destabilizing effects of market size in the dynamics of ...kmatsu...destabilizing effects of market...
TRANSCRIPT
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 1 of 29
Destabilizing Effects of Market Size in the Dynamics of Innovation
Kiminori Matsuyama
Northwestern University
Philip Ushchev
HSE University, St. Petersburg
2020-09-28; 7:01:04 PM
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 2 of 29
Introduction
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 3 of 29
Market Size Effect on the Dynamics of Innovation β’ Many existing studies on the market size effect on innovation & long-run growth. β’ Little is known about the market size effect on patterns of fluctuations in innovation. β’ In existing models of endogenous innovation cycles, market size merely alters the
amplitude of fluctuations without affecting the nature of fluctuations. o Due to CES homothetic demand system for innovated products, monopolistically
competitive firms sell their products at an exogenous markup rate o Procompetitive effect of market size is missing
β’ We extend a model of endogenous innovation cycles with a more general homothetic demand system, which allows for the procompetitive effect. o Judd (1985; section 4) o H.S.A. demand system
β’ We find that a larger market size/innovation cost ratio, which forces the innovators to
face more elastic demand and to sell their products at lower markup rates through the procompetitive effect, has destabilizing effects in the dynamics of innovation
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 4 of 29
Judd (1985) Dynamic Dixit-Stiglitz model, with symmetric CRS CES technology, where the masses of competitively and monopolistically supplied input varieties change over time through innovation, diffusion, and obsolescence o Innovators pay a one-time innovation cost to introduce a new variety. o Innovators keep monopoly power for a limited time. Then, their innovations become
supplied competitively. Thatβs when innovations reach their full potential. o This creates the force for temporary clustering of innovations. o All existing varieties, subject to idiosyncratic obsolescence shocks.
Judd (1985; Sec.3); Continuous time and monopoly lasting for 0 < ππ < β o Delayed differential equation (with an infinite dimensional state space) o For ππ > ππππ > 0, the economy alternates between the phases of active innovation
and of no innovation along any equilibrium path for almost all initial conditions.
Judd (1985; Sec.4); a simple yet rich set of dynamics, incl. endogen. fluctuations o Discrete time and one period monopoly for analytical tractability o 1D state space (the mass of competitive varieties inherited from past innovation
determines how saturated the market is) o Dynamics governed by a 1D PWL (skewed V-) map, fully characterized o Properties depend on the βdelayed impact of innovationsβ, a constant number,
determined solely by (exogenously) constant price elasticity, hence invariant of the market size and innovation cost.
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 5 of 29
H.S.A. (Homothetic with a Single Aggregator) Demand System The market share of each input depends solely on its single relative price, defined as its own price divided by the common input price aggregator
πππ‘π‘(ππ)ππ(π©π©ππ)
ππππ(π©π©ππ)πππππ‘π‘(ππ)
= π π οΏ½πππ‘π‘(ππ)π΄π΄(π©π©ππ)
οΏ½
β’ π π :β++ β β+: the market share function, decreasing in the relative price, βgross
substitutesβ with π π (0) = β & π π (β) = 0. β’ π΄π΄(π©π©): the common price aggregator against which the relative price of each input is
measured. Linear homogenous in π©π©, defined by
οΏ½ π π οΏ½πππ‘π‘(ππ)π΄π΄(π©π©ππ)
οΏ½ππππΞ©π‘π‘
β‘ 1
β’ ππ(π©π©): the unit cost function (the price index), related to π΄π΄(π©π©ππ) as:
lnππ(π©π©) = lnπ΄π΄(π©π©) βοΏ½ οΏ½οΏ½π π (ππ)ππ dππ
β
ππ(ππ)π΄π΄(π©π©)
οΏ½ ππππΞ©
+ πππππππ π ππ
Notes: β’ CES and translog are special cases. β’ π΄π΄(π©π©) β ππππ(π©π©) (with the sole exception of CES), π΄π΄(π©π©) captures the cross-effects in the demand system ππ(π©π©) captures the productivity consequences of price changes
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 6 of 29
Monopolistic Competition under H.S.A. Price elasticity of demand for an input depends only on its relative price, ππ οΏ½πππ‘π‘(ππ)
π΄π΄(π©π©ππ)οΏ½, where
ππ(π§π§) β‘ 1 βπ π β²(π§π§)π§π§π π (π§π§)
> 1
β’ ππ(β) is constant under CES β’ Under increasing ππ(β), Marshallβs 2nd law of demand, consistent with the
Procompetitive effect of market size and entry
Main Results: Dynamics of the Judd model under H.S.A.: β’ still governed by a 1D-PWL (skewed V-) map, hence remain equally tractable. β’ Delayed impact of innovations, still a constant number (with the same range of the
value if ππ(β) is increasing), but now depends on the market size/innovation cost ratio. β’ A larger market size/innovation cost ratio, by reducing the markup rate through the
procompetitive effect, increases the delayed impact, with destabilizing effects in the dynamics of innovation, o Under the log-concavity of ππ(β) β 1, i.e., if ππ(β) is βnot too convex.β o in two parametric families with choke prices, βGeneralized Translogβ and
βConstant Pass-Through,β even though the log-concavity does not hold.
