# Marginal Utility with Supply and Demand Curves

Can someone explain how the equilibrium points in the image below (source: The Foundations of Econometric Analysis) 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. • In the context the author is speaking about a specific demand curve; and modifies the definitions based on this new demand curve. – AnilB Jul 19 '15 at 18:32
• @OccupyGezi I don't understand how PQ, P'Q', P''Q'', etc., represent the marginal utilities corresponding with the lagging supply. – Amaziah Jul 19 '15 at 19:09
• please check my response – AnilB Jul 20 '15 at 8:29

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 $p'<\text{WTP}<p$. 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 $\text{MU}_x=p_x$.

• 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 Jul 18 '15 at 21:08
• @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 Jul 18 '15 at 21:40
• @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 Jul 18 '15 at 23:59

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); and it follows $$\Rightarrow\frac{\partial U}{\partial q_x}=p_x$$

• 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 Jul 21 '15 at 1:00