Utility-Maximization
Coconuts MRS = w Budget constraint; slope = w
π*
Labor Output supply demand 0 L*
C*
Given w, RC’s quantity supplied of labor is L* and output quantity demanded is C*. 24 Labor (hours)
Production Possibilities
Coconuts
Ppf’s slope is the marginal rate of product transformation. Increasingly negative MRPT ⇒ increasing opportunity cost to specialization.
Utility-Maximization & ProfitMaximization
Profit-maximization: Coconut and labor markets both clear. – w = MPL – quantity of output supplied = C* – quantity of labor demanded = L* Utility-maximization: – w = MRS – quantity of output demanded = C* – quantity of labor supplied = L*
Coconuts MRS = MPL C* Output Labor 0 24 L* Leisure 24 0 Labor (hours) Leisure (hours) Production function
Robinson Crusoe as a Firm
Now suppose RC is both a utilitymaximizing consumer and a profitmaximizing firm. Use coconuts as the numeraire good; i.e. price of a coconut = $1. RC’s wage rate is w. Coconut output level is C.
Non-Convex Technologies
Coconuts
MRS = MPL. The Pareto optimal allocation cannot be implemented by a competitive equilibrium.
0
24
Labor (hours)
Production Possibilities
Robinson Crusoe’s Preferences
Coconuts More preferred
0
24
Leisure (hours)
Robinson Crusoe’s Preferences
Coconuts More preferred
24
0
Labor (hours)
Robinson Crusoe’s Choice
0
24
Labor (hours)
First Fundamental Theorem of Welfare Economics
A competitive market equilibrium is Pareto efficient if – consumers’ preferences are convex – there are no externalities in consumption or production.
Robinson Crusoe as a Firm
RC’s firm’s profit is π = C - wL. π = C - wL ⇔ C = π + wL, the equation of an isoprofit line. Slope = + w . Intercept = π .
Isoprofit Lines
C=π *+w . L
Utility-Maximization & ProfitMaximization
Profit-maximization: – w = MPL – quantity of output supplied = C* – quantity of labor demanded = L*
π*
0
L*
24
Labor (hours)
Pareto Efficiency
Must have MRS = MPL.
Pareto Efficiency
Coconuts MRS ≠ MP(hours)
Pareto Efficiency
Coconuts MRS ≠ MPL
Preferred consumption bundles. 0 24 Labor (hours)
Pareto Efficiency
Coconuts MRS = MPL
0
24
Labor (hours)
Pareto Efficiency
Coconuts MRS = MPL. The common slope ⇒ relative wage rate w that implements the Pareto efficient plan by decentralized pricing.
π*
C*
π*= C*−wL*
Utility-Maximization
Now consider RC as a consumer endowed with $π* who can work for $w per hour. What is RC’s most preferred consumption bundle? Budget constraint is C =π *+w . L
24
Labor (hours)
Non-Convex Technologies
Do the Welfare Theorems hold if firms have non-convex technologies? The 2nd Theorem does require that firms’ technologies be convex.
Production Possibilities
Coconuts
Production possibility frontier (ppf) Production possibility set
Fish
Production Possibilities
Coconuts
Feasible and efficient Infeasible Feasible but inefficient Fish
Coconuts Higher profit; π1 <π2 <π3
C =π + w L
π3 π2 π1
0 24
Slopes = + w
Labor (hours)
Profit-Maximization
Coconuts Isoprofit slope = production function slope i.e. w = MPL = 1× MPL = MRPL. × Production function Labor Output demand supply 0 RC gets L* 24 Given w, RC’s firm’s quantity demanded of labor is L* and output quantity supplied is C*. Labor (hours)
Resource and technological limitations restrict what an economy can produce. The set of all feasible output bundles is the economy’s production possibility set. The set’s outer boundary is the production possibility frontier.
Non-Convex Technologies
Coconuts
MRS = MPL The common slope ⇒ relative wage rate w that implements the Pareto efficient plan by decentralized pricing.
0
Non-Convex Technologies
Do the Welfare Theorems hold if firms have non-convex technologies?
Non-Convex Technologies
Do the Welfare Theorems hold if firms have non-convex technologies? The 1st Theorem does not rely upon firms’ technologies being convex.
Utility-Maximization & ProfitMaximization Coconuts
MRS = w = MPL C* Given w, RC’s quantity supplied of labor = quantity demanded of labor = L* and output quantity demanded = output quantity supplied = C*.
Second Fundamental Theorem of Welfare Economics