<p>You can solve it sequentially by noting the nesting structure of the utility function <span class="math-container">$U$</span>.</p>
<p>So first note that the utility function combines utility functions you are probably already familiar with <span class="math-container">$U=\min\{u_1,u_2\}$</span> of complements and <span class="math-container">$u_1=\sqrt{x+y}$</span> and <span class="math-container">$u_2 = z+w$</span> both of which are perfect substitutes. Where <span class="math-container">$u_1$</span> and <span class="math-container">$u_2$</span> are nested within <span class="math-container">$U$</span>.</p>
<p>The solution procedure is based on the following argument: However much money <span class="math-container">$m_{u_1}$</span> the agent decides to spend on <span class="math-container">$u_1 =\sqrt{x+y}$</span> she must be minimizing the cost of getting <span class="math-container">$u_1$</span> utility points. Similarly, however much money <span class="math-container">$m_{u_2}$</span> the agent spends on <span class="math-container">$u_2= z + w$</span> it must be the case that the agent minimize cost. So first find the expenditure functions for these inner problem.</p>
<p>Secondly the agent must the maximize the outer utility <span class="math-container">$\min\{u_1,u_2\}$</span> subject to the expenditures of the inner problems adding up to total income.</p>
<p><strong>Solve the second inner problem:</strong></p>
<p>The second inner problem is
<span class="math-container">$$max_{z,w} \{z+w \lvert p_zz+p_ww\leq m_{u_2}\},$$</span></p>
<p>where <span class="math-container">$m_{u_2}$</span> is money spend on <span class="math-container">$u_2$</span>.</p>
<p>How do you get the most utility for the buck when utility is <span class="math-container">$u_2 = z + w$</span>? You simply buy the cheapest good so the price of one point of <span class="math-container">$u_2$</span> utility is <span class="math-container">$p_{u_2} := \min\{p_z,p_w\}$</span>. The demand for <span class="math-container">$z$</span> and <span class="math-container">$w$</span> depends on the amount of money you decide to spend on <span class="math-container">$u_2$</span> assumed for now to be <span class="math-container">$m_{u_2}$</span>. Given this amount demand for <span class="math-container">$z$</span> is</p>
<p><span class="math-container">$$z = \frac{m_{u_2}}{p_z} 1[p_z &lt;p_w]$$</span></p>
<p>and</p>
<p><span class="math-container">$$w = \frac{m_{u_2}}{p_w} 1[p_w &lt;p_z].$$</span></p>
<p>I leave it to you to handle the threshold case where <span class="math-container">$p_z=p_w$</span>. How much utility <span class="math-container">$u_2$</span> do you get? Well this must be <span class="math-container">$u_2 = m_{u_2}/\min\{p_z,p_w\}$</span> - this is the value function for this inner utility maximization problem - hence it must be the case that</p>
<p><span class="math-container">$$m_{u_2} = \min\{p_z,p_w\} u_2,$$</span></p>
<p>which is the expenditure function for this inner problem.</p>
<p><strong>Solve the first inner problem:</strong></p>
<p>The first inner problem is
<span class="math-container">$$max_{x,y} \{\sqrt{x+y} \lvert p_xx+p_yy\leq m_{u_1}\},$$</span></p>
<p>where <span class="math-container">$m_{u_1}$</span> is money spend on <span class="math-container">$u_1$</span>.</p>
<p>How do you get the most utility for the buck when utility is <span class="math-container">$u_1 = \sqrt{x+y}$</span>? It is perfect substitutes so you buy the cheapest good so demand is</p>
<p><span class="math-container">$$x = \frac{m_{u_1}}{p_x} 1[p_x &lt;p_y]$$</span></p>
<p>and</p>
<p><span class="math-container">$$y = \frac{m_{u_1}}{p_y} 1[p_y &lt;p_x],$$</span></p>
<p>where <span class="math-container">$m_{u_1}$</span> is amount of money spend on <span class="math-container">$u_1$</span> assumed known. How much utility <span class="math-container">$u_1$</span> do you get? That must be <span class="math-container">$u_1 = \sqrt{m_{u_1}/\min\{p_x,p_y\}}$</span> such that</p>
<p><span class="math-container">$$m_{u_1} = \min\{p_x,p_y\} u_1^2$$</span></p>
<p><strong>Solve outer problem:</strong></p>
<p>The outer problem is</p>
<p><span class="math-container">$$\max_{u_1,u_2}: \ \ \  \min\{u_1,u_2\}  \\[8pt] 
s.t. \ \ \ \ m_{u_1} + m_{u_2} = m $$</span></p>
<p>for which we rewrite the budget constraint to</p>
<p><span class="math-container">$$\min\{p_x,p_y\} u_1^2 + \min\{p_z,p_w\} u_2 = m$$</span></p>
<p>and then use that <span class="math-container">$u_1=u_2$</span> because the problem is perfect complements. You then have two equations in two unknowns <span class="math-container">$u_1$</span> and <span class="math-container">$u_2$</span>. Solve for <span class="math-container">$u_1$</span> and <span class="math-container">$u_2$</span> and plug solutions back into expenditure functions for inner problems to get <span class="math-container">$m_{u_1}$</span> and <span class="math-container">$m_{u_2}$</span> which you then plug into the equations for the demand.</p>

