<ol>
<li>
<blockquote>
<p>If <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are perfect substitutes such that a unit of <span class="math-container">$X$</span> can be replaced by <span class="math-container">$n$</span> units of <span class="math-container">$Y$</span> [...] I know [...] <span class="math-container">$U=b(nx+y)$</span> for this case</p>
</blockquote>
</li>
</ol>
<p>This is a possible representation but not the only one. E.g., <span class="math-container">$\tilde{U}=(nx+y)^2$</span> represents the same preferences.</p>
<ol start="2">
<li>
<blockquote>
<p>how do we arrive at that?</p>
</blockquote>
</li>
</ol>
<p>Perfect substitutes (with a substitution ratio of <span class="math-container">$1:n$</span>) defines a preference relation <span class="math-container">$\succeq$</span> over the baskets of goods. A possible definition of the perfect substitutes preference relation over <span class="math-container">$\mathbb{R}^2$</span> is that the relation fulfills the following two properties:
<br>
i) <span class="math-container">$(x_1,y_1) \sim (x_2,y_2) \text{ iff } nx_1 + y_1 = nx_2 + y_2$</span> <br>
ii) monotonicity</p>
<p>Though ii) is generally implied, it is not strictly necessary - e.g., two bads can also be perfect subtitutes.</p>
<p>As I already pointed it out above, there are many utility functions that <em>represent</em> this preference.</p>
<ol start="3">
<li>
<blockquote>
<p>can we say that any function <span class="math-container">$U$</span> satisfying <span class="math-container">$U(x+1,y) = U(x,y+n)$</span> will describe this relation [...]?</p>
</blockquote>
</li>
</ol>
<p>If you restrict yourself to integer numbers, sure. A possible reformulation of
<span class="math-container">$$(x_1,y_1) \sim (x_2,y_2) \text{ iff } nx_1 + y_1 = nx_2 + y_2$$</span>
is
<span class="math-container">$$(x_1,y_1) \sim (x_2,y_2) \text{ iff } n(x_1-x_2) = y_2-y_1$$</span>
which is indeed fulfilled by the equation above.</p>

CC BY-SA 4.0
2022-02-09T07:40:00.343Z
Restrict to integers, as in, $n$ being an integer or $\text{domain}(U) = \mathbb{Z} \times \mathbb{Z}$?
not tdm's twin
CC BY-SA 4.0
2022-02-09T07:43:45.520Z
Not just $U=(nx+y)^2$ but *probably* every $g(nx+y)$ where $g$ is non-decreasing considering the definition you gave.
not tdm's twin
CC BY-SA 4.0
2022-02-09T07:46:36.277Z
Indeed they are.
Giskard
CC BY-SA 4.0
2022-02-09T07:48:25.290Z
Thanks. And what about the "restrict to integers"?
not tdm's twin
CC BY-SA 4.0
2022-02-09T08:11:37.277Z
I may have misunderstood you there. I thought you meant for $x,y,n$ all to be integers, in which case your definition only covers $U$'s domain if it is $\mathbb{Z}^2$. If $x,y,n$ can have any real value then your definition seems general enough.
Giskard
CC BY-SA 4.0
2022-02-09T08:23:14.803Z
Okay. If that is general enough, then it actually leads to a problem. Because then it *should* be equivalent to the PDE I wrote [in the other question](https://economics.stackexchange.com/questions/50349/marginal-utility-meaning-and-properties). But this recurrence and that PDE are not *generally* equivalent.
not tdm's twin
CC BY-SA 4.0
2022-02-09T08:27:28.077Z
What is a 'PDE'? In your other question only 3. comes close to this, but it only covers the line $(t,nt)$ in the space of good baskets, not the entire space, and as such, it is not at all equivalent to your definition here.
Giskard
CC BY-SA 4.0
2022-02-09T08:56:48.720Z
PDE is partial differential equation. It's a commonly used acronym (in English textbooks). If two elements are perfect substitutes w/ ratio $1:n$, then we can say that $U(x+1,y) = U(x,y+n)$. In the [other thread](https://economics.stackexchange.com/a/50354/33107), you say the same $X$ and $Y$ can be represented by $\frac{\partial U}{\partial x} = n \cdot \frac{\partial U}{\partial y}$. But this recurrence and this PDE are not equivalent in general. 

If two elements are perfect substitutes then they always satisfy the recurrence you said.
not tdm's twin
CC BY-SA 4.0
2022-02-09T08:57:48.897Z
(Cont. of the previous comment) The PDE is equivalent to $g(nx+y) \implies U(x+1,y) = U(x,y+n)$. But the converse, that is, $U(x+1,y) = U(x,y+n) \implies \frac{\partial U}{\partial x} = n \cdot \frac{\partial U}{\partial y}$ probably does not hold in general.
not tdm's twin
CC BY-SA 4.0
2022-02-09T07:23:48.867Z
2022-02-09T07:23:48.867Z
Giskard
<p>If <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are perfect substitutes such that a unit of <span class="math-container">$X$</span> can be replaced by <span class="math-container">$n$</span> units of <span class="math-container">$Y$</span>, how do we get the mathematical equation from it? I know the equation is of the form <span class="math-container">$ax+by$</span> (and <span class="math-container">$U = b(nx+y)$</span> for this case), but how do we arrive at that?</p>
<p>If the utility function is <span class="math-container">$U(x,y)$</span>, can we say that any function <span class="math-container">$U$</span> satisfying <span class="math-container">$U(x+1,y) = U(x,y+n)$</span> will describe this relation between <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>? (Assume that <span class="math-container">$U$</span> is increasing to ensure the non-satiation property.)</p>

