(Question) Suppose two individuals form an economy where each devotes $10$ hours of labour to produce goods $x$ and $y$. The utilities of the agents $S$ and $J$ are $U_S(x,y) = x^{0.3}y^{0.7}$ and $U_J(x,y)=x^{0.5}y^{0.5}$. If the individuals do not care whether they produce $x$ or $y$ and the production function for each good is given by $x=2l$ and $y=3l$ where $l$ is the total labour devoted to production of each good, find the price ratio $\frac{p_x}{p_y}$.
I have two different solutions for this:
(Solution 1) Suppose the price of labour is $p_l$ in the economy. To maximize profits on $x$, $$\pi_x(l_x^J, l_x^S, p_x,p_l) = p_xx - p_ll = 2p_x(l_x^J+l_x^S)-p_l(l_x^J+l_x^S)$$
Differentiation gives us $p_l = 2p_x$ and if we do the same on $y$'s maximization problem, we get $$p_l = 2p_x = 3p_y$$ The price ratio can derived from this as $3/2$.
(Solution 2) Given $x$, the production possibility frontier is given by the equation $x/2 + y/3 = l_x + l_y = 20$. The slope of this equation at the optimal point gives us the price ratio and it's very clearly $3$.
My questions regarding this problem:
- Are both my solutions correct?
- Why is the slope of the Production-Possibility Frontier (PPF) at the optimal $(x^*, y^*)$ equal to the price ratio? It appears to be a a cost maximization problem subject to the PPF.
- Once I know the PPF, how do I find the optimal $(x^*, y^*)$, let's say for this problem? I am aware of the usual way of equating the demand and supply and also maximizing the profits. But is there any way our knowledge of the PPF can be used to find $(x^*, y^*)$?