<p>I will refer to perfect substitutes with ratio <span class="math-container">$1:n$</span> by <span class="math-container">$U_{PS,n}$</span>.</p>
<hr />
<blockquote>
<ol>
<li><span class="math-container">$\text{MU}_X(1) = \text{MU}_Y(n)$</span>,</li>
</ol>
</blockquote>
<p><span class="math-container">$U_{PS,n}$</span> has this property, but other preferences do as well, e.g., the quasi-linear preference
<span class="math-container">$$
U(x,y) = \ln x + y,
$$</span>
thus this is not equivalent.</p>
<hr />
<blockquote>
<ol start="2">
<li>or <span class="math-container">$\text{MU}_X(t) = n \cdot \text{MU}_Y(t)$</span> for <span class="math-container">$t \in X \cap Y$</span>?</li>
</ol>
</blockquote>
<p><span class="math-container">$U_{PS,n}$</span> has this property because we know the indifference curves have slope (times <span class="math-container">$-1$</span>)</p>
<p><span class="math-container">$$
\frac{\text{MU}_X(x,y)}{\text{MU}_Y(x,y)} = n
$$</span></p>
<p>for every <span class="math-container">$(x,y)$</span>. But there are other preferences that satisfy 2., e.g.,</p>
<p><span class="math-container">$$
U(x,y) = nx^2 + y^2,
$$</span></p>
<p>thus this is not equivalent.</p>
<hr />
<blockquote>
<ol start="3">
<li>or <span class="math-container">$\text{MU}_X(t) = \text{MU}_Y(nt)$</span> for <span class="math-container">$(t,nt) \in X \times Y$</span>?</li>
</ol>
</blockquote>
<p><span class="math-container">$U_{PS,n}$</span> has this property, but other preferences do as well, e.g., the Cobb-Douglas preference
<span class="math-container">$$
U(x,y) = \ln x + n\ln y,
$$</span>
thus is not equivalent.</p>
<hr />
<p>Note:</p>
<ol start="2">
<li>Could be modified by demanding <span class="math-container">$\text{MU}_X(x,y) = n \cdot \text{MU}_Y(x,y)$</span> for all <span class="math-container">$(x,y) \in \mathbb{R}^2$</span>. This would cover the entire space, rather than just the 45 degree line that <span class="math-container">$(t,t)$</span> did.</li>
</ol>

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2022-02-09T08:46:48.187Z
Thank you. This is a satisfactory answer.
not tdm's twin
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2022-02-09T08:49:33.710Z
See my new note at the end!
Giskard
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2022-02-09T08:51:42.143Z
My bad, I meant for all (and not fixed) $t$. So that's the same thing. Edited the question.
not tdm's twin
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2022-02-09T08:52:20.853Z
By $t \in X \cap Y$ you did not mean $(t,t)$? Something to consider when abusing math notation in the future.
Giskard
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2022-02-09T08:53:28.760Z
Never mind. It's still different (despite the edit). The one you gave considers all points. Mine still does not.
not tdm's twin
CC BY-SA 4.0
2022-02-09T08:21:58.983Z
2022-02-09T08:47:06.923Z
2022-02-09T08:47:06.923Z
Giskard
<p>Consider goods <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> such that the marginal utility of a unit of good <span class="math-container">$X$</span> is always that of <span class="math-container">$n$</span> units of good <span class="math-container">$Y$</span>. <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are perfect substitutes.</p>
<p><strong>Question 1:</strong> What does the above mean mathematically:</p>
<ol>
<li><span class="math-container">$\text{MU}_X(1) = \text{MU}_Y(n)$</span>,</li>
<li>or <span class="math-container">$\text{MU}_X(t) = n \cdot \text{MU}_Y(t)$</span> for all <span class="math-container">$t \in X \cap Y$</span>?</li>
<li>or <span class="math-container">$\text{MU}_X(t) = \text{MU}_Y(nt)$</span> for all <span class="math-container">$(t,nt) \in X \times Y$</span>?</li>
</ol>
<p>To me, (2) seems unlikely, and (3) is the generalization of (1) which seems the most likely.</p>
<p><strong>Question 2:</strong> Is the marginal utility of <span class="math-container">$n$</span> units of good <span class="math-container">$Y$</span> same as <span class="math-container">$n$</span> times the marginal utility of <span class="math-container">$1$</span> unit of good <span class="math-container">$Y$</span> keeping the quantity of <span class="math-container">$X$</span> fixed?</p>
<p>This is generally not true, for example, if we take <span class="math-container">$U(x,y) = y^3$</span>. But this utility function does not probably satisfy the given condition, hence this question. In fact, if this is not true in general, it would tell that (2) and (3) in Question 1 are not equivalent.</p>
<p>I didn't find a mathematical formula that directly translates the informal definition of the utility function, that is, the added satisfaction from an additional unit of good. The doubts arise from this.</p>

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2022-02-08T23:06:06.437Z
Without *context* or a *source*, it is not possible to divine what a poorly phrased sentence means. (My guess is 3.)
Giskard
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2022-02-08T23:14:26.293Z
@Giskard That was the sentence which I found sort of confusing as well. Can we write that using recurrence relation? Like, $U(x+1,y) = U(x,y+n)$ (which would probably mean (3) in question 1 to be equivalent to this recurrence.. which they're probably not)? If this recurrence is not true, can you tell a better recurrence to model this?
not tdm's twin
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2022-02-08T23:35:16.637Z
@Giskard I am trying to derive the utility function of perfect substitutes. I have edited the first line in the post and added this info as well.
not tdm's twin
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2022-02-09T00:26:56.217Z
What do $X\cap Y$ and $X\times Y$ mean?  $X$ and $Y$ are goods, not sets.
WillO
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2022-02-09T04:23:58.063Z
@WillO It's an abuse of notation. $x \in X$ denotes the quantity of good $X$ and similarly define $Y$. (What else could they possibly mean?)
not tdm's twin
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2022-02-09T04:31:52.950Z
Then what does $t\in X \cap Y$ mean?
WillO
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2022-02-09T05:03:48.940Z
@WillO If $t \in X \cap Y$, then $t \in X$ and $t \in Y$ which means we are considering $t$ units of good $X$ (and/or $Y$). Here, $X$ and $Y$ are both reals, so it does not matter. But it would, if they were different (possibly, finite) sets.
not tdm's twin
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2022-02-09T07:02:21.397Z
Does this answer your question? [Mathematical definition of perfect substitutes](https://economics.stackexchange.com/questions/50352/mathematical-definition-of-perfect-substitutes)
Giskard
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2022-02-09T07:03:47.857Z
I know it does not, that is just the autogenerated-text for duplicate. You essentially posted the same question, somewhat better phrased. That is fine, but then delete the old one; there is no answer yet.
Giskard
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2022-02-09T08:04:51.077Z
@Giskard I am afraid it does not. I am sort of directly trying to translate the definition to a mathematical form here. You can think of this question as which relation (among 1,2 and 3) would two perfect substitutes satisfy (over the reals)? So the main point of this question is the relation itself. (I'm waiting for someone to answer this. It would clear a doubt that's been bothering me for sometime now.)
not tdm's twin
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2022-02-09T08:12:39.803Z
@Gang'sBigBoss You are not trying to translate an exact definition of perfect substitutes, or if you are, you do not refer to it. You are trying to translate a somewhat vague sentence.
Giskard
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2022-02-09T08:15:32.513Z
@Giskard True (and sorry, it's not definition but a statement.. I miswrote that part), so you can consider this instead: $X$ and $Y$ are perfect substitutes with ratio $1:n$. Now which of 1,2 and 3 would be equivalent to this definition?
not tdm's twin
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2022-02-09T08:16:26.787Z
If that is your question, than I think [I have answered it here](https://economics.stackexchange.com/a/50353/1601). I will write a short 'applied' answer here.
Giskard
CC BY-SA 4.0
2022-02-09T08:19:40.377Z
@Giskard Yes, but this is using partial derivatives (which I should have mentioned). So if two are perfect substitutes, would it satisfy (all the time): $\frac{\partial U}{\partial x} = n \cdot \frac{\partial U}{\partial y}$ (which is (2) in Q1) or $\frac{\partial U}{\partial x} \bigg|_{x=t} = n \cdot \frac{\partial U}{\partial y} \bigg|_{x = 3t}$ (which is (3) in Q1)?
not tdm's twin
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2022-02-09T08:24:32.550Z
You were right, this question was somewhat different.
Giskard
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2022-02-09T12:30:29.467Z
If $X$ is eggs (measured in cartons) and $Y$ is wine (measured in bottles) and $t\in X\cap Y$ be both a number of cartons and a number of bottles at the same time?
WillO
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2022-02-09T13:18:41.910Z
@WillO No; we are only considering the quantity here. So $t$ does not have a unit attached to it. It's just the number. So in your example, it would simply denote the "number" of cartons and/or the "number" of bottles.
not tdm's twin
CC BY-SA 4.0
2022-02-09T13:32:16.537Z
So if you suddenly decide to measure wine in cases instead of bottles, condition 2 can switch from true to false.  Nothing that depends on your arbitrary choice of units can be economically meaningful.
WillO
<utility><consumer-theory><preferences><choice-theory>
Marginal utility meaning and properties
not tdm's twin
# Marginal utility meaning and properties

(Asked by: username_0 on 2022-02-08)

Consider goods $X$ and $Y$ such that the marginal utility of a unit of good $X$ is always that of $n$ units of good $Y$. $X$ and $Y$ are perfect substitutes.

**Question 1:** What does the above mean mathematically:

1. $\text{MU}\_X(1) = \text{MU}\_Y(n)$,
2. or $\text{MU}\_X(t) = n \cdot \text{MU}\_Y(t)$ for all $t \in X \cap Y$?
3. or $\text{MU}\_X(t) = \text{MU}\_Y(nt)$ for all $(t,nt) \in X \times Y$?

To me, (2) seems unlikely, and (3) is the generalization of (1) which seems the most likely.

**Question 2:** Is the marginal utility of $n$ units of good $Y$ same as $n$ times the marginal utility of $1$ unit of good $Y$ keeping the quantity of $X$ fixed?

This is generally not true, for example, if we take $U(x,y) = y^3$. But this utility function does not probably satisfy the given condition, hence this question. In fact, if this is not true in general, it would tell that (2) and (3) in Question 1 are not equivalent.

I didn't find a mathematical formula that directly translates the informal definition of the utility function, that is, the added satisfaction from an additional unit of good. The doubts arise from this.

**Tags:** utility, consumer-theory, preferences, choice-theory
> **Comments:**
> **username_1**: Without *context* or a *source*, it is not possible to divine what a poorly phrased sentence means. (My guess is 3.)  
> **username_0**: @Giskard That was the sentence which I found sort of confusing as well. Can we write that using recurrence relation? Like, $U(x+1,y) = U(x,y+n)$ (which would probably mean (3) in question 1 to be equivalent to this recurrence.. which they're probably not)? If this recurrence is not true, can you tell a better recurrence to model this?  
> **username_0**: @Giskard I am trying to derive the utility function of perfect substitutes. I have edited the first line in the post and added this info as well.  
> **username_2**: What do $X\cap Y$ and $X\times Y$ mean?  $X$ and $Y$ are goods, not sets.  
> **username_0**: @WillO It's an abuse of notation. $x \in X$ denotes the quantity of good $X$ and similarly define $Y$. (What else could they possibly mean?)  
> **username_2**: Then what does $t\in X \cap Y$ mean?  
> **username_0**: @WillO If $t \in X \cap Y$, then $t \in X$ and $t \in Y$ which means we are considering $t$ units of good $X$ (and/or $Y$). Here, $X$ and $Y$ are both reals, so it does not matter. But it would, if they were different (possibly, finite) sets.  
> **username_1**: Does this answer your question? [Mathematical definition of perfect substitutes](https://economics.stackexchange.com/questions/50352/mathematical-definition-of-perfect-substitutes)  
> **username_1**: I know it does not, that is just the autogenerated-text for duplicate. You essentially posted the same question, somewhat better phrased. That is fine, but then delete the old one; there is no answer yet.  
> **username_0**: @Giskard I am afraid it does not. I am sort of directly trying to translate the definition to a mathematical form here. You can think of this question as which relation (among 1,2 and 3) would two perfect substitutes satisfy (over the reals)? So the main point of this question is the relation itself. (I'm waiting for someone to answer this. It would clear a doubt that's been bothering me for sometime now.)  
