<p>Assume that a consumer can spend his money on commodity X or retain it with himself as money. The utility function becomes
$$U=U(q_x,M)$$
where M is for monetary asset. The budget constraint is as below
$$p_xq_x+M=I$$
where I is the income. To maximize its utility below condition must be satisfied
$$dU=\frac{\partial U}{\partial q_x}dq_x+\frac{\partial U}{\partial M}dM=0$$
and budget constraint becomes
$$p_xdq_x+dM=0\rightarrow dM=-p_xdq_x$$
Replacing $dM$ to utility maximizer
$$\frac{\partial U}{\partial q_x}dq_x-\frac{\partial U}{\partial M}p_xdq_x=0$$
$$\bigg(\frac{\partial U}{\partial q_x}-\frac{\partial U}{\partial M}p_x\bigg)dq_x=0$$
$$\Rightarrow\frac{\partial U}{\partial q_x}-\frac{\partial U}{\partial M}p_x=0$$
The term $\frac{\partial U}{\partial M}$ stands for marginal utility of money. Marshall assumed the marginal utility for money as constant and equal to one (cardinal utility approach: <a href="https://books.google.com.tr/books?id=6M7r-gGAV24C&amp;lpg=PA122&amp;ots=Pne8B_DEgK&amp;dq=consumer&#39;s%20equilibrium%20one%20commodity%20model&amp;hl=tr&amp;pg=PA122#v=onepage&amp;q&amp;f=false" rel="nofollow">ref</a>); and it follows
$$\Rightarrow\frac{\partial U}{\partial q_x}=p_x$$</p>

CC BY-SA 3.0
2015-07-21T01:00:59.710Z
Thank you for your response.  I have not seen marginal utility derived without budget lines and indifference curves.  That is where my confusion lies.  How does the concept of marginal utility fit in with supply and demand curves?  Do we think of demand curves as aggregated indifference curves?
Amaziah
CC BY-SA 3.0
2015-07-19T21:37:38.737Z
2015-07-19T21:37:38.737Z
AnilB
<p>Suppose there is only one good and I fix a price, $p$. The demand curve implies a corresponding quantity $q$.</p>

<p>Now, reduce the price by a <em>tiny</em> amount to the new level $p'$. The demand curve tells us that this reduction in price will cause an increase in quantity to $q'$.</p>

<p>Now, let's think about what happens here. Our consumer was willing to buy the some extra units at the price of $p'$, but not willing to buy those units at the price of $p$. We can infer that the consumer's willingness to pay for these units satisfies $p'&lt;\text{WTP}&lt;p$.</p>

<p><img src="https://i.sstatic.net/IqNXM.png" alt="enter image description here"></p>

<p>Remember that I said the change in $p$ was <em>tiny</em>? If this change is almost zero then the number of extra units demanded will also be small and we can get an exact value for the consumer's willingness to pay for these units: $p\approx \text{WTP}\approx p'$.</p>

<p>So the willingness to pay for the <em>marginal unit</em> is just the price. But the price is also the height of the demand curve!$^†$ Note that if I had started with a different price and performed the same exercise then I could have calculated the willingness to pay for a different unit of the good. So the height of the demand curve at a given point measures the consumer's willingness to pay for the unit of the good below that point.</p>

<p>Lastly, how much should the consumer be willing to pay for a unit? The answer is that he should be willing to pay as much utility as that unit gives him (a.k.a. the marginal utility). Thus, the demand curve's height measures willingness to pay, which is just another way of saying it measures marginal utility.</p>

<p>$^†$ Note that the distance $PQ$ in your figure is just the height of the demand curve too.</p>

<hr>

<h1>Appendix</h1>

<p>To see how this works in maths and with an arbitrary number of goods: note that the optimal solution to the consumer choice problem occurs where the marginal rate of substitution between each pair of demanded goods is equal to their relative price
$$\frac{\text{MU}_x}{\text{MU}_y}=\frac{p_x}{p_y}.$$
Rearranging:
$$\frac{\text{MU}_x}{p_x}=\frac{\text{MU}_y}{p_y}=\text{some number}.$$
Rearranging
$$\frac{1}{\text{some number}}\text{MU}_x=p_x$$
so the price is just some constant multiplied by the marginal utility. By appropriate normalisation of either the units in which money is measured or the utility function (which is invariant to monotone transformations) we can make $\text{some number}=1$ so that $\text{MU}_x=p_x$.</p>