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 7 of 29
Innovation Cycle under CES: Revisiting Judd (1985)
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 8 of 29
Time: ππ β {0, 1,2, β¦ } Representative Household: supply L units of labor (numeraire), consume the single perishable consumption good, πΆπΆπ‘π‘, with πππ‘π‘πΆπΆπ‘π‘ = πΏπΏ. No need to specify intertemporal preferences. With no means to save in the model, the interest rate adjusts endogenously to force the household to spend its income each period. Competitive Final Goods Producers: assemble differentiated inputs using symmetric CES with gross substitutes
πΆπΆπ‘π‘ = πππ‘π‘ = πΉπΉ(π±π±ππ) = ππ οΏ½οΏ½ [π₯π₯π‘π‘(ππ)]1β1ππππππ
Ξ©π‘π‘οΏ½
ππππβ1
(ππ > 1),
Unit cost function: πππ‘π‘ = ππ(π©π©ππ) =1πποΏ½β« [πππ‘π‘(ππ)]1βππππππΞ©π‘π‘ οΏ½
11βππ
Demand for a Differentiated Input: π₯π₯π‘π‘(ππ) =
[πππ‘π‘(ππ)]βπππΏπΏ
β« [πππ‘π‘(ππβ²)]1βππππππβ²Ξ©π‘π‘
Sets of differentiated inputs: change due to innovation, diffusion, obsolescence Ξ©π‘π‘ = Ξ©π‘π‘ππ + Ξ©π‘π‘ππ: Set of all differentiated inputs available in ππ, partitioned into: Ξ©π‘π‘ππ: Set of new inputs introduced & sold exclusively by the innovators for one period. Ξ©π‘π‘ππ: Set of competitively supplied inputs, which were innovated before ππ.
