How to find the price ratio $\frac{p_x}{p_y}$?

(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:

1. (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 $$3/2$$.

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 $$(x^*, y^*)$$?

To find competitive equilibrium prices $$(p_X, p_Y, w=1)$$, we need to find the supply of $$X$$ and $$Y$$ and labor $$L$$ demand of the two firms:

$$\displaystyle\max_{x\geq 0,l_X\geq 0} p_Xx - l_X$$ subject to $$x \leq 2l_X$$

$$\displaystyle\max_{y\geq 0,l_Y\geq 0} p_Yy - l_Y$$ subject to $$y \leq 3l_Y$$

Solving the above problems give the supply of $$X$$ and $$Y$$ as:

$$\begin{eqnarray*} x^s \in \begin{cases} \emptyset &\text{if } p_X > \frac{1}{2} \\ [0,\infty) &\text{if } p_X = \frac{1}{2} \\ \{0\} &\text{if } p_X < \frac{1}{2} \end{cases} \end{eqnarray*}$$ and corresponding labor demand for $$X$$ is $$l^d_X = \dfrac{x^s}{2}$$. $$\begin{eqnarray*} y^s \in \begin{cases} \emptyset &\text{if } p_Y > \frac{1}{3} \\ [0,\infty) &\text{if } p_Y = \frac{1}{3} \\ \{0\} &\text{if } p_Y < \frac{1}{3} \end{cases} \end{eqnarray*}$$ and corresponding labor demand for $$Y$$ is $$l^d_Y = \dfrac{y^s}{3}$$.

This rules out the possibility that $$p_X > \frac{1}{2}$$ or $$p_Y > \frac{1}{3}$$ in equilibrium. Therefore, if the equilibrium exists, optimal profits will be $$0$$.

Now we can find the demand for $$X$$ and $$Y$$ by solving the following problems:

$$\displaystyle\max_{x\geq 0,y\geq 0} x^{0.3}y^{0.7}$$ subject to $$p_Xx + p_Yy \leq 10$$

$$\displaystyle\max_{x\geq 0,y\geq 0} x^{0.5}y^{0.5}$$ subject to $$p_Xx + p_Yy \leq 10$$

Solving the above problems give the demand for $$X$$ and $$Y$$ as:

$$(x^d_S, y^d_S) = \left(\dfrac{3}{p_X}, \dfrac{7}{p_Y}\right)$$

$$(x^d_J, y^d_J) = \left(\dfrac{5}{p_X}, \dfrac{5}{p_Y}\right)$$

Since demands for both the goods are always strictly positive, observing the supply rules out the possibility of $$p_X < \frac{1}{2}$$ or $$p_Y < \frac{1}{3}$$ in equilibrium.

Therefore, if the equilibrium exists, it must be the case that $$p_X = \frac{1}{2}$$ and $$p_Y = \frac{1}{3}$$. We can now verify that it is indeed the equilibrium.

Equilibrium prices are $$(p_X=\frac{1}{2}, p_Y=\frac{1}{3}, w=1)$$, and the corresponding equilibrium quantities of $$X$$, $$Y$$ and $$L$$ are:

$$x^d_S + x^d_J = 6 + 10 = 16 = x^s$$

$$y^d_S + y^d_J = 21 + 15 = 36 = y^s$$

$$l^d_X + l^d_Y = \dfrac{x^s}{2} + \dfrac{y^s}{3} = 8 + 12 = 20 = 10+10=l^s_S + l^s_J$$

$$\frac{p_x}{p_y}=\frac{3}{2}$$ Because if $$\frac{p_x}{p_y}>\frac{3}{2}$$, both the persons will have an incentive to produce only x and then use the income to purchase y from the market. This will result in a rise in $$p_y$$. The process will continue until $$\frac{p_x}{p_y}=\frac{3}{2}$$. Analogously, if $$\frac{p_x}{p_y}<\frac{3}{2}$$, every-one has an incentive to produce only y and use the income to purchase x from the market. This will result in a rise in $$p_x$$ until we reach the ratio $$\frac{p_x}{p_y}=\frac{3}{2}$$ This explanation says why the price ratio must be the slope of the PPC. Both the solutions are correct except that in the second one you have probably made a typo, the slope of PPC is $$\frac{3}{2}$$.