General Equilibrium and Welfare

• General Equilibrium
  • Pure Exchange Economy
    • Preliminaries and Assumptions
    • Prices, Income, and Competitive Equilibria
  • Production Economies
    • Autarky
    • Trade
• Efficiency
  • Pareto Efficiency
  • Characterizing a Pareto-Efficient Outcome
  • The Invisible Hand and the Role of Competitive Prices

General Equilibrium

The Fundamental Theorem of Exchange: Voluntary trade between two individuals is beneficial to both parties.

• Exceptions include scams, whimsical purchases, or vices
• Benefits are derived from the rearrangement of consumption goods and production
  • In other words, each person can obstain a consumption basket more preferred to what is given, and people can specialize in production which increases the goods available for consumption

Pure Exchange Economy

Characteristics:

• Several consumers, each described by their preferences and their endowments (what goods they possess)
• Consumers trade among themselves, acting in their self-interests
• No production means that there is a fixed supply of goods
• Income becomes endogenous with prices, even when endowments are exogenous and fixed

Preliminaries and Assumptions

Consider an economy with two individuals, i = A, B, and two goods, X (apples) and Y (bananas)

$$\text{Endowments: } E_A = (\bar{x_A}, \bar{y_A}), E_B = (\bar{x_B}, \bar{y_B}) \\ \bar{x} = \bar{x_A} + \bar{x_B}, \bar{y} = \bar{y_A} + \bar{y_B} \\ \text{Assume that } \bar{x_A} < \bar{x_B} \text{ and } \bar{y_A} > \bar{y_B} \\ \text{Preferences: Defined by a utility function and its corresponding indifference curve.}$$

Note that at point E, where both parties consume their initial endowments, there are multiple indifference curves for both parties that are better than their current ones. Thus, the middle of the two indifference curves that pass through E gives the region of mutually advantageous trade. At points of mutual tangency of the two curves, the parties cannot trade any further or else at least one of them will be at a loss.

Prices, Income, and Competitive Equilibria

The budget constraints of both parties give an equilibrium for trade.

$$\text{Expenditures must equal income for an individual.} \\ E_i = p_x x_i + p_y y_i = I_i = p_x \bar{x_i} + p_y \bar{y_i} \\ dE_i = p_x dx_i + p_y dy_i = 0 \rightarrow \frac{dy_i}{dx_i} = - \frac{p_x}{p_y}$$

The budget line with slope of -p_x/p_y indicates for a given endowment/income how much of one good would have to be sacrificed to increase consumption of the other good.

To find the competitive equilibrium, we must find the point at which:

1. Tangent to both indifference curves
2. On the budget line

$$\text{At the equilibrium, the following holds:} \\ p_x x_A + p_y y_A = p_x \bar{x_A} + p_y \bar{y_A} \\ p_x x_B + p_y y_B = p_x \bar{x_B} + p_y \bar{y_B} \\ | MRS^A_{xy} | = p_x / p_y =| MRS^B_{xy} \\ \text{Solution: } \\ \text{Let } p_y = 1 \text{. x is now measured in units of y.} \\ \text{max } U_i (x_i, y_i) \text{ such that } p_x x_i + y_i = I_i \\ \text{Assuming that x > 0 and y > 0: } y_i = I_i - p_x x_i \\ dy_i / dx_i = -p_x$$

Simple steps:

1. Find MRS
2. Plug into Expenditures = Income equation
3. Plug into Endowments equation: x_A + x_B = x

Production Economies

Characteristics:

• Assume individuals are endowed with a set of resources that can be combined to produce goods for consumption
• Production precedes trade
• Individuals participate in both production and trade

Autarky

Under autarky, an individual can create a set of bundles underneath their production-possibilities frontier, whose boundary can be written as Q(x, y).

The cost of specializing in the production of x in terms of y can be seen in the slope of the PPF. This is also known as the marginal rate of transformation, and it can be calculated by dividing the marginal cost of x by the marginal cost of y.

Similarly to the problem shown to consumers with budget constraints, the optimal set to produce will be the point of tangency between the PPF and the highest indifference curve.

Trade

With trade, the individual will choose the production bundle that maximizes income given prices p_x and p_y.

