Post on 16-Dec-2015
Lecture 5: Externalities (chapter 9)
• Relation to lectures 1-4
• Negative externalities
• Positive externalities: public goods and empathy
• Efficiency wage relation
• Nonconvexities
• Risk and uncertainty
Aim of lecture 5
• Show how positive and negative externalities can be included in general equilibrium models: public goods, empathy, pollution
• With specific emphasis on the effects of the introduction of an efficiency wage relation and social security arrangements
Relation to lectures 1-4
• Competitive equilibrium can be represented in various formats
• Time dependency can be introduced through commodity classification (implicit) or by explicitly describing dynamics of producer and consumer behavior (OLG and dynastic models)
• Agents are selfish (not fully so in dynastic models)
• Goods are not simultaneously effective in consumption and production
• Mathematical programs are assumed to be convex
• Until now: public consumption levels are set exogenously (lecture 3) and taxes are levied to finance this
• No welfare justification for level of public consumption
Externalities
• Individual utilities and technology sets directly depend on demand and supply by other agents
• Positive externalities:– Non-rival consumption
– Empathy
– Interdependent consumption/production
– Efficiency wage relation
– Economies of scope and learning-by-doing
• Negative externalities:– Pollution: effects on health and production
Negative externalities: pollution
• Interpretation of pollution as use of inputs instead of as joint output: “goods” instead of “bads”
• Then, analysis on positive externalities applies
• Recall lecture 3: previously free resources that are used as are priced: “double dividend” as efficiency of resource use is restored and revenue enables government to reduce other (distortionary) taxes
Welfare program with non-rival commodities
• Where first-order conditions include:
, 0, 0max , subject to
i i gx g x i i i ii
i g j ii j i
i g i
j j
u x g
x x y pg xy Y
0
0i i i i i
i gi
u g g
p x
Lindahl equilibrium
• Negishi equilibrium where:– Consumers agree on provision of non-rival commodities:
– Consumers jointly finance them:
– Each consumer pays share
– Since share falls as marginal utility of income increases, the rich pay more than the poor
• Remarks:– Public goods not desired by high income groups will have to be
financed by low income groups, which may not be feasible
– Implementation of Lindahl equilibrium: willingness of pay by the consumers. Underreporting will lead to undersupply
– If level is determined, financing can be implemented by direct taxes
iip
i gg x
ik ( * )ik k i k i ikp p u g
Interdependency in utilities: empathy
Where first-order conditions include:
0, 0, all i, 0, all i and h, all j
1
max
subject to
( ,( ,..., )) ( )
( )
i i ih ju x v y i ii
i i i i im i
ih h ih
i j ii j i
u
u W x v v
v u
x y
( )
j j
p
y Y
if 0
if 0ih i i ih ih
i i ih ih
W v v
u
Interdependence in consumptions: external effects
0, all i, 0, all i and h, all j max ( , )
subject to
( )
( )
i ih jx y i i i ii
i j ii j i
ih h ih
j j
W x
x y p
x
y Y
Where first order conditions include:
,ih k i i hkW
Interdependency in utilities and consumptions
• Utility of other consumers cannot be observed; therefore, consumers imagine being the other person. Rawls’ “veil of ignorance”
• In a dynamic context, this other person could also represent the agent himself at an older age: savings result as transfers to this other person
• Note: consumption is a flow variable: consumers can also value the presence of stocks of commodities being available for (non-rival) consumption, such as nature parks. This requires representation of empathy within dynamic models of lecture 4
Interdependence in production
0, all i, , 0, 0, all j max ( )
subject to
( )
( )
( , ) 0
( )
i g g j ji i ix y y g y i
i j g g ii j i
j g j
j j j
g g g
u x
x y y y p
g y
F y g
F y y
0
Efficiency wage relation
• Efficiency wage: endowments need to be produced:
i (xi)
xi
wlow
xO
( )i ix
Migration model with efficiency wage
• Welfare program (compare lecture 1):
• Budgets:
i i in 0;x 0,all i i i i i
i ii i i i i
i i
max n u ( x )
subject to
n x n (x ) (p)
n =N
i i i ipx p ( x ) T
Efficiency wage
• In absence of transfers
• Hence, wage worker receives which is spent on consumption and decomposes into a payment for utility and a payment for work efficiency • Worker pays for his own health, education and nutrition• No external effects (e.g. public health, public safety).
i ii i i i i i
i i
upx x p x p ( x )
x x
i ip ( x )ipx
Efficiency wage and nonconvexities
• Shape of labor production function is decisive
• If is concave and homogeneous, there will be migration to different destinations until marginal productivities become equal. No need for interventions.
• If is concave with a set-up cost: specialize on a few: workers who are identical ex ante will end up differently ex post. (“better a few strong workers than many weak ones”). This is the Dasgupta (1997) result.
• If jobs are integer-valued ( ), the central planner sets the optimal values and uses premia and rations to implement solution
( )i ix
( )i ix
( )i ix
Social security
• If in welfare program, and all budgets consolidate:
• this represents social security: there will be transfers across destinations and efficiency is preserved
• Equilibrium utilities and consumption levels, are not equal, even if preferences are identical but the differences only serve to feed the workers better:
i 0
i ii ip x p
i i i0 k ik
ik ik
u( x ) ( x )p p , for x 0
x x
Efficiency wage and taxes
• Proportional tax on income:
• If endowments are given, no distortion
• However, in present formulation, consumer choice of consumption level is affected.
• In general, if consumption falls below critical level, productivity of consumers falls, and economy is trapped in underdevelopment equilibrium (Mirrlees, 1975).
(1 )( ( ) )i i i i ipx p x
Non-convexities in production
• Within firm nonconvexities:– , with F a CRTS production function. Even if F is
not concave, divisibility ensures ensures compact, nonempty production set
• Indivisibilities at firm level: – instead of continuous n, , with profit maximization program:
.
Supply response not usc, convex valued, so equilibrium may not exist
– Note: welfare program with indivisibilities has solution.
• Nonconvexities at above-firm level– Large firms
– Firms may need non-rival input supplied or used in non-convex way
0 1, 0max ( , )n v pF v n cv
0,1
0,1 ,max
y Ypy
Above-firm non-convexities
• Recall welfare program with production and non-rival good:
• If are strictly quasiconvex (sqc), we are back in previous situation• If , program is non-convex and and have
to be set centrally• If , then program is non-convex and all
the g have to be set equal to by the central planner
0, all i, , 0, 0, all j max ( )
subject to
( )
( )
( , ) 0
( )
i g g j ji i ix y y g y i
i j g g ii j i
j g j
j j j
g g g
u x
x y y y p
g y
F y g
F y y
0
(.), (.)j gF F (.) is sqc, but (.) is notj gF F
(.) is sqc but (.) is sqc in onlyg j jF F y
gygy
gy
Risk and uncertainty
• Welfare program with groups i and possible destinations s:
• is the probability of group i to be in s
i i i sn 0;x 0,all i i is is is
i s i sis is is is is
s is i i
max n u ( x )
subject to
n x n (x ) (p)
n =N ( )
is is iP n N
Risk and uncertainty (continued)• Risks
– Idiosyncratic risk: probabilities materialize fully in each period
– Aggregate risk: not all probabilities materialize (extreme: there is only a single draw of the distribution in each period)
• In both cases, risks are equal for all individuals: is s in P N
s is is i sP 0,all s,n 0,x 0,all i,s,y 0 i is is is
i s i sis is is is is
s is i i
is s
max n u ( x )
subject to
n x y n (x ) (p)
n =N ( )
n P
i is
s s s
N ( )
0 P P ( y ) ( )