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Increasing Returns and Economic Efficiency

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Increasing Returns and Economic Efficiency Powered By Docstoc
					Increasing Returns and
  Economic Efficiency
            More Seriously
• Learning costs, indivisibilities giving rise
  to sizable fixed costs. Information and
  knowledge in production makes IR
  prevalent (Wilson 1975, Radner &
  Stiglitz 1984, Arrow 1995).
• Empirical evidence for IR (Ades &
  Glaeser 1999, Antweiler & Trefler 2002).
     IR and Economic Analysis
• Largely ignored in the core, esp. in GE
  analysis.
• Frequently discussed in IO, development
  economics, international trade.
• Surveys: Yang & S. Ng (1998); Yang
  (2001), Cheng & Yang (2003).
• Reading: G. Sun (forthcoming).
‘One clear conclusion is that there are many
important areas of economics in which the
recognition of increasing returns makes a big
difference, and changes the established
wisdom significantly. …we have not yet
reached the point of diminishing returns in the
study of increasing returns: there is a long way
to go, and the results of the work yet to be
completed will be interesting’ (Geoffrey Heal,
1999).
        Different Types of IR
• IR at the firm level, economies of scale.
• IR at the industry level due to external
  economies (Marshall 1920, Chipman 1970,
  Romer 1986).
• IR at the economy level due to economies of
  specialization (ES)/division of labour (DL)
  (Smith, Young, Rosen, Yang, Buchanan).
• IR at the world level (Ethier 1979, Chandra
  et al. 2002).
    Different Analytical Sources of IR
• Property of the production function;
  indivisibilities.
• External economies due to knowledge
  transmission, economies internal to another
  industry, etc.
• Higher productivity from the use of more
  intermediate inputs (Ethier 1979, 1982).
• ES at the individual level (Smith, Yang).
Many Traditional Results Have to be
       Drastically Revised
• Significant IR at the firm level is
  inconsistent with perfect competition.
• Perfect competition also inconsistent
  with the virtually omnipresent product
  differentiation.
• But allowing for non-perfect
  competition play havocs to many
  traditional results.
  A.Neutrality of money may not hold
• Even no time lags, money illusion, and other
  frictions, just non-perfect competition may make a
  change in nominal aggregate demand affect either
  the price level (the monetarist case) or the real
  output (the Keynesian case).
• Expectation wonderland (outcome depends on
  expectation which will be self-fulfilling) and
  cumulative expansion/contraction are also possible,
  partly explaining some real-world phenomena like
  business cycles, importance of confidence, and
  difficulties of prediction (Ng 1977, 1980, 1986,
  1992, 1998, 1999; Ng & Wu 2002).
In the non-traditional cases, there exists
an interfirm macroeconomic externality
where the expansion by each firm benefits
other firms apart from the familiar income
multiplier effects. This is an area where
welfare economics, macroeconomics and
its micro-foundation intersect, an area still
not adequately studied.
       The crux of the difference
• Demand Side: A horizontal demand curve
  cannot shift left or right, only up or down. But
  an upward shift means an increase in price.
  With no time lags, money illusion, this leads
  to a similar shift in MC.
• Supply Side: MC upward-sloping under
  perfect competition; may be horizontal or
  downward sloping (esp. with IR) under non-
  perfect competition.
The analysis is based on a
representative firm but it takes account
of the influence of macro variables and
the interaction (including ‘infinite
number of feedback loops’) with the
rest of the economy, using a simplified
general-equilibrium method.
   Moreover, a fully general-equilibrium
        analysis is used to show
• (1) for any (exogenous) change in cost or demand,
  there exists a hypothetical representative firm whose
  response accurately represents that of the whole
  economy in aggregate output and average price;
• (2) a representative firm defined by a simple
  weighted average can be used as a good
  approximation of the whole economy to any
  economy-wide change in demand and/or costs that
  does not result in drastic inter-firm changes. (See Ng
  1986, App. 3I.)
  B. Pecuniary external effects may
  have real efficiency implications

