General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core. General Equilibrium (without Production) or Exchange (Chapter 31)

Transcription

1 General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core General Equilibrium (without Production) or Exchange (Chapter 31)

2 General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core General Equilibrium Events in one market have effects on other markets (spillovers) Demand for x depends upon prices of complements, substitutes; income Supply of x depends upon factor prices Previously, we ve taken these as given doing partial equilibrium analysis ut its important to understand interdependence of markets general equilibrium analysis Partial equilibrium analysis says that competitive markets yield efficient outcomes is this still true in general equilibrium?

3 General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core General Equilibrium Our approach: Simple environment the entire economy 2 kinds of goods 2 people Focus on exchange bstract away from production of new goods Give people endowments Specify preferences llow them to trade Make predictions about behavior of utility-maximizers Evaluate welfare

4 General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core Endowment Economy Consumers and ; goods 1 and 2 Endowments: ω = (ω 1, ω 2 ) and ω = (ω 1, ω 2 ) Example: ω = (6, 4) and ω = (2, 2) This means total endowment of good 1 is ω 1 + ω 1 = = 8 and of good 2 is ω 2 + ω 2 = = 6

5 General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core llocations Endowment represents where people start, but through trade, their allocations may change General allocation or consumption: x = (x 1, x 2 ) and x = (x 1, x 2 ) (x, x ) is feasible if it uses at most the aggregate endowment: x 1 + x 1 ω 1 + ω 1 and x 2 + x 2 ω 2 + ω 2 Helpful graphical tool: Edgeworth ox llows us to simply depict all feasible allocations

6 eworth ox General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core Edgeworth ox The Endowment llocation 2 O The endowment allocation O 6 8 ( 6, 4) ( 2, 2)

7 Edgeworth ox General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core Edgeworth ox Feasible Reallocations x 1 O 2 x 2 x 2 O x 1 1 1

8 General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core How do we think about equilibrium? In partial equilibrium analysis: Treat each good separately Find p and q that equate supply and demand ut this is general equilibrium analysis: where do supply and demand come from? and can trade with each other For everything to be balanced, the amount that gives up has to equal amount that receives (for each good, and vice versa) In other words Supply = Demand for each good This will determine prices for each good How do we find supply and demand curves? Go back to utility maximization problem Need to specify preferences to do this

9 General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core Utility maximization Preferences are given Given prices for each good, endowment bundle serves as income Can write down budget constraint p 1 x 1 + p 2 x 2 p 1 ω 1 + p 2 ω 2 Solve utility maximization problem Gives you optimal allocation, as a function of price ratio x 1 ω 1 > 0 means person demands more of good 1 x 1 ω 1 < 0 means person is willing to supply good 1 Key question: what prices will make it so that demand exactly as much of each good as supplies?

10 rences of General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core Consumer s Preferences dding Preferences to the ox x 2 For consumer. More preferred O 1 x 1

11 rences of General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core Consumer s Preferences dding Preferences to the ox x 2 For consumer. More preferred O 1 x 1

12 rences of General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core Consumer s Preferences dding Preferences to the ox x 1 For consumer. 1 O More preferred x 2

13 tting both of them together General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core x 2 Putting Them oth Together Edgeworth s ox x 1 1 O O 1 x1 x 2

14 General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core Pareto-improving allocations Given a particular allocation, a Pareto-improving allocation improves the welfare of at least one consumer without reducing the welfare of another. How do we depict Pareto-improving allocations in the Edgeworth box?

15 tting both of them together General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core x 2 Pareto-Improving llocations Edgeworth s ox x 1 1 O O 1 x1 x 2

16 reto-improving allocations General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core x 2 Pareto-Improving llocations Pareto-Improvements x 1 1 O O The set of Paretoimproving allocations 1 x 2 x 1

17 General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core Pareto-optimal allocations n allocation is Pareto-optimal if it is feasible and there is other feasible allocation that is a Pareto-improvement over it. In other words, there is no was to make anyone better off without making someone worse off. The set of all Pareto-optimal allocations is called the contract curve.

18 areto-optimal llocation General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core oth and are worse off Pareto-Optimal llocation Pareto-Optimality is strictly better off but is strictly worse off is strictly better off but is strictly worse off oth and are worse off

19 reto-optimal llocations General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core The Set of Pareto-Optimal llocations x 2 x 1 Pareto-Optimality ll the allocations marked by a are Pareto-optimal. 1 O O The contract curve 1 x1 x 2

20 General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core Pareto-optimal llocations From the figures, we can see that an allocation at which the indifference curves of the two consumers are tangent must be Pareto-optimal Tangency implies they have the same slope What is the slope of an indifference curve? The Marginal rate of substitution (MRS)! Condition for Pareto-optimality: MRS = u (x 1,x 2 ) x 1 u (x 1,x 2 ) x 2 We also require feasibility: = u (x 1,x 2 ) x 1 u (x 1,x 2 ) x 2 = MRS x 1 + x 1 = ω 1 + ω 1 and x 2 + x 2 = ω 2 + ω 2

21 General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core Example Identifying Pareto-optimal allocations Recall total endoments: ω1 + ω 1 = = 8 and ω2 + ω 2 = = 6 Let u (x1, x 2 ) = ln(x 1 ) + 2 ln(x 2 ) and u (x1, x 2 ) = ln(x 1 ) + 2 ln(x 2 ). MRS of consumer : Similarly, MRS = MRS = 1 x 1 2 x 2 1 x 1 2 x 2 = x 2 2x 1 = x 2 2x 1 So a Pareto-optimum is a feasible allocation for which x 2 2x 1 = x 2 2x 1

22 General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core Example Finding all Pareto-optimal allocations (deriving the contract curve) We can simplify tangency condition to: x2 x1 = x 2 x 1 Recall endowment/feasibility requirement: x 1 + x 1 = 8 and x 2 + x 2 = 6 Re-write tangency condition, substituting x1 = 8 x 1 and x2 = 6 x 2 : x2 x1 = 6 x 2 8 x1 or x2 = 3 4 x 1 This is the equation of the contract curve. In this case, it s just the diagonal of the rectangle

23 General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core Core llocations How can we narrow down our prediction about the outcome resulting from trade? We ve derived the contract curve: x 2 = 3 4 x 1 Note that clicker options D and E are both on this curve. However, Pareto Optimal, but given her endowment, (ω = (2, 2), Person would never agree to trade to allocation E = (6,4.5,2,1.5). We need to restrict attention to Pareto-optimal allocations that are Pareto-improvements over the initial endowment. These allocations are called core allocations, or the core the set of all PO allocations that are welfare improving for both consumers relative to their own endowments n allocation will be in the core if it is feasible and it is not blocked by any consumer

24 re llocations General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core x 2 x 1 Core llocations The Core Pareto-optimal trades blocked by 1 O O 1 x1 Pareto-optimal trades blocked by x 2

25 e llocations General vs. Partial Equilibrium The Edgeworth ox The Contract Curve The Core x 1 x 2 The Core The Core Pareto-optimal trades not blocked by or are the core. 1 O O 1 x1 x 2