Walrasian Equilibrium & Price Indeterminacy

Consider a perfectly competitive exchange economy with no production, and two different goods 1 and 2. Let p₁ and p₂ be the prices of the goods. The economy is populated by two people A and B. A's initial endowment is given by (w, w²) and B's initial endowment is (wb, wh). A can choose any bundle (x1, x2) and B can choose any bundle (x1, x2). In this pure exchange economy write out the conditions for a Walrasian equilibrium. Show that for such an economy for any equilibrium set of prices that the absolute price level is indeterminate.

How to Approach

This question tests understanding of Walrasian equilibrium in a simple exchange economy and the concept of price indeterminacy. The approach should involve first defining Walrasian equilibrium conditions – excess demand being zero for all goods. Then, demonstrate how scaling prices by a constant factor doesn't alter the equilibrium conditions, thus proving price indeterminacy. Focus on mathematical representation and logical explanation. The answer should be concise and directly address the question's requirements.

Model Answer

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Introduction

A Walrasian equilibrium represents a state in a market where supply equals demand for all goods, leading to market clearing. This concept, foundational to general equilibrium theory, assumes rational agents maximizing utility subject to budget constraints. In a pure exchange economy, without production, the equilibrium is determined by the initial endowments of consumers and their preferences. The question asks us to define the conditions for such an equilibrium in a two-person, two-good economy and demonstrate the inherent indeterminacy of the absolute price level.

Walrasian Equilibrium Conditions

Let x_i represent the consumption bundle of individual i (where i = A, B), and ω_i represent their initial endowment. The budget constraint for individual i is given by: p₁x_1i + p₂x_2i = p₁ω_1i + p₂ω_2i. The utility maximization problem leads to demand functions x_1i(p₁, p₂, ω_1i, ω_2i) and x_2i(p₁, p₂, ω_1i, ω_2i).

The Walrasian equilibrium conditions require that the total demand for each good equals the total supply (endowment):

Market Clearing for Good 1: x_1A(p₁, p₂, ω_1A, ω_2A) + x_1B(p₁, p₂, ω_1B, ω_2B) = ω_1A + ω_1B
Market Clearing for Good 2: x_2A(p₁, p₂, ω_1A, ω_2A) + x_2B(p₁, p₂, ω_1B, ω_2B) = ω_2A + ω_2B

Price Indeterminacy

To demonstrate price indeterminacy, consider a set of equilibrium prices (p₁^*, p₂^*) that satisfy the market clearing conditions. Now, let's multiply both prices by a positive scalar, λ > 0, to obtain a new set of prices (λp₁^*, λp₂^*). We need to show that this new price set also satisfies the market clearing conditions.

Substituting the new prices into the market clearing equations:

x_1A(λp₁^*, λp₂^*, ω_1A, ω_2A) + x_1B(λp₁^*, λp₂^*, ω_1B, ω_2B) = ω_1A + ω_1B
x_2A(λp₁^*, λp₂^*, ω_1A, ω_2A) + x_2B(λp₁^*, λp₂^*, ω_1B, ω_2B) = ω_2A + ω_2B

Since demand functions are homogeneous of degree zero in prices (due to utility maximization), scaling the prices by a constant factor does not change the quantities demanded. Therefore, x_1A(λp₁^*, λp₂^*, ω_1A, ω_2A) = x_1A(p₁^*, p₂^*, ω_1A, ω_2A) and similarly for other demands. Thus, the market clearing conditions still hold with the new prices (λp₁^*, λp₂^*).

This demonstrates that for any equilibrium price set, there are infinitely many other equilibrium price sets obtained by multiplying by a positive scalar. Consequently, the absolute price level is indeterminate in this pure exchange economy.

Conclusion

In conclusion, the Walrasian equilibrium in a pure exchange economy is defined by the equality of total demand and total supply for each good. However, the solution is not unique in terms of absolute price levels. The homogeneity of demand functions allows for price scaling without altering the equilibrium conditions, leading to price indeterminacy. This highlights a key characteristic of general equilibrium models and the need for further restrictions to uniquely determine prices.

Answer Length

This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.

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Additional Resources

Key Definitions

Walrasian Equilibrium

A state of economic equilibrium where supply equals demand in all markets, and no economic agent has an incentive to change their behavior, given the prices.

Key Statistics

According to the World Bank, global trade as a percentage of GDP was approximately 28.5% in 2022, indicating the importance of market equilibrium on a global scale.

Source: World Bank, 2023

India's agricultural sector contributes approximately 18.8% to the country's GDP (2022-23), highlighting the importance of understanding market equilibrium in this sector.

Source: National Statistical Office (NSO), 2023

Examples

Farmers Market

A local farmers market can be seen as a simplified example of a Walrasian equilibrium. If the price of apples is set such that the quantity supplied by farmers equals the quantity demanded by consumers, the apple market is in equilibrium.

Frequently Asked Questions

▶What happens if the initial endowments are not equally distributed?

Unequal endowments affect the equilibrium allocation of goods. While the equilibrium conditions remain the same, the resulting consumption bundles for individuals A and B will differ, reflecting their initial resource constraints.

Topics Covered

EconomicsMicroeconomicsGeneral EquilibriumWelfare EconomicsPrice Theory

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