General Equilibrium: Introductory Notes, Exercises of Statics

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Definition: Let E = ((ui,˚xi))ni=1 be an economy consisting of n consumers, all of whose demand functions xi(·) : Rl + → Rl^ are well-defined and single-valued on Rl +, and let (^) X∆ (·) : Rl + → Rl^ denote the corresponding market net demand function. A market equilibrium of E is a price-list p ∈ Rl + that satisfies the equilibrium condition

∀k = 1,... , l :

∆ Xk (p)^5 0 and^

∆ Xk (p) = 0 if^ pk >^0.^ (Clr)

We’ll also refer to a price-list that satisfies (Clr) as an equilibrium of the net demand function ∆ X (·).

We’ll use this equilibrium condition throughout the course, so we give it a name that we’ll use to refer to it: (Clr), which is an abbreviation for Clear, since the condition says that all markets clear.

A market equilibrium is also called a Walrasian equilibrium. An essential feature of this equilib- rium concept is the assumption — implicit in the definition — that all consumers are price takers. Each consumer, in solving his consumer maximization problem, treats the prices as parameters that will be unaffected by his decision about which consumption bundle he will choose.

Some Remarks

(1) Note the analogy with optimization: Here the equilibrium conditions are equations that deter- mine the values of the variables, and in optimization the first-order conditions are equations that determine the values of the variables.

(2) With l goods we will have l − 1 independent equilibrium conditions (equations), with Wal- ras’s Law accounting for the remaining market, so only l − 1 relative prices are determined by equilibrium. (We could, for example, use one of the prices as “numeraire.”) Because the demand functions are homogeneous of degree zero, they also depend only on the relative prices.

(3) What if, unlike in our Cobb-Douglas examples, we can’t get a closed-form solution (i.e., an explicit expression) for the state variables in terms of the parameters? How do we do comparative statics in that case? We can apply the Implicit Function Theorem to the equilibrium equations, just as we apply the IFT to the first-order equations to do comparative statics for optimization.

(4) The approach in (3) is often not good enough: for example, we often need to determine the actual equilibrium prices and/or quantities, not just the comparative statics derivatives. If we can’t get closed-form solutions (which is the typical situation), we can try to compute the equilibrium values. How do we do that?

(5) What if there is no equilibrium? Under what conditions will there be an equilibrium?

(6) What if there are multiple equilibria? Under what conditions will there be a unique equilibrium?

(7) What if the system is not in equilibrium? What are the stability properties of the equilibrium?

(8) Is the equilibrium outcome a good outcome?

(9) What if markets and prices aren’t used? Under what conditions will they be used?

(10) What if not everyone is a price-taker?

We will address all of these questions in the course, some in depth, and others only in passing.