I encountered the following economic model.
Consider the following general equilibrium model with only two households, two consumer goods ($x$ and $y$) and two inputs (capital $k$ and labor $l$). Each household has an endowment of capital and labor that it can choose to retain or sell in the market. These endowments are denoted by $k_1, l_1$ and $k_2, l_2$ respectively. Households obtain utility from the amounts of the consumer goods they purchase and from the amount of labor they do not sell into the market (i.e., leisure $= \overline{l_i} - l_i$). The households have Cobb–Douglas utility functions: $$U_1 = x_1^{0.5}y_1^{0.3}(\overline{l_1} - l_1)^{0.2} \ \ \ \text{ and } \ \ \ U_2 = x_2^{0.4}y_2^{0.4}(\overline{l_2} - l_2)^{0.2}.$$ Each household provides its entire endowment of capital to the marketplace. Households retain some labor because leisure provides utility directly. Production of goods $x$ and $y$ is characterized by Cobb–Douglas technologies: $$x = k_x^{0.2}l_x^{0.8} \ \ \ \text{ and } \ \ \ y = k_y^{0.8}l_y^{0.2}.$$ The initial endowments are $\overline{k_1} = 40, \overline{k_2} = 10, \overline{l_1} = \overline{l_2} = 24$.
Assume further that $p_x, p_y$ are the prices of goods $x$ and $y$, $w$ is the wage rate (cost of labor) and $r$ is the cost of capital.
1. Do labor and capital for production come from these two households only or there are other agents that we are unaware of?
2. How to calculate the equilibrium prices and allocations?
Here's my attempt at determining the equilibrium. The income and budget constraint of household $i$ will be $I_i = wl_i+r(\overline{k_i}-k_i)$ and $p_x x_i + p_y y_i \leq wl_i + r (\overline{k_i} - k_i)$ respectively. Since capital doesn't affect utility, $\overline{k_i} = k_i$ (that is, every ounce of capital will be provided to the market) and the utility maximization problems will be: $$\max [U_i(x_i, y_i, l_i)] \text{ subject to } p_x x_i + p_y y_i = wl_i$$
Now what to do with the production functions?
Note to potential answerers: You don't need to do the tedious calculations. Just help me construct the maximization problem.
