<blockquote>
<p>I just started learning economics and the textbook says <span class="math-container">$MU_X/P_X=MU_Y/P_Y$</span> for a buyer with a fixed budget to spend on two goods, X and Y.</p>
</blockquote>
<p>This is correct statement. Proof:</p>
<p>Start with:</p>
<p><span class="math-container">$$U(x,y), st: p_x x + p_y y =I$$</span></p>
<p>Use Lagrangian:</p>
<p><span class="math-container">$$\mathcal{L} = U(x,y) - \lambda (p_x x + p_y y -I)$$</span></p>
<p>Derive FOCs:</p>
<p><span class="math-container">$$MU_x - \lambda p_x = 0$$</span></p>
<p><span class="math-container">$$MU_y - \lambda p_y = 0$$</span></p>
<p><span class="math-container">$$p_x x + p_y y -I=0$$</span></p>
<p>Combination of first two FOC's tells you that in optimum:</p>
<p><span class="math-container">$$MU_x/p_x = MU_y/p_y$$</span></p>
<p>This condition must be satisfied for consumer to consume optimum bundles of <span class="math-container">$x$</span> and <span class="math-container">$y$</span>.</p>
<blockquote>
<p>Let's say goods X and Y both cost \$1. X's utility is 6 and decreases by 2 with each additional purchase, and Y's utility is 5 and decreases by 2 with each additional purchase. The buyer has \$2 to spend. Then it is obviously the optimal choice is one X and one Y. But then MUXPX=6 and MUYPY=5. The rule is wrong.</p>
</blockquote>
<p>Well you can't just make up numbers in science problem like this. For example, in physics we know that distance equals rate times time:</p>
<p><span class="math-container">$$d=rt$$</span></p>
<p>You can't just then say; &quot;but what if distance is 100km rate is 55km/h and time is 100 hours?&quot;. Or to be clear you can ask the question of course, but the answer is simply that numbers you made up are inconsistent with physical relationship between distance and rate and time.</p>
<p>Similarly, in your case you made up a series of numbers that are inconsistent with a rational consumer behavior subject to budget constraint.  That combination of numbers would simply not occur in optimum in this model.</p>

CC BY-SA 4.0
2023-01-10T12:03:09.993Z
If quantities are in discrete units, as the example seems to imply, the numbers *are* consistent with rational consumer behavior subject to the stated budget constraint.
VARulle
CC BY-SA 4.0
2023-01-10T12:32:01.880Z
@VARulle where does the user imply that?
1muflon1
CC BY-SA 4.0
2023-01-10T14:11:25.900Z
With saying "$X$'s utility is $6$ and decreases by $2$ with each additional purchase". And also by saying that it is "obviously optimal" to buy $(1,1)$.
VARulle
CC BY-SA 4.0
2023-01-10T14:16:28.100Z
@VARulle but those are just examples, if I say its obvious that min of x^2 is 0 it does not mean I am saying x is discrete and if  U'(x)=-2x then utility would decrease by -2 by each additional 1 unit increase in x yet x remains continuous. I think you are simply reading to much into OP's question. That is fine, if there are ambiguities different people may interpret the question differently, I do think OP is trying to understand textbook working with continuous variables otherwise the textbook would not proclaim that $MU_x/p_x = MU_y/p_y$. If you think otherwise you can provide discrete A
1muflon1
CC BY-SA 4.0
2023-01-11T01:16:01.377Z
My textbook uses the example of choosing between pairs of jeans and T-shirts. Is it wrong because those are discrete objects?
TFR
CC BY-SA 4.0
2023-01-11T08:54:05.967Z
@1muflon1, but the OP says nothing that would compare to "it is obvious that the min of $x^2$ is $0$". The OP says that if you have $u(2,0)=10$, $u(1,1)=11$, and $u(0,2)=8$, then it is "obvious" that $u(x,2-x)$ is maximized at $(1,1)$. This implies that the OP is only considering discrete units $x\in\{0, 1, 2\}$, because for these values the statement is indeed true and obvious, while for continuous values the problem would quite obviously be underdetermined.
VARulle
CC BY-SA 4.0
2023-01-11T09:25:12.667Z
@TFR for discrete case it won't be always true as it is possible that at that condition optimal y and x won't be whole numbers, although it would be still a point around which the actual solution would be found, but I am then pretty sure your textbook just uses bad examples or pretends the goods are continuous. You seem to be talking about about some undergraduate textbook. I never in my life saw an undergraduate textbook explain optimization with discrete bundles that is usually graduate level topic (i mean in rigorous way showing how to derive it)
1muflon1
CC BY-SA 4.0
2023-01-10T11:03:15.280Z
2023-01-10T11:03:15.280Z
1muflon1
<p>The rule is derived from the condition that in the optimum the budget line is tangent to the indifference curve. This implicitly requires that quantities <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are real numbers that can vary continuously. Your example uses discrete quantities instead, for which the rule doesn't apply.</p>
<p>To embed your discrete values example into a continuous framework you can use <span class="math-container">$u_x=7x-x^2$</span> and <span class="math-container">$u_y=6y-y^2$</span> with additive bundle utility <span class="math-container">$u(x,y)=u_x+u_y$</span>. Then <span class="math-container">$MU_x=7-2x$</span> and <span class="math-container">$MU_y=6-2y$</span>. The optimality condition then gives you <span class="math-container">$x^*=5/4$</span> and <span class="math-container">$y^*=3/4$</span>, with equal marginal utility of <span class="math-container">$9/2$</span>. The rule is correct.</p>