• Even where the supply/demand
  analysis is still applicable, if the supply
  curve is downward sloping due to
  whatever source of IR, the traditional
  analysis showing the absence of
  inefficiency of pecuniary ext. effects is
  no longer valid.
      C. Market equilibrium no longer
                Pareto Optimal
• Well-known that IR may give rise to efficiency
  problems (e.g. Arrow 1987, 2000, Guesnerie 1975,
  Heal 1999, Quinzii 1992, Villar 1996).
• Pigou (1920) advocated tax/subsidy on goods with
  up/downward sloping supply curves. Further
  discussion (reprinted in AEA 1952) revealed
  problems. Pigou’s example: a non-congested, wide
  but uneven road and a congested, narrow but good
  road, to illustrate the over-use of the narrow road
  with increasing costs. Knight (1924): failure of
  pricing the congested road. With optimal pricing, no
  overuse.
• Dixit & Stiglitz’ (1977) show that no
  general conclusion can be made.
• The more specific models of Heal (1980,
  1999) show that the combination of
  imperfection competition and IR leads to
  the over-serving of large markets and
  under-serving of small markets.
• See also Spence 1976 on optimal product
  variety.
• Even abstracting from monopolistic output
  restriction by assuming AC-pricing from
  contestability, Section 2 shows that goods with
  higher degrees of IR are under-expanded.
• Subsidies on goods with (high degrees of) IR
  financed by taxes on goods with non and lower IR
  may increase efficiency.
• But may open up a flood-gate of rent-seeking
  activities. Perhaps it is optimal to continue to
  pretend that IR do not exist. Ha!
• Unlikely true for all issues; otherwise, policies like
  encouraging the development of the great western
  region in China does not make sense.
        Existence of AC-Pricing
          Equilibria with IR
• Theorem 1 (Brown-Heal generalized): A
  productively efficient AC-pricing equilibrium
  exists.
• Proof: While the production possibility set need
  not be convex, the production possibility frontier
  (super-surface) is topologically equivalent to a
  compact and convex set. From Brouwer’s fixed
  point theorem, any continuous mapping of a set
  homeomorphic to a compact and convex set onto
  itself possess a fixed point. Thus, any continuous
  mapping of the PPF has a fixed point.
• Consider the following continuous mapping Ф of
  PPF into PPF: (G1, …, Gg)0 → (W1/WR, …, WR-
  1 /WR)0 ; (P1, …, PG)0 → (Gd1, …, Gdg)1 → (G1, …,
  Gg)2, where
• (i) (G1, …, Gg)0 is an arbitrary point on the PPF.
• (ii) (W1/WR, …, WR-1/WR)0 is the set of relative
  resource prices determined by the common (to all
  firms using the same pair of resources) marginal
  rates of technical substitution as specified in (2.7)
  above at the point Gg(R1g, …, RRg) = G0g; g = 1, …,
  G, i.e. the same point as (G1, …, Gg)0.
• (iii) (P1, …, PG)0 = product prices at AC, i.e. P0g =
  Cg(W1, …, WR, Gg)/Gg; g = 1, …, G, at the given
  production levels given by (G1, …, Gg)0.
• (iv) (Gd1, …, Gdg)1 is the market demand for the
  various goods, i.e. Gdg = G1g + …, GIg; g = 1, …, G,
  where each Gig is the individual utility-maximization
  quantity of the g good demanded by individual i at the
  set of product prices (P1, …, PG)0 and resource prices
  (W1/WR, …, WR-1/WR)0.
• (v) (G1, …, Gg)2 is the intersection of the ray through
  the point (Gd1, …, Gdg)1 and the PPF.
• Since the mapping Ф is continuous and the PPF is
  homeopmorphic to a compact and convex set, the
  mapping has a fixed point which is denoted as (G1, …,
  Gg)*. At this fixed point, (G1, …, Gg)0 = (Gd1, …, Gdg)1
  = (G1, …, Gg)2 = (G1, …, Gg)*. Hence, demand for goods
  (G1, …, Gg)1 equals supply (G1, …, Gg)0 at the product
  prices (P1, …, PG)*, the AC of producing the various
  goods at (G1, …, Gg)0. The equilibrium relative resource
  prices (W1/WR, …, WR-1/WR)* is the common MRTS
  specified in (2.7). This production point gives equilibrium
  values of resource demand Rrg; r = 1, …, R; g = 1, …, G
  satisfying (2.11). Finally, the individual demands for
  products Gig; i = 1, …, I; g = 1, …, G total to satisfy
  (2.10). Since the production point is on the PPF, it is
  productively efficient. This completes the proof.
    Efficiency of Encouraging Goods with
             High Degrees of IR
•   (3.1)    Pg = Ag for all good g,
•   Define the (local) degree of IR
•   (3.2)    Ig  - (Ag/Gg)Gg/Ag
•   (3.5)    Uig/Uih = Pg/Ph (Utility max.)
•   Pareto optimality: Uig/Uih = Fg/Fh = Mg/Mh
•   From (3.1) and (3.5), a market equilibrium:
•   (3.8)    Uig/Uih = Ag/Ah
•   (3.9)    Ig  - (Ag/Gg)Gg/Ag = 1 – Mg/Ag
•   (3.10) Ig > Ih iff Ag/Ah > Mg/Mh.
• From (3.8) and (3.10): For any market equilibrium P:
  (3.11)    MRSgh > Mg/Mh iff Ig > Ih.
• The (absolute) slope of PPF in the g/h plane equals Mg/Mh
  (with good g on the horizontal axis).
• If good g has a higher degree of IR than good h, the
  market-equilibrium MRSgh > slope of PPF.
• A movement down the (downward-sloping) PPF involving
  more good g and less good h must reach a higher
  indifference curve/surface.
• Theorem 2A: At an AC-pricing market equilibrium, if
  the degree of IR for good g is larger than that for good
  h, a point of higher efficiency (Pareto-superior) could
  be reached by increasing good g and decreasing h.
• Next, a cost-benefit analysis is used to show
• Theorem 2B: At an AC-pricing market
  equilibrium, if the production/consumption of
  a good with a lower IR is decreased to allow for
  a     corresponding        increase       in   the
  production/consumption of another good with
  a       higher         IR,        holding      the
  consumption/production of other goods
  unchanged, the aggregate net benefits of the
  change is positive.
• A specific example confirms the above results.
• But, Information; rent-seeking.
         Joint paper with D.S. Zhang
• The analysis of economies of specialization at the individual
  level by Yang & Shi (1992) and Yang & Ng (1993) is
  combined with the Dixit & Stiglitz (1977) analysis of
  monopolistic-competitive firms to show that, even if both
  the home and the market sectors have IR and there are no
  pre-existing taxes, it is still efficient to tax the home sector
  to finance a subsidy on the market sector to offset the under-
  production of the latter due to the failure of price-taking
  consumers to take account of the effects of higher
  consumption in reducing the average costs and hence prices,
  through IR or the publicness nature of fixed costs.
• But offset by environmental concerns.
     Average-cost pricing,
increasing returns, and optimal
  output in a model with home
    and market production
                Yew-Kwang Ng
    Dept. of Economics, Monash University