CC BY-SA 3.0
2015-07-18T21:08:10.900Z
While I understand your answer I do not see how it explains the figure displayed in the question. What makes $AB$ a demand curve and what on Earth is $D'''$?
Giskard
CC BY-SA 3.0
2015-07-18T21:40:25.940Z
@densep My interpretation are that $D$, $D'$, $D''$ are demand curves that are progressively shifted. Although, on closer inspection, I see that axes are labells $X$ and $Y$, so my interpretation of the figure may be wrong.
Ubiquitous
CC BY-SA 3.0
2015-07-18T23:59:56.770Z
@denesp The author argues that the "new type" of demand curve is just a supply curve.  The figure shows a progressively shifting demand curve with a lagging supply.
Amaziah
CC BY-SA 3.0
2015-07-18T08:08:25.517Z
2015-07-18T08:46:58.467Z
2015-07-18T08:46:58.467Z
Ubiquitous
<p>Can someone explain how the equilibrium points in the image below (source: <a href="https://books.google.hu/books?id=kS5c5LVNPPMC&amp;pg=PA210&amp;lpg=PA210&amp;dq=%22etc.,%20represent%20the%20conditions%20of%20increasing%20demand%22&amp;source=bl&amp;ots=p7oLrCiG0C&amp;sig=Ya-z-MBctsmhZ4bWlFwAqPOdbGs&amp;hl=en&amp;sa=X&amp;ved=0CB8Q6AEwAGoVChMI7ry2_ZfkxgIVRO0UCh0jiADZ#v=onepage&amp;q=%22etc.%2C%20represent%20the%20conditions%20of%20increasing%20demand%22&amp;f=false" rel="nofollow noreferrer">The Foundations of Econometric Analysis</a>) represent marginal utilities.  I have read about the relationship between the marginal rate of substitution and the ratio of marginal utilities in terms of the theory of consumer behavior but have not seen marginal utility in this context.  I imagine this use is similar but my logic is failing me.</p>

<p>The figure represents a demand curve progressively shifting to the right with a lagging supply.</p>

<p><img src="https://i.sstatic.net/GE328.png" alt="enter image description here"></p>

CC BY-SA 3.0
2015-07-19T19:09:39.010Z
@OccupyGezi I don't understand how PQ, P'Q', P''Q'', etc., represent the marginal utilities corresponding with the lagging supply.
Amaziah
CC BY-SA 3.0
2015-07-20T08:29:02.113Z
please check my response
AnilB
CC BY-SA 3.0
2015-07-19T18:32:41.070Z
In the context the author is speaking about a specific demand curve; and modifies the definitions based on this new demand curve.
AnilB
<microeconomics><utility><supply-and-demand><self-study>
Marginal Utility with Supply and Demand Curves
Amaziah
# Marginal Utility with Supply and Demand Curves

(Asked by: username_0 on 2015-07-18)

Can someone explain how the equilibrium points in the image below (source: [The Foundations of Econometric Analysis](https://books.google.hu/books?id=kS5c5LVNPPMC&pg=PA210&lpg=PA210&dq=%22etc.,%20represent%20the%20conditions%20of%20increasing%20demand%22&source=bl&ots=p7oLrCiG0C&sig=Ya-z-MBctsmhZ4bWlFwAqPOdbGs&hl=en&sa=X&ved=0CB8Q6AEwAGoVChMI7ry2_ZfkxgIVRO0UCh0jiADZ#v=onepage&q=%22etc.%2C%20represent%20the%20conditions%20of%20increasing%20demand%22&f=false)) represent marginal utilities. I have read about the relationship between the marginal rate of substitution and the ratio of marginal utilities in terms of the theory of consumer behavior but have not seen marginal utility in this context. I imagine this use is similar but my logic is failing me.

The figure represents a demand curve progressively shifting to the right with a lagging supply.

