How do I find the Walrasian equilibrium price?

Consider a production economy with 1 consumer and 2 firms. The consumer's utility is $$u(x,y)=xy$$. The first firm makes $$x$$ using capital and labor according to the production function $$x=k^\alpha l^{1-\alpha}$$ and the second firm makes $$y$$ using $$k,l$$ by $$y=k^\beta l^{1-\beta}$$. The total amount of capital and labor available in the economy are fixed at $$K$$ and $$L$$ respectively and are owned by the consumer. The consumer is also the owner of both firms.

How do I find the equilibrium prices for $$x$$ and $$y$$?

Suppose we normalize the price of $$y$$ to be 1. The first firm maximizes profits: $$px-rk_1-wl_1\;\;s.t. x=k_1^\alpha l_1^{1-\alpha}\implies \max (pk_1^\alpha l_1^{1-\alpha}-rk_1-wl_1)$$ By first order condition, $$\alpha p k_1^{\alpha-1}l_1^{1-\alpha}=r$$ and $$(1-\alpha)p k_1^\alpha l_1^{-\alpha}=w$$. Thus, $$k_1=\frac{\alpha l_1 w}{(1-\alpha)r}$$ substitute this back into $$x=k_1^\alpha l_1^{1-\alpha}$$ we have $$l_1=\frac{x}{\left(\frac{\alpha w}{(1-\alpha r)}\right)^\alpha}$$. So $$k_1=\frac{\alpha l_1 w}{(1-\alpha)r}=(\frac{\alpha}{1-\alpha})^{1-\alpha}(\frac{w}{r})^{1-\alpha}x$$. But I cannot get the $$x$$ out. How do I proceed?

1. Your equation system may look scary, but it is not, it is linear in $$x$$, so things could still be fine.
1. Use some formulas? Just memorizing formulas is bad form, and you may also have to 'show your work' but if you have done several Lagrangians for Cobb-Douglas functions you might notice that the formula $$k_1=\frac{\alpha l_1 w}{(1-\alpha)r}$$ also has this (frequently used) form: $$|\text{MRTS}(k_1,l_1)| = \frac{\alpha}{(1-\alpha)}\frac{l_1}{k_1}= \frac{r}{w}$$ which is useful because it is easy to remember, and using these formulas you don't need to do the basic calculations for Cobb-Douglas production functions. (Similarly see MRS for C-D utility functions.)
• hi - what i'm struggling with is that since the $k_1, l_1$ I got still have $x$ in it, I cannot calculate the profit without having $x$ in the expression. Thus I cannot do the utility maximization problem. Or can I? Nov 15, 2023 at 14:05
• @LudwigGershwin Having $x$ in the expression might complicate things but it is not in itself a problem, you can treat as a parameter in the utility maximization. Or you can try to do the other things (firm 2 profitmax and resource equilibrum) which will hopefully resolve $x$ before dealing with utility maximization. Nov 16, 2023 at 10:08