I have two equations: $$N_sc_s + p_s L =N_s \gamma^sw$$ and $$\frac{(1+g)N_s c_s}{\gamma} + \frac{(1+h)p_s L}{(1+n)\gamma} = N_s \gamma^s w$$ Where the variables are $c_s, p_s$. Obviously One way for both these equations to be true is for: $$ \tag{1} N_sc_s = \frac{(1+g)N_s c_s}{\gamma} \text{ and } p_s L = \frac{(1+h)p_s L}{(1+n)\gamma}$$ which can be solved. However would solving $(1)$ necessarily give me the solution, or could be there other solutions (intuitively, I'm thinking that it's possible that instead of having the equalities in $(1)$, perhaps one term is less than it's corresponding term and the other one greater...) In fact, I would guess that there are multiple solutions...
However... if I add two restrictions: that $c_s, p_s >0$, then I have no clue whether there would be multiple solutions or not...
Also, for completeness, $g,h,n >0 \text{ and } < 1$, $\gamma >1$, $L,N$ are some large, positive constants.
Thanks