CC BY-SA 4.0
2023-01-11T01:13:51.840Z
My textbook uses the example of choosing between pairs of jeans and T-shirts. Is it wrong because those are discrete objects?
TFR
CC BY-SA 4.0
2023-01-11T09:42:10.997Z
@TFR, if it says that the optimal choice of jeans and T-shirts always equates the ratio of marginal utilities with the ratio of prices then yes, it's wrong.
VARulle
CC BY-SA 4.0
2023-01-10T11:38:37.550Z
2023-01-10T11:38:37.550Z
VARulle
<p>I just started learning economics and the textbook says <span class="math-container">$\frac{MU_X}{P_X}=\frac{MU_Y}{P_Y}$</span> for a buyer with a fixed budget to spend on two goods, <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>.</p>
<p>Let's say goods <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> both cost <span class="math-container">$\\\$1$</span>. <span class="math-container">$X$</span>'s utility is <span class="math-container">$6$</span> and decreases by <span class="math-container">$2$</span> with each additional purchase, and <span class="math-container">$Y$</span>'s utility is <span class="math-container">$5$</span> and decreases by <span class="math-container">$2$</span> with each additional purchase. The buyer has <span class="math-container">$\\\$2$</span> to spend. Then it is obviously the optimal choice is one <span class="math-container">$X$</span> and one <span class="math-container">$Y$</span>. But then <span class="math-container">$\frac{MU_X}{P_X}=6$</span> and <span class="math-container">$\frac{MU_Y}{P_Y}=5$</span>. The rule is wrong.</p>

<microeconomics><optimization><marginal>
Why does $\frac{MU_x}{P_x}=\frac{MU_y}{P_y}$?
TFR
# Why does $\frac{MU_x}{P_x}=\frac{MU_y}{P_y}$?

(Asked by: username_0 on 2023-01-10)

I just started learning economics and the textbook says $\frac{MU\_X}{P\_X}=\frac{MU\_Y}{P\_Y}$ for a buyer with a fixed budget to spend on two goods, $X$ and $Y$.

Let's say goods $X$ and $Y$ both cost $\\\$1$. $X$'s utility is $6$ and decreases by $2$ with each additional purchase, and $Y$'s utility is $5$ and decreases by $2$ with each additional purchase. The buyer has $\\\$2$ to spend. Then it is obviously the optimal choice is one $X$ and one $Y$. But then $\frac{MU\_X}{P\_X}=6$ and $\frac{MU\_Y}{P\_Y}=5$. The rule is wrong.

**Tags:** microeconomics, optimization, marginal


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## 2 answers

### Answer by username_1 on 2023-01-10. Score: 1

> I just started learning economics and the textbook says $MU\_X/P\_X=MU\_Y/P\_Y$ for a buyer with a fixed budget to spend on two goods, X and Y.

