Lecture notes on Public Goods and Externalities

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Public Goods and Externalities

Measuring Welfare
 CS: difference between WTP and amount paid
 EV: income required at old prices to maintain new utility
 CV: income required at new prices to maintain old utility
Quasilinear Preferences
 Demand for nonlinear good independent of m – no income
effect except for low incomes – indifference curve vertical shift
 EV=CV=∆CS – aggregate indirect utility = sum of indirect utilities
k 1 ∂C k
 M comes from fact consumer owns firm: m=π =Σ pk y −C ( y , … y ) ⇒ = pk
∂y
1 k
 u=u ( x , … x )+ π −Σ k p k ⇒ in eqbm consumption = production and m=π
 u=u ( x1 , … , x k ) −C ( x 1 , … , x k ) ie benefit – cost so MB= p=MC ie P-efficient
Effect of Tax (not lump sum)
 Wedge between consumer and producer price prevents profitable
trades so DWL
 Raises revenue for government
 Δ p< 0 , Δ x< 0
Δp p
 u' ( x ) =p ⇒ u'' ( x )== <0 recall ε D = Δ x p >0
Δ x x εD Δp x
ΔC C

' ''
C ( x )= p ⇒C ( x )= = > 0 recall ε s= Δ x p > 0
Δ x xεs Δp x
'' −pΔx '' p Δx
 Δ p+ τ= Δ x u ( x )= ; Δ p=Δ x C ( x )=
x εD x εs
Δ xτ −1 τ 2 εD εs
 DWL=
2
=
2 p
px () (
ε s−ε D )
=tax ⋅ sales ⋅ constant

 Increasing in: square of tax, sales, elasticities
 Burden depends on who has relatively more inelastic curve
Optimal Commodity Taxation
k
 Raise revenue, min DWL: min Σ k DW Lk s . t . Σ k τ k x ≥ G
τi/ pi ε j
 Assume: linear D, constant MC, XED=0, no lump sum tax ⇒ = ⇒ tax inelastic
τ j / p j εi
goods more, but all goods in proportion to required revenue. Optimal taxes regressive
Monopolist Inefficiency
−1
 Constant MC⇒ ε s=∞, mark up of μ ⇒ DWL= μΔx
2
¿ x
p ( 1+ε D )
¿ ∗¿
¿∗¿
¿ −1 p −1 ¿∗¿ ε ¿
 ¿∗¿= p + μ= ⇒ DWL= Δ x= p ¿¿ ¿
εD 2 εD 2
p
 As ε D evaluated at p¿∗¿, x
¿∗¿¿
¿

Cost-Benefit Analysis
Σ ( Bt −Ct )
 NPV =
( 1+r )t
 Can't value at market prices as gen eq effects, non-marginal changes of p and q, market
failure / shadow prices, non-market valuations
General Equilibrium Effects
 Project will directly affect some quantities (inputs), and indirectly affect all others
 U =u ( x1 , … , x k ) −Σ p k x k + Σ p k y k −C ( y 1 ,… , y k )

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k k k k
dU ∂u d x dx dy ∂C d y
 =Σ k −Σ +Σ −Σ k
d pk ∂x d p k d p k d pk ∂ y d pk
∂u ∂C
 dU =Σ ( y k −x k ) d p k + Σ
k k
∂x
k(− pk d x k −Σ
∂y )k (
− pk d y k = Δ p + Δ q
)
 Price change: if y =x just a transfer, no ∆W – do not value unless i) social value of income
different for gainers / losers (depends on W ( u ) ) ii) if y k ≠ x k then terms of trade effect: gain if
price increase on exported good (Stolper-Samuelson)
 Quantity change: if perfect economy ie u' = p=c ' then no social value – ignore gen eq
effect in other markets; if imperfect, positive social value to expanding for u' >c ' , positive
value to contracting if u' <c ' - need to know valuations and quantity changes
Market Failure and Shadow Prices
 Monopoly: p> MC ⇒ MC is resource cost, p−MC is transfer to monopolist, not real RC
 Unemployed labour: if unemployed people, shadow price is op. cost of employing worker; if
otherwise employed, shadow price = wage = VMPL in other job; if unemployed, w=0? Or
value of leisure foregone?
 General principle: identify opportunity cost
Non-Marginal Changes
 Project increases output by non-marginal amount x ' → x' ' ⇒ p' → p' '
'' '' ' ( p' − p' ' ) ( x ' ' −x ' )
 B = social value of extra output: p ( x −x ) + ie Δ TR+ Δ CS
2
 Commercial decisions do not account for ∆CS – very small for marginal changes
Social Discount Rate
' '
 Value of marginal consumption today ¿ u ( xt ) e ⇒−log u ( x t ) + ρt [ ]
−ρt

' '' ''
d −1 ∂ u ( x t ) dx −u ( x t ) dx −u ( x t ) dx 1
 = ' + ρ= ' + ρ= ' xt ⋅ +ρ
dt u ( xt ) ∂ x dt Time u (Rate dtMU of
x t )at which uElasticity
( x t ) of dt Consumption
xt
preference consumption falls MU of growth rate
consumption



Social Discount Rate
''
−u ( xt ) dx 1
 r= '
xt ⋅ +ρ
u ( xt ) dt x t
''
−u ( x t )
 '
xt : determines how fast MU falls: prefer equality over space and time
u ( xt )
dx 1
 : less weight on future generations if they are much richer: C growing fast
dt x t
 ρ : impatience, extinction probability, generational bias
Externalities
 Action of an agent directly affects utility of another. Pecuniary externalities relate to prices
and do matter with frictions, but don’t exist under perfect competition
 Taxonomy:
Rival Non-Rival
Excludable Private Good Club Good
Non-Excludable Commons Good Public Good
 Consider: scale, scope, time, distribution, reversibility
Pure Public Goods Model
 x , y are public and private goods, u=v i ( x ) + y i s .t . M i ≥ x + y i
 MRS=v 'i ( x )=M Bi ( x )=1
Under-provision