Does the Samuelson condition assume all the prices for the private goods are the same?

Question

Suppose I have two agents with utility functions $U_i$ and $U_j$ and budgets $p_x x + p_g g = m_i$ and $p_y y + p_g g = m_j$.

If I solve a social planner UMP, I will get for my g FOC (after plugging in the other FOCs)

$$ p_x MRS_i + p_y MRS_j = p_g$$

If I assume $p_x = p_y$ then I have after rearrangement

$$MRS_i + MRS_j = MRTS =\frac{p_g}{p_x} $$

But suppose $p_x \neq p_y$. Does the Samuelson condition hold?

---

As I wrote the question, I noticed the following:

I can see one would write for $n$ agents

$$\sum_{i} p_{x_i}\frac{MU^i_g}{MU^i_{x_j}} = p_g$$

Now we can write that as

$$\frac{MU^i_g}{MU^i_{x_i}} + \sum_{j\neq i} \frac{p_{x_j}}{p_{x_i}} \frac{MU^j_g}{MU^j_{x_j}} = \frac{p_g}{p_{x_i}} $$

Answer 1

I thought it might be helpful to discuss the LS condition in general as it pertains to the LS equilibrium:

A Lind./Sam. equilibrium allows for a unique equilibrium price $q_i, \forall i \in {1,...,n} \equiv N$, the set of agents, for the public good. The clearing condition for the public good price vector, given by $\sum MRS=MRTS$ (as stated in another answer here) is an aggregate condition. Individuals value public goods differently, which is reflected by vector $q$.

To make it a bit more intuitive: If it were necessary that each individual contribute equally to a public good, then our idiosyncratic valuations would prevent some (or maybe many) of us from contributing to a public goods project (because the fixed contribution exceeds our individual, unique valuation for that public goods project). This could cause us to scrap projects that are actually socially desirable. And in easily conceivable circumstances, it can cause us to scrap projects that would be beneficial to every person in our society.

The price vector $q$ does not necessarily comprise distinct prices. However, the question is can it? The answer to that question is, clearly, yes.

Answer 2

No as we can derive the condition outside of the framework of prices.

Consider 2 person case:

$\max U_1=U_1(x_1,y_1)$ subject to

$\bar U=U_2(x_2,y_2)$ and

$F(x_1+x_2, y_1+y_2)=0$

Solving this gets the $\sum MRS=MRT$ condition. Think of the Samuelson condition as one of efficiency of resources and utility, so prices don't enter.