> **username_1**: @Gang'sBigBoss You are not trying to translate an exact definition of perfect substitutes, or if you are, you do not refer to it. You are trying to translate a somewhat vague sentence.  
> **username_0**: @Giskard True (and sorry, it's not definition but a statement.. I miswrote that part), so you can consider this instead: $X$ and $Y$ are perfect substitutes with ratio $1:n$. Now which of 1,2 and 3 would be equivalent to this definition?  
> **username_1**: If that is your question, than I think [I have answered it here](https://economics.stackexchange.com/a/50353/1601). I will write a short 'applied' answer here.  
> **username_0**: @Giskard Yes, but this is using partial derivatives (which I should have mentioned). So if two are perfect substitutes, would it satisfy (all the time): $\frac{\partial U}{\partial x} = n \cdot \frac{\partial U}{\partial y}$ (which is (2) in Q1) or $\frac{\partial U}{\partial x} \bigg|_{x=t} = n \cdot \frac{\partial U}{\partial y} \bigg|_{x = 3t}$ (which is (3) in Q1)?  
> **username_1**: You were right, this question was somewhat different.  
> **username_2**: If $X$ is eggs (measured in cartons) and $Y$ is wine (measured in bottles) and $t\in X\cap Y$ be both a number of cartons and a number of bottles at the same time?  
> **username_0**: @WillO No; we are only considering the quantity here. So $t$ does not have a unit attached to it. It's just the number. So in your example, it would simply denote the "number" of cartons and/or the "number" of bottles.  
> **username_2**: So if you suddenly decide to measure wine in cases instead of bottles, condition 2 can switch from true to false.  Nothing that depends on your arbitrary choice of units can be economically meaningful.  

---

## 1 answers

### Answer by username_1 on 2022-02-09. Score: 1 | Accepted answer

I will refer to perfect substitutes with ratio $1:n$ by $U\_{PS,n}$.

---

> 1. $\text{MU}\_X(1) = \text{MU}\_Y(n)$,

$U\_{PS,n}$ has this property, but other preferences do as well, e.g., the quasi-linear preference
$$
U(x,y) = \ln x + y,
$$
thus this is not equivalent.

---

> 2. or $\text{MU}\_X(t) = n \cdot \text{MU}\_Y(t)$ for $t \in X \cap Y$?

$U\_{PS,n}$ has this property because we know the indifference curves have slope (times $-1$)

$$
\frac{\text{MU}\_X(x,y)}{\text{MU}\_Y(x,y)} = n
$$

for every $(x,y)$. But there are other preferences that satisfy 2., e.g.,

$$
U(x,y) = nx^2 + y^2,
$$

thus this is not equivalent.

---

> 3. or $\text{MU}\_X(t) = \text{MU}\_Y(nt)$ for $(t,nt) \in X \times Y$?

$U\_{PS,n}$ has this property, but other preferences do as well, e.g., the Cobb-Douglas preference
$$
U(x,y) = \ln x + n\ln y,
$$
thus is not equivalent.

---

Note:

2. Could be modified by demanding $\text{MU}\_X(x,y) = n \cdot \text{MU}\_Y(x,y)$ for all $(x,y) \in \mathbb{R}^2$. This would cover the entire space, rather than just the 45 degree line that $(t,t)$ did.

> **Comments:**
> **username_0**: Thank you. This is a satisfactory answer.  
> **username_1**: See my new note at the end!  
> **username_0**: My bad, I meant for all (and not fixed) $t$. So that's the same thing. Edited the question.  
> **username_1**: By $t \in X \cap Y$ you did not mean $(t,t)$? Something to consider when abusing math notation in the future.  
> **username_0**: Never mind. It's still different (despite the edit). The one you gave considers all points. Mine still does not.  

---