Additionally, the consumer purchases the bundle that maximizes their utility, which often lies on a higher indifference curve than the autarkic optimum.

Notice how that the introduction of trade incentives the individual to specialize in the production of a good.

Thus, under trade, there are two separate problems to solve: maximizing income, and maximizing utility. First, find the production bundle that gives the highest budget line and budget constraint. Then, find the highest indifference curve according to that budget constraint.

For consumers, trade allows for a more balanced, preferred bundle, but for the economy, trade allows for specialization which means that there are more goods in the economy.

Efficiency

Pareto Efficiency

Suppose there are two individuals, A and B, and each individual’s real income is a reasonable measure for their well-being

• Social opportunity set: Depicts the entire range of achievable income, I_A, I_B
• Social opportunity frontier: The boundary of the social opportunity set

There are various ways to gauge efficiency based on the sum of real income, I_A + I_B

• Iso-income line: Combinations of I_A and I_B which implies a constant sum with a slope of -1
• Maximal income: Where the social opportunity frontier is tangent with the outermost iso-income line
• Very similar to budget constraints and isoquant lines

Pareto efficiency is based off of the observation that, on the frontier, you cannot make one person better off without making the other worse off.

• An allocation of goods X is Pareto-preferred to another allocation Y if, under X, everyone is at least as well off as under Y AND at least one individual is better off
• Allocation X is Pareto-efficient or Pareto-optimal if no feasible alternative is Pareto-preferred to it
  • All points along the frontier are therefore Pareto-efficient

Characterizing a Pareto-Efficient Outcome

Under perfect competition, utility-maximizing behavior by individuals combined with profit maximizing behavior by firms leads to a Pareto-efficient outcome

An allocation of resources is efficient in production or technically efficient if no further reallocation would permit more of one good to be produced without necessarily reducing the output of some other good.

• Pareto-efficiency implies technical efficiency, as not being technically efficient implies that the allocation does not lie on the frontier
• Technical efficiency does NOT imply Pareto-efficiency

Efficient choice of inputs for a single firm: Requires that all inputs held by a firm be fully employed and that the rate of technical substitution between inputs (or the ratio of the inputs’ marginal productivities) be the same across goods produced by the firm.

• If one firm has too much labor, then the other will have too much capital. The social planner will then shift some labor and capital to make the ratio of TRS equal.

Efficient allocation of resources across firms: Requires that the marginal product of any resource in the production of a particular good be the same no matter which firm produces that good.

• If the marginal product of capital at firm 1 was greater than the marginal product of labor, then the planner would give firm 1 more capital, and vice versa.

Efficient choice of output across firms: Requires that the marginal rate of transformation between two products be equalized across firms.

• If the marginal rate of transformation for a certain good is higher at firm 1 than firm 2, then the planner will shift production of that good away from firm 2 towards firm 1

Efficiency in consumption: Under trade, the MRS are equal between two individuals; in other words, no individual has an incentive to trade any more than they already have.

Efficient balance in consumption and production: Both production and consumption must be efficient. Mathematically, the MRS between two goods should be the same as the corresponding MRT.

The Invisible Hand and the Role of Competitive Prices

The theorem of the invisible hand, or the first theorem of welfare economics, states that no social planner is needed because individuals and firms will naturally reach efficient outcomes.

1. Profit maximizing production results in technical rates of substitution between inputs being equal, satisfying the clause for an efficient choice of inputs.
2. The marginal products for each firm will be equal to each other for all firms producing x and y, as a firm not doing so would lose profits on one of the goods, satisfying the clause for efficient allocation of resources.
3. The marginal cost for firms in a competitive market must be equal to the price it faces. In other words, MC_x / MC_y = p_x / p_y = MRT_xy. Thus, this satisfies the clause for efficient choice of output.

Utility maximizing individuals will maximize their utility by default, and at that bundle, they won’t be willing to give up either good to gain more of the other, satisfying efficiency in consumption.

Recall that p_x / p_y is the isocost/isoquant line, and profit/utility maximization means that the bundle lies on that line. Thus, the MRT and MRS are both equal to the slope of that line, satisfying the clause of efficient balance in consumption and production.