Increasing Returns and Economic Efficiency

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Increasing Returns and Economic Efficiency
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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).

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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).

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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).
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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).

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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).

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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.

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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).

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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.
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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.

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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.
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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.)

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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.

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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. Pigous 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.

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• 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.

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• 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.

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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
  Brouwers 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.

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• Consider the following continuous mapping ? 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.

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• (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.

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• Since the mapping ? 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.

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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 gt Ih iff Ag/Ah gt Mg/Mh.

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• From (3.8) and (3.10) For any market equilibrium
  P (3.11) MRSgh gt Mg/Mh iff Ig gt 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 gt 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.

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• 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/consumpti
  on 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.

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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-competiti
  ve 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.

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Average-cost pricing, increasing returns, and
optimal output in a model with home and market
production

• Yew-Kwang NgDept. of Economics, Monash
  University
• Dingsheng Zhang
• Dept. of Economics, Monash University
• Institute for Advanced Economic Studies, Wuhan
  University

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The Model

• Consider an economy with M identical consumers.
  Each of them has the following decision problem
  for consumption, working, and home production.
• Max
• s.t.

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The above optimization problem gives the
following solutions
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• We assume that the market structure is
  monopolistic competition. The production function
  of good r is
• The first-order condition for the monopolist to
  maximize profit with respect to output level or
  price implies that

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The general equilibrium values of the various
variables are
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Comparative statics analysis
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Equilibrium utility value
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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 consumers problem is
• S.t.

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• In addition, denoting the subsidy rate per unit
  of market product as , the zero-profit
  condition for each firm is
• Governments budget constrain is

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we can get the equilibrium level of utility as
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• where

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• 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 ,
  is given by

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• 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,

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• 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.

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• 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.

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• 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.

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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.