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 9 of 29
Production & pricing of different. inputs: ππ units of labor per unit of each variety πππ‘π‘(ππ) = ππ = ππππ , π₯π₯π‘π‘ππ(ππ) β‘ π₯π₯π‘π‘ππ ππππππ ππ β Ξ©π‘π‘ππ β Ξ©π‘π‘
πππ‘π‘(ππ) =ππ
1 β 1/ππβ‘ ππππ, π₯π₯π‘π‘ππ(ππ) β‘ π₯π₯π‘π‘ππ ππππππ ππ β Ξ©π‘π‘ππ β Ξ©π‘π‘
ππππ
ππππ = 1 β1ππ
< 1,π₯π₯π‘π‘ππ
π₯π₯π‘π‘ππ= οΏ½1 β
1πποΏ½βππ
> 1,πππππ₯π₯π‘π‘ππ
πππππ₯π₯π‘π‘ππ= οΏ½1 β
1πποΏ½1βππ
β‘ ππ β (1, ππ)
Note: in equilibrium, each variety faces the demand curve,
π₯π₯π‘π‘(ππ) =πΏπΏοΏ½πππ‘π‘(ππ)οΏ½
βππ
πππ‘π‘ππ(ππππ)1βππ + πππ‘π‘ππ(ππππ)1βππ=
πΏπΏπππ‘π‘(ππ)1βππ
οΏ½πππ‘π‘(ππ)οΏ½βππ
and πππ‘π‘πΏπΏ =
1πππ‘π‘
= ππ[πππ‘π‘ππ(πππ‘π‘ππ)1βππ + πππ‘π‘ππ(πππ‘π‘ππ)1βππ]1
ππβ1 =ππππ
(πππ‘π‘)1
ππβ1
where πππ‘π‘ππ (πππ‘π‘ππ) is the measure of Ξ©π‘π‘ππ (Ξ©π‘π‘ππ) and
πππ‘π‘ β‘ πππ‘π‘ππ + πππ‘π‘ππ ππβ β’ One competitive variety has the same effect with ππ > 1 monopolistic varieties. β’ Impact of an innovation reach its full potential after its innovator loses monopoly β’ Past innovations discourage more than contemporaneous innovations, which would
give an incentive for temporal clustering of innovations
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 10 of 29
Introduction of New Varieties: Innovation cost per variety, F
πππ‘π‘ππ β₯ 0; (ππππ β ππ)π₯π₯π‘π‘ππ =πππππ₯π₯π‘π‘ππ
ππ =πππππ₯π₯π‘π‘ππ
ππππ β€ πΉπΉ Complementary Slackness: Either net profit or innovation is zero in equilibrium Resource Constraint:
πΏπΏ = πππ‘π‘ππ(πππ₯π₯π‘π‘ππ) + πππ‘π‘ππ(πππ₯π₯π‘π‘ππ + πΉπΉ) = πππ‘π‘ππ(πππππ₯π₯π‘π‘ππ) + πππ‘π‘ππ(πππππ₯π₯π‘π‘ππ) = πππ‘π‘(πππππ₯π₯π‘π‘ππ)
βΉ πππ‘π‘ β‘ πππ‘π‘ππ +πππ‘π‘ππ
ππ = πππππ₯π₯ οΏ½πΏπΏπππππΉπΉ
,πππ‘π‘πποΏ½ βΊ πππ‘π‘ππ = πππππ₯π₯ οΏ½πΏπΏπππΉπΉ
β πππππ‘π‘ππ , 0οΏ½
β’ When innovations are active (πππ‘π‘ππ > 0), o one competitive variety crowds out ΞΈ > 1 innovations. o π₯π₯π‘π‘ππ = ππππ(πΉπΉ ππβ ) and π₯π₯π‘π‘ππ = (ππ β 1)(πΉπΉ ππβ ); independent of πΏπΏ, which affect only how
much innovation takes place. β’ πΏπΏ (πππππΉπΉ)β : the saturation level of competitive varieties, which kills incentive to
innovate. Idiosyncratic Obsolescence Shock: πΏπΏ β (0,1), the survival rate
πππ‘π‘+1ππ = πΏπΏ(πππ‘π‘ππ + πππ‘π‘ππ) = πΏπΏπππππ₯π₯ οΏ½πΏπΏπππΉπΉ + (1 β ππ)πππ‘π‘
ππ ,πππ‘π‘πποΏ½
Define πππ‘π‘ β‘ οΏ½πππππππΏπΏοΏ½ πππ‘π‘ππ: (the normalized measure of) competitive varieties (=the market
share of the competitive varieties, πππ‘π‘ < 1)
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 11 of 29
Dynamical System
πππ‘π‘+1 = ππ(πππ‘π‘) β‘ οΏ½πππΏπΏ(πππ‘π‘) β‘ πΏπΏ(ππ + (1 β ππ)πππ‘π‘) ππππππ πππ»π»(πππ‘π‘) β‘ πΏπΏπππ‘π‘ ππππππ
πππ‘π‘ < 1πππ‘π‘ > 1
πΏπΏ β (0,1), Survival rate of each variety
ππ β‘ οΏ½1 β 1πποΏ½1βππ
β (1, ππ), increasing in ππ Market share multiplier due to the loss of monopoly power by its innovator ππ β 1 > 0: Delayed impact of innovations πππ‘π‘ππ =
πΏπΏππππππ
πππ‘π‘; πππ‘π‘ππ =πΏπΏπππππππππ₯π₯{1 β πππ‘π‘ , 0};
πππ‘π‘ β‘ πππ‘π‘ππ +πππ‘π‘ππ
ππ= πΏπΏ
πππππππππππ₯π₯{1,πππ‘π‘};
πππ‘π‘ =ππππ
(πππ‘π‘)1
1βππ; πππ‘π‘πΏπΏ
= 1πππ‘π‘
= (πππ‘π‘)1
ππβ1ππππ
Key Features of Dynamical System: β’ 1D-PWL noninvertible (a skewed V-shaped) map β’ ππ depends solely on ππ; independent of L, F and ππ. β’ πΏπΏ/πΉπΉ merely affects the amplitude of fluctuations.
