The Edgeworth Box and Competitive Equilibrium
The box holds everything two people own. Endowments, offer curves, and the contract curve all appear in one picture.
The contract curve collects points where neither person can do better without hurting the other. At the equilibrium price, one budget line touches both indifference curves at once. That tangency is the competitive equilibrium. The thick part of the curve is the core, which no one can block.
For example, say you hold 7 🍎 apples and 3 🍌 bananas. Your 🧑 friend holds 7 bananas and 3 apples. You like bananas more, and your friend likes apples more. Trading makes both of you happier. When no trade helps anymore, you land on a point of the contract curve.
p 1 = a 1 e 2 1 + a 2 e 2 2 ( 1 − a 1 ) e 1 1 + ( 1 − a 2 ) e 1 2 p_1 = \frac{a_1 e^1_2 + a_2 e^2_2}{(1-a_1) e^1_1 + (1-a_2) e^2_1} p 1 = ( 1 − a 1 ) e 1 1 + ( 1 − a 2 ) e 1 2 a 1 e 2 1 + a 2 e 2 2 M R S 1 = M R S 2 (contract curve) MRS_1 = MRS_2 \quad \text{(contract curve)} M R S 1 = M R S 2 (contract curve) Part of these learning paths
Examples & Exercises
Consider a 2-person, 2-good economy with a 1 = a 2 = 1 / 2 a_1 = a_2 = 1/2 a 1 = a 2 = 1/2 W = ( 10 , 10 ) W = (10, 10) W = ( 10 , 10 ) e 1 = ( 7 , 3 ) e^1 = (7, 3) e 1 = ( 7 , 3 )
Compute the competitive equilibrium price and allocation directly.
p 1 = a 1 e 2 1 + a 2 e 2 2 ( 1 − a 1 ) e 1 1 + ( 1 − a 2 ) e 1 2 p_1 = \frac{a_1 e^1_2 + a_2 e^2_2}{(1-a_1) e^1_1 + (1-a_2) e^2_1} p 1 = ( 1 − a 1 ) e 1 1 + ( 1 − a 2 ) e 1 2 a 1 e 2 1 + a 2 e 2 2 x i ( p , w i ) = ( a i w i p 1 , ( 1 − a i ) w i ) , w i = p ⋅ e i x^i(p, w_i) = \left(\frac{a_i w_i}{p_1}, (1-a_i) w_i\right), \quad w_i = p \cdot e^i x i ( p , w i ) = ( p 1 a i w i , ( 1 − a i ) w i ) , w i = p ⋅ e i Symmetric economy — equilibrium price p 1 = 1 p_1 = 1 p 1 = 1 ( 5 , 5 ) (5, 5) ( 5 , 5 ) Show proof
e 2 = ( 3 , 7 ) e^2 = (3, 7) e 2 = ( 3 , 7 ) p 1 = 0.5 ⋅ 3 + 0.5 ⋅ 7 0.5 ⋅ 7 + 0.5 ⋅ 3 = 5 5 = 1 p_1 = \frac{0.5 \cdot 3 + 0.5 \cdot 7}{0.5 \cdot 7 + 0.5 \cdot 3} = \frac{5}{5} = 1 p 1 = 0.5 ⋅ 7 + 0.5 ⋅ 3 0.5 ⋅ 3 + 0.5 ⋅ 7 = 5 5 = 1 p 2 = 1 p_2 = 1 p 2 = 1
Wealths are w 1 = 7 + 3 = 10 w_1 = 7 + 3 = 10 w 1 = 7 + 3 = 10 w 2 = 3 + 7 = 10 w_2 = 3 + 7 = 10 w 2 = 3 + 7 = 10
Cobb-Douglas demands are x 1 = ( 0.5 ⋅ 10 / 1 , 0.5 ⋅ 10 ) = ( 5 , 5 ) x^1 = (0.5 \cdot 10/1,\ 0.5 \cdot 10) = (5, 5) x 1 = ( 0.5 ⋅ 10/1 , 0.5 ⋅ 10 ) = ( 5 , 5 ) x 2 = ( 5 , 5 ) x^2 = (5, 5) x 2 = ( 5 , 5 )
They sum to ( 10 , 10 ) = W (10, 10) = W ( 10 , 10 ) = W
In the same economy, find the contract curve and the core, and verify that the competitive equilibrium lies in the core.
M R S 1 = M R S 2 ⟺ x 2 1 x 1 1 = x 2 2 x 1 2 MRS_1 = MRS_2 \iff \frac{x^1_2}{x^1_1} = \frac{x^2_2}{x^2_1} M R S 1 = M R S 2 ⟺ x 1 1 x 2 1 = x 1 2 x 2 2 core = { x on contract curve : u i ( x i ) ≥ u i ( e i ) } \text{core} = \{x \text{ on contract curve} : u_i(x^i) \ge u_i(e^i)\} core = { x on contract curve : u i ( x i ) ≥ u i ( e i )} Contract curve is the diagonal; core is the [ 4.58 , 5.42 ] [4.58, 5.42] [ 4.58 , 5.42 ] Show proof
Since a 1 = a 2 a_1 = a_2 a 1 = a 2 M R S 1 = M R S 2 MRS_1 = MRS_2 M R S 1 = M R S 2 x 2 1 / x 1 1 = x 2 2 / x 1 2 x^1_2/x^1_1 = x^2_2/x^2_1 x 2 1 / x 1 1 = x 2 2 / x 1 2 x 2 = W − x 1 x^2 = W - x^1 x 2 = W − x 1 x 2 1 = x 1 1 x^1_2 = x^1_1 x 2 1 = x 1 1
Endowment utilities are u 1 ( e 1 ) = 21 ≈ 4.58 u_1(e^1) = \sqrt{21} \approx 4.58 u 1 ( e 1 ) = 21 ≈ 4.58 u 2 ( e 2 ) = 21 ≈ 4.58 u_2(e^2) = \sqrt{21} \approx 4.58 u 2 ( e 2 ) = 21 ≈ 4.58
On the diagonal, ( x , x ) (x, x) ( x , x ) u 1 = x ≥ 4.58 u_1 = x \ge 4.58 u 1 = x ≥ 4.58 u 2 = 10 − x ≥ 4.58 u_2 = 10 - x \ge 4.58 u 2 = 10 − x ≥ 4.58 x ∈ [ 4.58 , 5.42 ] x \in [4.58, 5.42] x ∈ [ 4.58 , 5.42 ]
The competitive equilibrium x 1 = ( 5 , 5 ) x^1 = (5, 5) x 1 = ( 5 , 5 )
12. Equilibrium and Welfare