CC BY-SA 4.0
2021-05-18T20:50:38.727Z
It is just an indicator function that is $0$ if condition in brackets is not satisfied and $1$ when condition is satisfied. Hence $z = (m/p_z) 1[p_z<p_w]$ is just $z=m/p_z$ when $p_z<p_w$  because $1[p_z<p_w]=1$ and  $z = (m/p_z) 1[p_z<p_w]$ then becomes $0$ when$p_w<p_z$ because $1[p_z<p_w]$ then becomes $0$. So it is just a way to write a fork function and insure that demand is 0 for the good that is the most expensive of the two perfect substitutes.
Jesper Hybel
CC BY-SA 4.0
2021-05-18T21:04:40.797Z
Yes, I was merely using the more abstract notation to illustrate the general procedure.
Jesper Hybel
CC BY-SA 4.0
2021-05-19T10:21:27.663Z
All the $m_{u_i}$ would just be my known wage 4, if I understand your approach correctly? It isn’t a partial ammount since we are dealing with perfect substitutes.
user879295
CC BY-SA 4.0
2021-05-19T11:52:50.620Z
Or should it be distributed evenly between the 2 inner problems? I.e 2 each?
user879295
CC BY-SA 4.0
2021-05-19T11:58:43.517Z
$m_{u_1}$ is income used on $x$ and $y$, $m_{u_2}$ is income used on $z$ and $w$. With the prices you have consumer will only buy $y$ and $w$. The distribution will most likely not be even split because $\sqrt y = w$ since $u_1=u_2$ due to outer utility being complements. Yes total income is distributed $m = m_{u_1} + m_{u_2}$ but no requirement that $m_{u_1}=m_{u_2}$.
Jesper Hybel
CC BY-SA 4.0
2021-05-19T12:12:53.710Z
Ah so actually, $m_{u_2}=u_2$ since the price is $p_w=1$ (as the minimum of the $2$ prices is $1$ and the wage is $4$ according to $m_{u_2} = \min\{p_z,p_w\} u_2$)
user879295
CC BY-SA 4.0
2021-05-19T12:21:43.167Z
I see that we got $2$ unknowns in both inner utility functions but only $1$ equation. Unless I understood it correctly that we let both constrain for the inners be $m_i=4$
user879295
CC BY-SA 4.0
2021-05-19T13:00:13.623Z
You have two equations in two unknowns: $\min\{p_x,p_y\} u_1^2  + \min\{p_z,p_w\} u_2 = m$ this reduces to $2 u_1^2  +  u_2 = m$ using values of prices as given. The other equation is that $u_1 = u_2$ because the outer problem is complements $U = \max_{u_1,u_2} \min\{u_1,u_2\}$. So you have $2 u_1^2  +  u_1 = m$ which because you know $m$ you can solve for $u_1$ having found $u_1$ you can find $u_2$ which is the same as $u_1$ using $u_1=u_2$ equation. You have equations for $m_{u_1}$ and $m_{u_2}$ as function of $u_1$,$u_2$ and prices all of which you now know.
Jesper Hybel
CC BY-SA 4.0
2021-05-19T13:43:36.640Z
I am unsure on the approach. Could we move to a discussion chat? i.e in the second inner problem we essentially just end up with $m_{u_2}=m_{u_2}$ since the price is $1$ which does not seem to do anything for me.
user879295
CC BY-SA 4.0
2021-05-19T14:18:13.897Z
I could have made a mistake let me just go through it.
Jesper Hybel
CC BY-SA 4.0
2021-05-19T14:20:23.110Z
I used that $u_1=u_2$ and thus get a polynomial of degree $n=2$ (by plugging it into the equation $2u_1^2+u_2=m$. Solving it gives $u_1=-1$ and $u_1=1/2$. Im sorry but $m=1$ and not $m=4$ which are quite a bit simpler***
user879295
CC BY-SA 4.0
2021-05-19T14:23:49.753Z
Check my edit for $m$. That might have been it.
user879295
CC BY-SA 4.0
2021-05-19T14:43:45.843Z
Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/124435/discussion-between-jesper-hybel-and-user879295).
Jesper Hybel
CC BY-SA 4.0
2021-05-18T20:01:20.297Z
2021-05-18T20:01:20.297Z
Jesper Hybel
<p>The consumer has the utility function with <span class="math-container">$4$</span> goods <span class="math-container">$$U=\min\left \{ \sqrt{x+y},z+w \right \}$$</span> The prices are <span class="math-container">$p=(3,2,2,1)$</span> with wage <span class="math-container">$m=1$</span>.</p>
<ul>
<li>Find the demand.</li>
</ul>
<p>So far I have observed that it is perfect complements BUT it must be between the 4 goods. So some combination of <span class="math-container">$x,y$</span> to <span class="math-container">$z,w$</span>. Usually we know that the optimal choice is when <span class="math-container">$x_2=x_1$</span> with demand <span class="math-container">$x_{1,2}=m/(p_1+p_2)$</span>(but with 4 goods, I lose my grib. How do I approach this? Do I just square the right side and let both equal eachother?</p>

