# General Equilibrium Theory: Examples

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1 General Equilibrium Theory: Examples

2 3 examples of GE: pure exchange (Edgeworth box) 1 producer - 1 consumer several producers and an example illustrating the limits of the partial equilibrium approach

3 First example: Edgeworth Box A pure exchange economy (no production possibilities): 2 consumers i = A, B 2 commodities l = 1, 2 individual endowments ω i = (ω i1, ω i2 ) global endowment ϖ = ω A + ω B allocation x = (x A, x B ) with x i = (x i1, x i2 ), x i 0 price p = (p 1, p 2 ) Edgeworth box = allocations such that x A + x B = ϖ Endogenous wealth: w i = p.ω i The budget line splits the box into the 2 budget sets

4 Individual Preferences represented by a utility function u i continuous (the representation of preferences by a utility function requires transitive, complete, continuous preferences) strictly quasi-concave (unique optimum) strictly monotonic (stronger than locally non satiated) Offer curve of i = optima of i (parameterized by p)

5 Definition : a Walrasian equilibrium is (x, p) such that 1. individual optimality : i, x i solves max u i (x), p.x p.ω i 2. market clearing x i = ϖ i = intersection points of the two offer curves (other than the endowment point) GE determines the relative price only ( one defines a numeraire, a good with price 1, without loss of generality) Uniqueness is not guaranteed Examples : Cobb-Douglas, linear, Leontief preferences

6 Two examples of non existence: 1. An important one: non convexity of one u i : no intersection of the offer curves because of a discontinuity 2. A more subtle one: non strict monotonicity of one u i : impossible to clear the markets by adjusting the prices

7 Illustration of the 2 Welfare Theorems Th 1 : The allocation x of an equilibrium (x, p) is Pareto-optimal Definition : Equilibrium with nominal transfers = (x, p, t A, t B ) s.t. 1. t A + t B = 0 (t A, t B IR) 2. i, x i maximizes u i (x i ) under p.x i p.ω i + t i 3. x A + x B = ϖ Th 2 : If x is a PO allocation, then x is the allocation of an equilibrium with transfers (compute the relative price p 2 /p 1 = MRS, then define the transfer: t i = p.x i p.ω i ) Th 2 requires the convexity of preferences (not Th 1) real transfers can be considered as well (example: p.x i p.ω i + p 1 t i, transfer of good 1)

8 Second example: 1 consumer + 1 producer 2 commodities: leisure (price w), consumption good (price p) firm: production function q = f (z) (f > 0 > f ) max pq wz consumer: utility u (l, x) endowment (L, 0) owns the firm

9 Definition: A Walrasian equilibrium is (l, x ),(q, z ),(w, p) 1. individual optimality: (q, z ) solves (l, x ) solves 2. market clearing max pq wz q=f (z) max u (l, x), with π = wl+px wl+π pq wz l + z = L and x = q In this example, equilibrium is unique.

10 Illustration of the 2 Welfare Theorems Th 1 : The (unique) equilibrium allocation is PO Th 2 : The (unique) PO allocation is the equilibrium allocation (no transfer is needed in this example) Without the convexity assumptions (preferences and production set): An equilibrium is still PO ( Th 1 still holds ) A PO allocation may not be an equilibrium allocation, even with transfers ( Th 2 does not hold )

11 Remark: production function and production set Definition of the production set Y : y Y if and only if y = (y 1,..., y L ) is a technologically feasible vector convention sign: y l < 0 whenever l is an input, y l > 0 whenever l is an output For a technology defined by a production function f (the output is good L, inputs are goods 1,..., L 1): the associated production set Y is { } y IR L /y L f ( y 1,..., y L 1 ) Y convex f concave

12 Example: f (z) = Az α α < 1 : DRTS (f concave), everything is OK α = 1 : CRTS (f linear), technology determines the relative prices, π = 0, the production level is determined by demand (the supply is infinitely elastic) α > 1 : IRTS (f convex), no equilibrium

13 Remark: Returns to Scale For a technology defined by a production set Y : decreasing (DRTS) y Y, a [0, 1], ay Y increasing (IRTS) y Y, a 1, ay Y constant (CRTS) y Y, a 0, ay Y For a technology defined by a production function f : DRTS: z IR + L 1, a 1, f (az) af (z) IRTS: z IR + L 1, a 1, f (az) af (z) CRTS: z IR + L 1, a 0, f (az) = af (z) (f homogenous of degree 1)

14 Third example: J producers J firms use L inputs to produce one different output each global inputs endowment z = ( z 1,..., z L ) 0 technologies f j (C 2, > 0 and D 2 f j negative definite) f j z jl exogenous output prices p = (p 1,..., p J ) input prices w = (w 1,..., w L ) Definition : An equilibrium is (z, w) IR JL+L + j, z j j z j maximizes p j f j (z j ) w.z j = z

15 1st Order Conditions (for an interior equilibrium only, j, z j 0) = a system of equations with unknown (z, w) characterizes the equilibrium l, j, p j f j z jl (z j ) = w l, l, j z jl = z l. Illustration of Th 1: an equilibrium allocation z maximizes the production value: max p j f j (z j ) j z j = z j

16 Proof: the joint profit j (p jf j (z j ) w.z j ) is j p jf j (z j ) w. z. Hence, the max of joint profit and the max of the prod value have the same solution. the FOC of max joint profit (under the constraint j z j = z) are the same as the FOC of equilibrium l, j, p j f j z jl (z j ) = w l, l, j z jl = z l. hence z maximizes the joint profit. Producers (but not consumers) can be aggregated: a unique firm with J technologies make the same decisions (and gets the same profit) as J independent firms with one technology each.

17 GE versus partial equilibrium: a taxation example N towns, 1 firm/town (production function f ) Labor supply (inelastic) : M workers, Wage w, good = numeraire At equilibrium, w = f ( ) M N (from max profit : w = f (l n ) and market clearing: n l n = M - equilibrium is symmetric) Introduction of a tax t in town 1 : w + t = f (l 1 )

18 Partial equilibrium analysis in town 1: workers freely move between towns + w in towns 2,...,N w remains constant in town 1 hence l 1 determined by w + t = f (l 1 ) the profit decreases, not the wage the firm bears the whole burden of the tax t

19 GE Analysis: w and l 1,..., l N determined by: w + t = f (l 1 ) and n 2, w = f (l n ) l l N = M (hence l 2 =... = l N = M l 1 N 1 ) Introduction of a small tax dt dw + dt = f (l 1 ) dl 1 and dw = f (l) dl dl 1 + (N 1) dl = 0 (denote l = l 2 =... = l N )

20 Variation of the profit of a firm π 1 = f (l 1 ) (w + t) l 1 and π = f (l) wl dπ 1 = f (l 1 ) dl 1 (w + t) dl 1 l 1 (dw + dt) dπ = f (l) dl wdl ldw And ( ) M dπ 1 = f dl 1 wdl 1 M (dw + dt) N N ( ) M dπ = f dl wdl M N N dw with l 1 = l = M N at the no tax equilibrium (t = 0)

21 Variation of the aggregate profit dπ 1 + (N 1) dπ [ ( ) ] M = f w (dl 1 + (N 1) dl) N = 0 M N (dw + dt) (N 1) M N dw since dw + dt = f ( M N ) ( ) M dl 1 and dw = f dl N dl 1 + (N 1) dl = 0 the workers bear the whole burden of the tax (w decreases, not the profit)

22 The end of the chapter

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