This is correct statement. Proof:

Start with:

$$U(x,y), st: p\_x x + p\_y y =I$$

Use Lagrangian:

$$\mathcal{L} = U(x,y) - \lambda (p\_x x + p\_y y -I)$$

Derive FOCs:

$$MU\_x - \lambda p\_x = 0$$

$$MU\_y - \lambda p\_y = 0$$

$$p\_x x + p\_y y -I=0$$

Combination of first two FOC's tells you that in optimum:

$$MU\_x/p\_x = MU\_y/p\_y$$

This condition must be satisfied for consumer to consume optimum bundles of $x$ and $y$.

> Let's say goods X and Y both cost \$1. X's utility is 6 and decreases by 2 with each additional purchase, and Y's utility is 5 and decreases by 2 with each additional purchase. The buyer has \$2 to spend. Then it is obviously the optimal choice is one X and one Y. But then MUXPX=6 and MUYPY=5. The rule is wrong.

Well you can't just make up numbers in science problem like this. For example, in physics we know that distance equals rate times time:

$$d=rt$$

You can't just then say; "but what if distance is 100km rate is 55km/h and time is 100 hours?". Or to be clear you can ask the question of course, but the answer is simply that numbers you made up are inconsistent with physical relationship between distance and rate and time.

Similarly, in your case you made up a series of numbers that are inconsistent with a rational consumer behavior subject to budget constraint. That combination of numbers would simply not occur in optimum in this model.

> **Comments:**
> **username_2**: If quantities are in discrete units, as the example seems to imply, the numbers *are* consistent with rational consumer behavior subject to the stated budget constraint.  
> **username_1**: @username_2 where does the user imply that?  
> **username_2**: With saying "$X$'s utility is $6$ and decreases by $2$ with each additional purchase". And also by saying that it is "obviously optimal" to buy $(1,1)$.  
> **username_1**: @username_2 but those are just examples, if I say its obvious that min of x^2 is 0 it does not mean I am saying x is discrete and if  U'(x)=-2x then utility would decrease by -2 by each additional 1 unit increase in x yet x remains continuous. I think you are simply reading to much into OP's question. That is fine, if there are ambiguities different people may interpret the question differently, I do think OP is trying to understand textbook working with continuous variables otherwise the textbook would not proclaim that $MU_x/p_x = MU_y/p_y$. If you think otherwise you can provide discrete A  
> **username_0**: My textbook uses the example of choosing between pairs of jeans and T-shirts. Is it wrong because those are discrete objects?  
> **username_2**: @username_1, but the OP says nothing that would compare to "it is obvious that the min of $x^2$ is $0$". The OP says that if you have $u(2,0)=10$, $u(1,1)=11$, and $u(0,2)=8$, then it is "obvious" that $u(x,2-x)$ is maximized at $(1,1)$. This implies that the OP is only considering discrete units $x\in\{0, 1, 2\}$, because for these values the statement is indeed true and obvious, while for continuous values the problem would quite obviously be underdetermined.  
> **username_1**: @username_0 for discrete case it won't be always true as it is possible that at that condition optimal y and x won't be whole numbers, although it would be still a point around which the actual solution would be found, but I am then pretty sure your textbook just uses bad examples or pretends the goods are continuous. You seem to be talking about about some undergraduate textbook. I never in my life saw an undergraduate textbook explain optimization with discrete bundles that is usually graduate level topic (i mean in rigorous way showing how to derive it)  

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### Answer by username_2 on 2023-01-10. Score: 0

The rule is derived from the condition that in the optimum the budget line is tangent to the indifference curve. This implicitly requires that quantities $x$ and $y$ are real numbers that can vary continuously. Your example uses discrete quantities instead, for which the rule doesn't apply.

To embed your discrete values example into a continuous framework you can use $u\_x=7x-x^2$ and $u\_y=6y-y^2$ with additive bundle utility $u(x,y)=u\_x+u\_y$. Then $MU\_x=7-2x$ and $MU\_y=6-2y$. The optimality condition then gives you $x^\*=5/4$ and $y^\*=3/4$, with equal marginal utility of $9/2$. The rule is correct.

> **Comments:**
> **username_0**: My textbook uses the example of choosing between pairs of jeans and T-shirts. Is it wrong because those are discrete objects?  
> **username_2**: @username_0, if it says that the optimal choice of jeans and T-shirts always equates the ratio of marginal utilities with the ratio of prices then yes, it's wrong.  

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