πππΏπΏ(πππ‘π‘) β‘ πΏπΏππ β πΏπΏ(ππ β 1)πππ‘π‘
45ΒΊ
πππ‘π‘+1 πΏπΏππ
ππβ πππ»π»(πππ‘π‘) = πΏπΏπππ‘π‘ πΏπΏ
πππ‘π‘
πππΏπΏ(πΏπΏ)
O 1 Active Innovation
No Innovation
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 12 of 29
A Unique Attractor: β’ Stable steady state for πΏπΏ(ππ β 1) < 1, globally attracting For πΏπΏ(ππ β 1) < 1, unstable steady state, but the trajectory trapped into the red box. β’ Stable 2-cycle for πΏπΏ2(ππ β 1) < 1 < πΏπΏ(ππ β 1), to which almost always converging β’ Robust chaotic attractor with 2m intervals (m = 0, 1,β¦) for πΏπΏ2(ππ β 1) > 1 Effects of πΉπΉ (for ππ = 2.5) (In courtesy of L.Gardi and I.Sushko) Notes: Most existing examples of chaos
in econ are not attractors; βPeriod three does not imply chaotic attractors.β Most existing examples of chaotic
attractors in econ are not robust, since most applications use smooth dynamical systems. Judd model has a robust chaotic attractor due to its regime-switching (nonsmooth) feature.
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 13 of 29
In the (πΉπΉ,π½π½)-plane Endogenous fluctuations with β’ a higher Ο βΉ a higher ππ (past innovations are more discouraging than
contemporaneous innovations, stronger incentive for temporal clustering) β’ a higher Ξ΄ (more current innovation survives to discourage future innovation). (In courtesy of L.Gardi and I.Sushko) Note: In general, an attractor of a skewed V-shaped map could be a stable p-cycle or a robust chaotic attractor with p-intervals (where p can be any natural number.) But, this can be ruled out in the Judd model because of ππ < ππ. Even with a stable steady state, slower convergence with a higher Ο (ππ) and/or a higher Ξ΄.
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 14 of 29
Innovation Cycle under H.S.A.
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 15 of 29
Symmetric Homothetic Demand with A Single Aggregator (Symmetric H.S.A.):
πππ‘π‘(ππ)π₯π₯π‘π‘(ππ)ππ(π©π©ππ)πππ‘π‘
=πππ‘π‘(ππ)ππ(π©π©ππ)
ππππ(π©π©ππ)πππππ‘π‘(ππ)
= π π οΏ½πππ‘π‘(ππ)π΄π΄(π©π©ππ)
οΏ½
o π π :β++ β β+: the market share function, decreasing in the relative price, βgross substitutesβ with π π (0) = β & π π (β) = 0.
o π΄π΄(π©π©): the common price aggregator against which the relative price of each input is measured. Linear homogenous in π©π© defined implicitly by
οΏ½ π π οΏ½πππ‘π‘(ππ)π΄π΄(π©π©ππ)
οