<p>To provide some real world(ish) interpretation, you could consider the following:</p>

<p>Wallace enjoys eating cheese on its own. He doesn't much care for crackers on their own, but he especially loves eating crackers and cheese together, he makes nice little cracker n cheese sandwiches.</p>

<p>In this example, we can think of cheese (x) and crackers (y) as perfect complements, but cheese will be a perfect substitute for a cracker and cheese sandwich.</p>

<p>If a unit of cheese is cheaper than a unit of crackers he'll skip the sandwiches and spend his whole budget on cheese, but if crackers are cheaper he'll buy cheese and crackers in equal proportions.</p>

CC BY-SA 4.0
2019-10-21T01:59:15.777Z
2019-10-21T01:59:15.777Z
H Rogers
<p>I assume you know how does <span class="math-container">$\min\{x,y\}$</span> look like? In order to draw utility function of interest, you need to consider cases: <span class="math-container">$u(x,y)=x+\min\{x,y\}=\begin{cases}2x, \;\; \mathrm{for} \;\; x \leq y \\ x+y, \;\; \mathrm{for} \;\; x &gt; y\end{cases}$</span> </p>

<p>With <span class="math-container">$x$</span> on horizontal and <span class="math-container">$y$</span> on vertical axis:</p>

<p><a href="https://i.sstatic.net/i2S4a.gif" rel="noreferrer"><img src="https://i.sstatic.net/i2S4a.gif" alt="enter image description here"></a></p>

<p>Not sure about the "usual" perfect complements. It is more like a combination of substitutes (below <span class="math-container">$y=x$</span>) and complements (above <span class="math-container">$y=x$</span>).</p>

<p>Also, take a look here: <a href="https://economics.stackexchange.com/questions/32140/identifying-utility-function/32165#32165">Identifying utility function</a> and <a href="https://economics.stackexchange.com/questions/32309/algebraic-approach-towards-convexity/32311#32311">Algebraic approach towards convexity</a> where you can see more graphs. </p>

<p><strong>EDIT.</strong></p>

<p>As for the demand function, <a href="https://economics.stackexchange.com/questions/2969/finding-demand-function-given-a-utility-minx-y-function">Finding demand function given a utility min(x,y) function</a>.</p>

CC BY-SA 4.0
2019-10-21T00:28:29.603Z
Lovely answer. What tool or tools did you use to make this figure?
BKay
CC BY-SA 4.0
2019-10-21T01:12:46.650Z
Thank you for the answer! Will the optimal bundle always lie at the kink?
user708015
CC BY-SA 4.0
2019-10-20T22:51:14.007Z
2019-10-20T23:59:33.597Z
2019-10-20T23:59:33.597Z
bajun65537
<p>I have a utility function: <span class="math-container">$U = x + \min\{x,y\}$</span></p>

<p>I want to draw the indifference curve and find the demand functions. Will it be the case of the usual perfect complements? </p>

<p>Also, what preferences could such a utility function represent?</p>

<p>Will the optimal solution be at the kink? </p>

<microeconomics><utility><consumer-theory><demand>
Finding demand functions for an unusual utility function
user708015
# Finding demand functions for an unusual utility function

(Asked by: username_0 on 2019-10-20)

I have a utility function: $U = x + \min\{x,y\}$

I want to draw the indifference curve and find the demand functions. Will it be the case of the usual perfect complements?

Also, what preferences could such a utility function represent?

Will the optimal solution be at the kink?

**Tags:** microeconomics, utility, consumer-theory, demand

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## 2 answers

### Answer by username_2 on 2019-10-20. Score: 6 | Accepted answer

I assume you know how does $\min\{x,y\}$ look like? In order to draw utility function of interest, you need to consider cases: $u(x,y)=x+\min\{x,y\}=\begin{cases}2x, \;\; \mathrm{for} \;\; x \leq y \\ x+y, \;\; \mathrm{for} \;\; x > y\end{cases}$

With $x$ on horizontal and $y$ on vertical axis:

[![enter image description here](https://i.sstatic.net/i2S4a.gif)](https://i.sstatic.net/i2S4a.gif)

Not sure about the "usual" perfect complements. It is more like a combination of substitutes (below $y=x$) and complements (above $y=x$).

Also, take a look here: [Identifying utility function](https://economics.stackexchange.com/questions/32140/identifying-utility-function/32165#32165) and [Algebraic approach towards convexity](https://economics.stackexchange.com/questions/32309/algebraic-approach-towards-convexity/32311#32311) where you can see more graphs.

**EDIT.**

As for the demand function, [Finding demand function given a utility min(x,y) function](https://economics.stackexchange.com/questions/2969/finding-demand-function-given-a-utility-minx-y-function).

> **Comments:**
> **username_3**: Lovely answer. What tool or tools did you use to make this figure?  
> **username_0**: Thank you for the answer! Will the optimal bundle always lie at the kink?  

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### Answer by username_1 on 2019-10-21. Score: 4

To provide some real world(ish) interpretation, you could consider the following:

Wallace enjoys eating cheese on its own. He doesn't much care for crackers on their own, but he especially loves eating crackers and cheese together, he makes nice little cracker n cheese sandwiches.

In this example, we can think of cheese (x) and crackers (y) as perfect complements, but cheese will be a perfect substitute for a cracker and cheese sandwich.

If a unit of cheese is cheaper than a unit of crackers he'll skip the sandwiches and spend his whole budget on cheese, but if crackers are cheaper he'll buy cheese and crackers in equal proportions.


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