<microeconomics><utility><supply-and-demand><demand>
Demand for minimum of $4$ different goods
user879295
# Demand for minimum of $4$ different goods

(Asked by: username_0 on 2021-05-18)

The consumer has the utility function with $4$ goods $$U=\min\left \{ \sqrt{x+y},z+w \right \}$$ The prices are $p=(3,2,2,1)$ with wage $m=1$.

* Find the demand.

So far I have observed that it is perfect complements BUT it must be between the 4 goods. So some combination of $x,y$ to $z,w$. Usually we know that the optimal choice is when $x\_2=x\_1$ with demand $x\_{1,2}=m/(p\_1+p\_2)$(but with 4 goods, I lose my grib. How do I approach this? Do I just square the right side and let both equal eachother?

**Tags:** microeconomics, utility, supply-and-demand, demand

---

## 2 answers

### Answer by username_1 on 2021-05-18. Score: 7 | Accepted answer

You can solve it sequentially by noting the nesting structure of the utility function $U$.

So first note that the utility function combines utility functions you are probably already familiar with $U=\min\{u\_1,u\_2\}$ of complements and $u\_1=\sqrt{x+y}$ and $u\_2 = z+w$ both of which are perfect substitutes. Where $u\_1$ and $u\_2$ are nested within $U$.

