PUBLIC SECTOR ECONOMICS Lecture 3: The Fundamental Theorems of Welfare Economics

The Structure of the Model Assumptions: Two individuals: 1, 2 Two goods: X, Y Two factors of production: K, L Fundamental Elements of the Model: (1) Individual preferences: 1 & 2 get utility from consuming X & Y 1’s utility: U1 = U1(X,Y) 2’s utility: U2 = U2(X,Y) (2) Production technologies: X & Y made from combining K & L according to the production functions X = X(KX,LX) & Y = Y(KY,LY) → Assumed to be constant returns to scale (CRS)

(The Structure of the Model Continued…) (3) Market Clearance: Input and output markets must both clear → X = X1 + X2 & Y = Y1 + Y2 → K* = KX + KY & L* = LX + LY where K* = K1* + K2* & L* = L1* + L2* Why the CRS assumption? Guarantees that if all markets are perfectly competitive (1) Firm revenue equals firm expenditure (on K,L) → There is no economic profit (2) Consumer income equals consumer expenditure (on X,Y) → There is no individual saving

The First Fundamental Theorem of Welfare Economics A competitive economy can achieve a Pareto optimal allocation of resources Necessary conditions for a Pareto optimum: Consumption: Marginal rates of substitution between X & Y must be equal for 1 & 2 Production: Marginal rates of technical substitution between K & L must be equal for production of X & Y Consumption-production: Marginal rates of substitution between X & Y must also equal Marginal rates of transformation between X & Y

The Consumption Condition Intuition: To ensure Pareto-optimality, individuals 1 & 2 must X & Y in such combinations that their willingness to trade small amounts of Y for X are equal If this does not hold, individuals will be able to make trades that make both of them better off Example: Consider initial allocation (X1,Y1) & (X2,Y2) such that at that allocation MRS1X,Y > MRS2X,Y If 1 trades some Y to 2 in exchange for some X, then both individuals increase utility (i.e. move is Pareto superior) Initial allocation cannot have been Pareto-optimal

(The Consumption Condition continued) Pareto superior moves exist as long as MRS1X,Y > MRS2X,Y → Pareto superior moves only cease to exist when MRS1X,Y = MRS2X,Y Edgeworth Box Shows all potential Pareto optimal outcomes Bold line is contract curve All possible Pareto optimal allocations available to economy - If economy is not on contract curve, there is always a Pareto superior move available (A to B)

(The Consumption Condition continued) Further, to maximize utility, individuals consume until MRS1X,Y = Px / PY and MRS2X,Y = Px / PY MRSX,Y is rate at which individual values trade-off between X & Y PX / PY is rate at which market values trade-off between X & Y To maximize utility, these must be equal, otherwise, individuals could trade X for Y and make themselves better off At A, MRSX,Y = Px / PY Because both individuals face same prices (assuming perfect competition) it follows that it must be true that MRS1X,Y = MRS2X,Y

The Production Condition Marginal rates of technical substitution (MRTS) between K & L must be equal MRTS shows firm’ willingness to trade small amounts of K and L must be equal Bold line represents all efficient allocations of K & L Points on bold line correspond to points on PPF for X & Y If input markets are perfectly competitive, MRTSK,L are equal for X & Y

(The Production Condition continued) Further, to maximize utility, individuals consume until MRTSXK,L = PL / PK and MRTSYK,L = PL / PK MRTSXK,L is rate at which firms values trade-off between K & L PL / PK is rate at which firms value trade-off between K & L To maximize profit, these must be equal, otherwise, firms could trade K for L and make themselves better off → At A, MRTSK,L = PL / PK Because all firms face same input prices (assuming perfect competition) it follows that it must be true that MRTSXK,L = MRTSYK,L

The Consumption–Production Condition Combines previous two conditions MRS between two goods must be equal MRTS between two factors must be equal → economy must be on PPF for X & Y Requires that MRSX,Y = MRTX,Y MRTX,Y is slope of PPF for X & Y Economy must be on consumption contract curve (equal MRS → consumption efficiency) Economy must choose specific point on contract curve at which slope equals slope of PPF (i.e. MRSX,Y = MRTX,Y)

Perfect Competition and Production–Consumption Efficiency Under perfect competition PX = MCX & PY = MCY → MRTX, Y = (PX/PY) / (MCX/MCY) But we also know that MRSX,Y = PX/PY → Hence, MRS1X, Y = MRS2X, Y = MRTX, Y → A perfectly competitive market reaches its UPF when all of the assumptions for a well-functioning economy hold → This is simply a re-statement of the first fundamental theorem of welfare economics Note: This condition only holds if all consumers and firms face the same prices

The Second Fundamental Theorem of Welfare Economics Considers how society feels about end-results equity → Asks whether society is happy at C' C is on UPF so it is efficient, because all markets are competitive it satisfies process equity which should aid social mobility May not satisfy equity concerns as 2 has a lot more than 1 Society could accept C' but usually does not Usually taxes and transfers get more equitable outcome Second welfare theorem says that a new Pareto-optimal outcome can be achieved given existing resources