CC BY-SA 4.0
2022-02-09T07:24:11.007Z
Is this a `history-of-economic-thought` question, are you asking who first defined perfect substitutes and how?
Giskard
CC BY-SA 4.0
2022-02-09T07:45:03.960Z
@Giskard Not really.
not tdm's twin
<microeconomics><utility><preferences>
Mathematical definition of perfect substitutes
not tdm's twin
# Mathematical definition of perfect substitutes

(Asked by: username_0 on 2022-02-09)

If $X$ and $Y$ are perfect substitutes such that a unit of $X$ can be replaced by $n$ units of $Y$, how do we get the mathematical equation from it? I know the equation is of the form $ax+by$ (and $U = b(nx+y)$ for this case), but how do we arrive at that?

If the utility function is $U(x,y)$, can we say that any function $U$ satisfying $U(x+1,y) = U(x,y+n)$ will describe this relation between $X$ and $Y$? (Assume that $U$ is increasing to ensure the non-satiation property.)

**Tags:** microeconomics, utility, preferences
> **Comments:**
> **username_1**: Is this a `history-of-economic-thought` question, are you asking who first defined perfect substitutes and how?  
> **username_0**: @Giskard Not really.  

---

## 1 answers

### Answer by username_1 on 2022-02-09. Score: 4

1. > If $X$ and $Y$ are perfect substitutes such that a unit of $X$ can be replaced by $n$ units of $Y$ [...] I know [...] $U=b(nx+y)$ for this case

This is a possible representation but not the only one. E.g., $\tilde{U}=(nx+y)^2$ represents the same preferences.

2. > how do we arrive at that?

Perfect substitutes (with a substitution ratio of $1:n$) defines a preference relation $\succeq$ over the baskets of goods. A possible definition of the perfect substitutes preference relation over $\mathbb{R}^2$ is that the relation fulfills the following two properties:
  
i) $(x\_1,y\_1) \sim (x\_2,y\_2) \text{ iff } nx\_1 + y\_1 = nx\_2 + y\_2$   
ii) monotonicity

Though ii) is generally implied, it is not strictly necessary - e.g., two bads can also be perfect subtitutes.

As I already pointed it out above, there are many utility functions that *represent* this preference.

3. > can we say that any function $U$ satisfying $U(x+1,y) = U(x,y+n)$ will describe this relation [...]?

If you restrict yourself to integer numbers, sure. A possible reformulation of
$$(x\_1,y\_1) \sim (x\_2,y\_2) \text{ iff } nx\_1 + y\_1 = nx\_2 + y\_2$$
is
$$(x\_1,y\_1) \sim (x\_2,y\_2) \text{ iff } n(x\_1-x\_2) = y\_2-y\_1$$
which is indeed fulfilled by the equation above.

> **Comments:**
> **username_0**: Restrict to integers, as in, $n$ being an integer or $\text{domain}(U) = \mathbb{Z} \times \mathbb{Z}$?  
> **username_0**: Not just $U=(nx+y)^2$ but *probably* every $g(nx+y)$ where $g$ is non-decreasing considering the definition you gave.  
> **username_1**: Indeed they are.  
> **username_0**: Thanks. And what about the "restrict to integers"?  
> **username_1**: I may have misunderstood you there. I thought you meant for $x,y,n$ all to be integers, in which case your definition only covers $U$'s domain if it is $\mathbb{Z}^2$. If $x,y,n$ can have any real value then your definition seems general enough.  
> **username_0**: Okay. If that is general enough, then it actually leads to a problem. Because then it *should* be equivalent to the PDE I wrote [in the other question](https://economics.stackexchange.com/questions/50349/marginal-utility-meaning-and-properties). But this recurrence and that PDE are not *generally* equivalent.  
> **username_1**: What is a 'PDE'? In your other question only 3. comes close to this, but it only covers the line $(t,nt)$ in the space of good baskets, not the entire space, and as such, it is not at all equivalent to your definition here.  
> **username_0**: PDE is partial differential equation. It's a commonly used acronym (in English textbooks). If two elements are perfect substitutes w/ ratio $1:n$, then we can say that $U(x+1,y) = U(x,y+n)$. In the [other thread](https://economics.stackexchange.com/a/50354/33107), you say the same $X$ and $Y$ can be represented by $\frac{\partial U}{\partial x} = n \cdot \frac{\partial U}{\partial y}$. But this recurrence and this PDE are not equivalent in general.   If two elements are perfect substitutes then they always satisfy the recurrence you said.  
> **username_0**: (Cont. of the previous comment) The PDE is equivalent to $g(nx+y) \implies U(x+1,y) = U(x,y+n)$. But the converse, that is, $U(x+1,y) = U(x,y+n) \implies \frac{\partial U}{\partial x} = n \cdot \frac{\partial U}{\partial y}$ probably does not hold in general.  

---