The solution procedure is based on the following argument: However much money $m\_{u\_1}$ the agent decides to spend on $u\_1 =\sqrt{x+y}$ she must be minimizing the cost of getting $u\_1$ utility points. Similarly, however much money $m\_{u\_2}$ the agent spends on $u\_2= z + w$ it must be the case that the agent minimize cost. So first find the expenditure functions for these inner problem.

Secondly the agent must the maximize the outer utility $\min\{u\_1,u\_2\}$ subject to the expenditures of the inner problems adding up to total income.

**Solve the second inner problem:**

The second inner problem is
$$max\_{z,w} \{z+w \lvert p\_zz+p\_ww\leq m\_{u\_2}\},$$

where $m\_{u\_2}$ is money spend on $u\_2$.

How do you get the most utility for the buck when utility is $u\_2 = z + w$? You simply buy the cheapest good so the price of one point of $u\_2$ utility is $p\_{u\_2} := \min\{p\_z,p\_w\}$. The demand for $z$ and $w$ depends on the amount of money you decide to spend on $u\_2$ assumed for now to be $m\_{u\_2}$. Given this amount demand for $z$ is

$$z = \frac{m\_{u\_2}}{p\_z} 1[p\_z

and

$$w = \frac{m\_{u\_2}}{p\_w} 1[p\_w

I leave it to you to handle the threshold case where $p\_z=p\_w$. How much utility $u\_2$ do you get? Well this must be $u\_2 = m\_{u\_2}/\min\{p\_z,p\_w\}$ - this is the value function for this inner utility maximization problem - hence it must be the case that

$$m\_{u\_2} = \min\{p\_z,p\_w\} u\_2,$$

which is the expenditure function for this inner problem.

**Solve the first inner problem:**

The first inner problem is
$$max\_{x,y} \{\sqrt{x+y} \lvert p\_xx+p\_yy\leq m\_{u\_1}\},$$

where $m\_{u\_1}$ is money spend on $u\_1$.

How do you get the most utility for the buck when utility is $u\_1 = \sqrt{x+y}$? It is perfect substitutes so you buy the cheapest good so demand is

$$x = \frac{m\_{u\_1}}{p\_x} 1[p\_x

and

$$y = \frac{m\_{u\_1}}{p\_y} 1[p\_y

where $m\_{u\_1}$ is amount of money spend on $u\_1$ assumed known. How much utility $u\_1$ do you get? That must be $u\_1 = \sqrt{m\_{u\_1}/\min\{p\_x,p\_y\}}$ such that

$$m\_{u\_1} = \min\{p\_x,p\_y\} u\_1^2$$

**Solve outer problem:**

The outer problem is

$$\max\_{u\_1,u\_2}: \ \ \ \min\{u\_1,u\_2\} \\[8pt]
s.t. \ \ \ \ m\_{u\_1} + m\_{u\_2} = m $$

for which we rewrite the budget constraint to

$$\min\{p\_x,p\_y\} u\_1^2 + \min\{p\_z,p\_w\} u\_2 = m$$

and then use that $u\_1=u\_2$ because the problem is perfect complements. You then have two equations in two unknowns $u\_1$ and $u\_2$. Solve for $u\_1$ and $u\_2$ and plug solutions back into expenditure functions for inner problems to get $m\_{u\_1}$ and $m\_{u\_2}$ which you then plug into the equations for the demand.

