# Relationship between marginal utility and price?

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".