                 Dingsheng Zhang
    Dept. of Economics, Monash University
   Institute for Advanced Economic Studies,
                 Wuhan University
                 The Model
Consider an economy with M identical
consumers. Each of them has the following
decision problem for consumption, working,
and home production.
Max:                              
          1                
       u l       [   
                                    
                            xr 1 ] 1 [        xj 2 ]   2
                      rR                jJ
s.t.      pr xr  w(1  l   l j )
       rR                   jJ
                    lj  a
               xj 
                       c
  The above optimization problem
   gives the following solutions:
  lh 
          a           2 (1     )
                   l
       1  2         2   (1  2 )
                (1   2 )
       m
          a[  2   (1   2 )]
                        2
xr                                  1          1

       [  2   (1   2 )] pr ( s1 ps )
                                   1 1   n   1 1

                                   
We assume that the market structure is
 monopolistic competition. The production
 function of good r is:

            X r  (lr  A) / b
The first-order condition for the monopolist
 to maximize profit with respect to output
 level or price implies that
   The general equilibrium values of
         the various variables are :
           b(n  1 )                 1 A(n  1)
     p                         X
            1 (n  1)                bn(1  1 )
           1 A(n  1)                 A(n  1 )
   x                            lr 
        bn(1  1 ) M                  n(1  1 )
         2 (1     )                    a
   l                               lh 
        2   (1   2 )                1  2
     M  2 (1  1 )                      (1   2 )
n                          1 m 
    A[  2   (1   2 )]           a[  2   (1   2 )]
 Comparative statics analysis:

 n    2 (1  1 )
                          0,
M A[  2   (1   2 )]
n     M  2 (1  1 )
    2                       0,
A   A [  2   (1   2 )]
n    M  2 (1  1 )
                          0,
 A[  2   (1   2 )]
n    M  2 (1  1 )(1   2 )
                                0,
     A[  2   (1   2 )]2



n              M  2
     1                        0
1      A[  2   (1   2 )]

 m        2 (1   2 )
                            0,
  a[  2   (1   2 )]2
m         (1   2 )
    2                       0,
a   a [  2   (1   2 )]

m            
                             0,
 2 a[  2   (1   2 )]2
          Equilibrium utility value:


                     
             1      2
                    
ue  l1  n x m xh
                                                                 
                                                                   
               2                   2          1                2
           
    2 1  b  M  a
      1
                                      c A              (1   2 )
                                                               
                                                            1
                                                        1      2
    (1     )(1   ) [  2   (1   2 )]
                                                                                        
                                                                                           
                                                                                        1
    [ M  2  A(  2   (1   2 ))] {M  2 (1  1 )  A1[  2   (1   2 )]}
To analyse the welfare properties,
we introduce the government to the
              model
Assume that the tax rate of per unit home
labor is  , and then consumer’s problem is
                                                      
                       [ xr ] [ x j ]
                                  
             1           1                  2
 max u  l                            1                  2

                       rR                 jJ
S.t.
        pr xr   l j  w(1  l   l j )
       rR       jJ                jJ
                  lj  a
             xj 
                     c
In addition, denoting the subsidy rate per
  unit of market product as  , the zero-
  profit condition for each firm is

         pr X r  (b   ) X r  A
Government’s budget constrain is

           Mm lh  n X
we can get the equilibrium level of
             utility as
                     
             1      2 
ue  l1  n x m xh
                                                                              
               2                      2            2            1               2
          
   2 1  (b   ) (  1)
     1
                                            M   
                                                   a        c A           (1  2 )
                                                       1
                                                    1     2
    (1     )(1  ) [ 2   (1  2 )]
                                                                                              
                                                                                             1
    [ M 2  A( 2   (1  2 ))] {M 2 (1  1 )  A1[ 2   (1  2 )]}
                          
                          2
   B(b   ) (  1)
where
                                                               
                                                                 
               2
           
B   2 1  M  a
      1                          2  
                                   c A          1
                                                     (1   2 )   2
                                                                          (1     )(1   )
                                    
                                 1
    [  2   (1   2 )]    1      2
                                          [ M  2  A(  2   (1   2 ))]
                                                                   
                                                                      
    {M  2 (1  1 )  A1[  2   (1   2 )]}                  1
The effect of a change in tax rate  on the
 equilibrium value of utility with respect to
 the tax rate, and with the subsidy rate at
 whatever level that is allowed by the
 government budget constraint as  varies,
 evaluated at   0 , is given by

 due       Bb  [ M 2 (1  1 )  A1 ( 2   (1  2 )]
                                                             0
 d  0        12[M 2  A( 2   (1  2 ))]
This means that, starting from the original position
without any tax/subsidy, a tax on home production
which finances for a subsidy on market production
increases utility, ignoring administrative costs and
any possible side effects, such as rent-seeking
activities triggered by the subsidy. Since all firms
just break-even in equilibrium, we may base our
welfare comparisons simply on the utility levels
alone. We thus have,
Proposition 1: In our model with both home
  and market production under the conditions
  of increasing returns and average-cost
  pricing, a subsidy, if not excessive, on
  market production financed by a tax on home
  production improves efficiency even if the
  initial position involves no tax distortion,
  ignoring administrative costs and any
  possible side effects.
We extend the above model to allow for
 different sectors of market goods that may
 have different degrees of elasticity of
 substitution and different degrees of
 increasing returns (through different
 values of the fixed cost and marginal cost),
 we have similar results.
Proposition 2: In our model with both home
  and market production under the conditions
  of increasing returns and average-cost
  pricing with two sectors of different fixed
  costs, elasticities of substitution, and degree
  of importance in preference, it is efficient to
  tax the sector with lower fixed costs, higher
  elasticity of substitution and/or higher
  degree of importance in preference and
  subsidize the other.
      Concluding Remarks
Our conclusions’ applicability to the real
 economy is subject to important
 qualifications. First, the government may
 not have the information to differentiate
 which goods should be taxed (and by how
 much) and which subsidized. Secondly,
 we have not considered other factors
 (including rent seeking) causing imperfect
 efficiency in the real economy.

				
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