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

2. 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)

3. (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

4. 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

5. 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

6. (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)

7. (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

8. 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

9. (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

10. 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)

11. 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

12. 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