![enter image description here](https://i.sstatic.net/GE328.png)

**Tags:** microeconomics, utility, supply-and-demand, self-study
> **Comments:**
> **username_1**: In the context the author is speaking about a specific demand curve; and modifies the definitions based on this new demand curve.  
> **username_0**: @OccupyGezi I don't understand how PQ, P'Q', P''Q'', etc., represent the marginal utilities corresponding with the lagging supply.  
> **username_1**: please check my response  

---

## 2 answers

### Answer by username_1 on 2015-07-19. Score: 3

Assume that a consumer can spend his money on commodity X or retain it with himself as money. The utility function becomes
$$U=U(q\_x,M)$$
where M is for monetary asset. The budget constraint is as below
$$p\_xq\_x+M=I$$
where I is the income. To maximize its utility below condition must be satisfied
$$dU=\frac{\partial U}{\partial q\_x}dq\_x+\frac{\partial U}{\partial M}dM=0$$
and budget constraint becomes
$$p\_xdq\_x+dM=0\rightarrow dM=-p\_xdq\_x$$
Replacing $dM$ to utility maximizer
$$\frac{\partial U}{\partial q\_x}dq\_x-\frac{\partial U}{\partial M}p\_xdq\_x=0$$
$$\bigg(\frac{\partial U}{\partial q\_x}-\frac{\partial U}{\partial M}p\_x\bigg)dq\_x=0$$
$$\Rightarrow\frac{\partial U}{\partial q\_x}-\frac{\partial U}{\partial M}p\_x=0$$
The term $\frac{\partial U}{\partial M}$ stands for marginal utility of money. Marshall assumed the marginal utility for money as constant and equal to one (cardinal utility approach: [ref](https://books.google.com.tr/books?id=6M7r-gGAV24C&lpg=PA122&ots=Pne8B_DEgK&dq=consumer's%20equilibrium%20one%20commodity%20model&hl=tr&pg=PA122#v=onepage&q&f=false)); and it follows
$$\Rightarrow\frac{\partial U}{\partial q\_x}=p\_x$$

> **Comments:**
> **username_0**: Thank you for your response.  I have not seen marginal utility derived without budget lines and indifference curves.  That is where my confusion lies.  How does the concept of marginal utility fit in with supply and demand curves?  Do we think of demand curves as aggregated indifference curves?  

---

### Answer by username_2 on 2015-07-18. Score: 3

Suppose there is only one good and I fix a price, $p$. The demand curve implies a corresponding quantity $q$.

Now, reduce the price by a *tiny* amount to the new level $p'$. The demand curve tells us that this reduction in price will cause an increase in quantity to $q'$.

Now, let's think about what happens here. Our consumer was willing to buy the some extra units at the price of $p'$, but not willing to buy those units at the price of $p$. We can infer that the consumer's willingness to pay for these units satisfies $p'<\text{WTP}

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

Remember that I said the change in $p$ was *tiny*? If this change is almost zero then the number of extra units demanded will also be small and we can get an exact value for the consumer's willingness to pay for these units: $p\approx \text{WTP}\approx p'$.

So the willingness to pay for the *marginal unit* is just the price. But the price is also the height of the demand curve!$^†$ Note that if I had started with a different price and performed the same exercise then I could have calculated the willingness to pay for a different unit of the good. So the height of the demand curve at a given point measures the consumer's willingness to pay for the unit of the good below that point.

Lastly, how much should the consumer be willing to pay for a unit? The answer is that he should be willing to pay as much utility as that unit gives him (a.k.a. the marginal utility). Thus, the demand curve's height measures willingness to pay, which is just another way of saying it measures marginal utility.

$^†$ Note that the distance $PQ$ in your figure is just the height of the demand curve too.

---

# Appendix

To see how this works in maths and with an arbitrary number of goods: note that the optimal solution to the consumer choice problem occurs where the marginal rate of substitution between each pair of demanded goods is equal to their relative price
$$\frac{\text{MU}\_x}{\text{MU}\_y}=\frac{p\_x}{p\_y}.$$
Rearranging:
$$\frac{\text{MU}\_x}{p\_x}=\frac{\text{MU}\_y}{p\_y}=\text{some number}.$$
Rearranging
$$\frac{1}{\text{some number}}\text{MU}\_x=p\_x$$
so the price is just some constant multiplied by the marginal utility. By appropriate normalisation of either the units in which money is measured or the utility function (which is invariant to monotone transformations) we can make $\text{some number}=1$ so that $\text{MU}\_x=p\_x$.

> **Comments:**
> **username_3**: While I understand your answer I do not see how it explains the figure displayed in the question. What makes $AB$ a demand curve and what on Earth is $D'''$?  
> **username_2**: @densep My interpretation are that $D$, $D'$, $D''$ are demand curves that are progressively shifted. Although, on closer inspection, I see that axes are labells $X$ and $Y$, so my interpretation of the figure may be wrong.  
> **username_0**: @denesp The author argues that the "new type" of demand curve is just a supply curve.  The figure shows a progressively shifting demand curve with a lagging supply.  

---

