# Production economy general equilibrium

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.

• This answer might be of some help: economics.stackexchange.com/a/15165/11824
– Amit
Commented Sep 17, 2022 at 8:05
• @Amit Thank you. You wrote that the competitive equilibrium is Pareto Efficient by the first welfare theorem. We have to find the competitive equilibrium but why do we care about whether it's pareto efficient or not? Is pareto efficiency related to PPF which is why you mentioned this theorem? I don't understand the reference.
– Liza
Commented Sep 17, 2022 at 8:15
• @Amit I added a comment below. I think it's not a Walrasian equilibrium as at the optimal points, the demands don't match the supplies. Also, consumption is less than the budget.
– Liza
Commented Sep 17, 2022 at 9:13
• Approach 2 says that under some conditions set of competitive equilibrium allocations is a subset of set of Pareto efficient allocations. So, you can narrow down your search for competitive equilibrium among the set of feasible allocations to only Pareto efficient allocations if those conditions are met.
– Amit
Commented Sep 17, 2022 at 12:35

1. Only the households.

2. Constructing the maximization problems for equilibrium:

Feasibility conditions:
The amount of total labor used by industries $$x$$ and $$y$$ is not more than the amount of total labor supplied by the two households.
Same goes for capital.
The total amount of good $$x$$ consumed (by the households) is not more than the total amount supplied by the industry.
Same goes for good $$y$$.

Optimality conditions:
Given the prices (inc. wages and interest rate), households' decisions (subject to budget constraints) maximized utility.
Given the prices companies' decisions (subject to technological feasibility as outlined by the production function) maximized profit.

• At the optimal value, for both households, the demand is less than supply and the consumption is less than income from selling labor and capital. If that is the case, can I still apply Lagrange?
– Liza
Commented Sep 17, 2022 at 9:07