> **Comments:**
> **username_1**: It is just an indicator function that is $0$ if condition in brackets is not satisfied and $1$ when condition is satisfied. Hence $z = (m/p_z) 1[p_z<p_w]$ is just $z=m/p_z$ when $p_z<p_w$  because $1[p_z<p_w]=1$ and  $z = (m/p_z) 1[p_z<p_w]$ then becomes $0$ when$p_w<p_z$ because $1[p_z<p_w]$ then becomes $0$. So it is just a way to write a fork function and insure that demand is 0 for the good that is the most expensive of the two perfect substitutes.  
> **username_1**: Yes, I was merely using the more abstract notation to illustrate the general procedure.  
> **username_0**: All the $m_{u_i}$ would just be my known wage 4, if I understand your approach correctly? It isn’t a partial ammount since we are dealing with perfect substitutes.  
> **username_0**: Or should it be distributed evenly between the 2 inner problems? I.e 2 each?  
> **username_1**: $m_{u_1}$ is income used on $x$ and $y$, $m_{u_2}$ is income used on $z$ and $w$. With the prices you have consumer will only buy $y$ and $w$. The distribution will most likely not be even split because $\sqrt y = w$ since $u_1=u_2$ due to outer utility being complements. Yes total income is distributed $m = m_{u_1} + m_{u_2}$ but no requirement that $m_{u_1}=m_{u_2}$.  
> **username_0**: Ah so actually, $m_{u_2}=u_2$ since the price is $p_w=1$ (as the minimum of the $2$ prices is $1$ and the wage is $4$ according to $m_{u_2} = \min\{p_z,p_w\} u_2$)  
> **username_0**: I see that we got $2$ unknowns in both inner utility functions but only $1$ equation. Unless I understood it correctly that we let both constrain for the inners be $m_i=4$  
> **username_1**: You have two equations in two unknowns: $\min\{p_x,p_y\} u_1^2  + \min\{p_z,p_w\} u_2 = m$ this reduces to $2 u_1^2  +  u_2 = m$ using values of prices as given. The other equation is that $u_1 = u_2$ because the outer problem is complements $U = \max_{u_1,u_2} \min\{u_1,u_2\}$. So you have $2 u_1^2  +  u_1 = m$ which because you know $m$ you can solve for $u_1$ having found $u_1$ you can find $u_2$ which is the same as $u_1$ using $u_1=u_2$ equation. You have equations for $m_{u_1}$ and $m_{u_2}$ as function of $u_1$,$u_2$ and prices all of which you now know.  
> **username_0**: I am unsure on the approach. Could we move to a discussion chat? i.e in the second inner problem we essentially just end up with $m_{u_2}=m_{u_2}$ since the price is $1$ which does not seem to do anything for me.  
> **username_1**: I could have made a mistake let me just go through it.  
> **username_0**: I used that $u_1=u_2$ and thus get a polynomial of degree $n=2$ (by plugging it into the equation $2u_1^2+u_2=m$. Solving it gives $u_1=-1$ and $u_1=1/2$. Im sorry but $m=1$ and not $m=4$ which are quite a bit simpler***  
> **username_0**: Check my edit for $m$. That might have been it.  
> **username_1**: Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/124435/discussion-between-jesper-hybel-and-user879295).  

---

### Answer by username_1 on 2021-05-19. Score: 4

To provide another answer with less equations:

Consider first that the inner utility function $u\_1 = \sqrt{x+y}$ and $u\_2 = z + w$ are perfect substitutes implying that consumer only buy the cheapest of $x$ and $y$ and similarly only the cheapest of $z$ and $w$. Because prices are given we know that $y$ is cheaper than $x$ and $w$ is cheaper than $z$. Using this we can simply set $x=0$ and $z=0$ and get the utility maximization problem

Maximize
$$U(y,w) = \min\{\sqrt y,w\}$$

which implies that $\sqrt y = w$ for any solution.

Therefore there are two equations in two unknowns

$$p\_y y + p\_w w = m \Rightarrow 2y + w = 1$$

and
$$\sqrt y = w$$

Hence it follows that

$$2y + \sqrt y = 1$$

which is satisfied for $y=1/4$ and since $\sqrt y = w$ it follows that $w = \sqrt(1/4) = 1/2$. Adding that $x=0$ and $z=0$ we have all demands.


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