<p>Forget the historical &quot;utility theory of value&quot;. In modern microeconomics' consumer theory,  utility is ordinal and therefore marginal utility of a good <em>per se</em> is meaningless. The only meaningful terms involving marginal utility are <em>ratios</em> of marginal utilities of different goods in a consumption bundle, since these are unaffected by positive transformations of the utility function. In the consumption optimum, the ratio of marginal utilities equals the ratio of prices of these goods, and also the marginal rate of substitution between the goods.</p>
<p>However, in these models prices are exogenous, so marginal utilities never &quot;decide&quot; prices in any way. It's rather the other way round: Prices (i.e., price ratios) &quot;decide&quot; the ratios of marginal utilities.</p>

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2024-07-31T15:42:58.523Z
2024-07-31T15:42:58.523Z
VARulle
<p>It is said that marginal utility &quot;decides&quot; the price, one of the ideas of the utility theory of value. However, the only proof I've seen is like this (in my elementary microeconomics course):</p>
<p>Suppose there are multiple commodities <span class="math-container">$\mathbb{R}^n$</span> and a preference <span class="math-container">$\geq$</span> defined on it. Suppose <span class="math-container">$\geq$</span> is rational and it induces a utility function <span class="math-container">$U(\mathbf{X})$</span>. On the budget line <span class="math-container">$\mathbf{P}\cdot\mathbf{X}=I$</span>, by the Lagrangian multiplier, we obtain the result:
<span class="math-container">$$\frac{M_iU}{P_i}=\lambda$$</span>
where <span class="math-container">$M_iU=\partial U/\partial X_i$</span> is the marginal utility. <span class="math-container">$\lambda$</span> is called the marginal utility of money.</p>
<p>Now, somehow we managed to make <span class="math-container">$\lambda$</span> a constant. Then, since the induced utility function can be chosen arbitrarily as long as the preference doesn't change, we set <span class="math-container">$\lambda=1$</span>. Then the result is:
<span class="math-container">$$MU=P$$</span>
and therefore marginal utility decides the price.</p>
<p>Now the problem is, This proof does not mathematically show that <span class="math-container">$\lambda$</span> can be set to <span class="math-container">$1$</span>. In fact, as I did some searching, this heavily relies on the assumption that the preference is quasilinear. If it isn't, then we really can't say that the marginal utility decides the price, but can only say that &quot;the price decides the point of the equilibrium where the proportion of the maginal utility is the proportion of the price&quot;.</p>

<utility><price><value><marginal>
Relationship between marginal utility and price?
ZhenRanZR
# Relationship between marginal utility and price?

(Asked by: username_0 on 2024-07-31)

It is said that marginal utility "decides" the price, one of the ideas of the utility theory of value. However, the only proof I've seen is like this (in my elementary microeconomics course):

Suppose there are multiple commodities $\mathbb{R}^n$ and a preference $\geq$ defined on it. Suppose $\geq$ is rational and it induces a utility function $U(\mathbf{X})$. On the budget line $\mathbf{P}\cdot\mathbf{X}=I$, by the Lagrangian multiplier, we obtain the result:
$$\frac{M\_iU}{P\_i}=\lambda$$
where $M\_iU=\partial U/\partial X\_i$ is the marginal utility. $\lambda$ is called the marginal utility of money.

Now, somehow we managed to make $\lambda$ a constant. Then, since the induced utility function can be chosen arbitrarily as long as the preference doesn't change, we set $\lambda=1$. Then the result is:
$$MU=P$$
and therefore marginal utility decides the price.

Now the problem is, This proof does not mathematically show that $\lambda$ can be set to $1$. In fact, as I did some searching, this heavily relies on the assumption that the preference is quasilinear. If it isn't, then we really can't say that the marginal utility decides the price, but can only say that "the price decides the point of the equilibrium where the proportion of the maginal utility is the proportion of the price".

**Tags:** utility, price, value, marginal

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## 1 answers

### Answer by username_1 on 2024-07-31. Score: 1 | Accepted answer

Forget the historical "utility theory of value". In modern microeconomics' consumer theory, utility is ordinal and therefore marginal utility of a good *per se* is meaningless. The only meaningful terms involving marginal utility are *ratios* of marginal utilities of different goods in a consumption bundle, since these are unaffected by positive transformations of the utility function. In the consumption optimum, the ratio of marginal utilities equals the ratio of prices of these goods, and also the marginal rate of substitution between the goods.

However, in these models prices are exogenous, so marginal utilities never "decide" prices in any way. It's rather the other way round: Prices (i.e., price ratios) "decide" the ratios of marginal utilities.


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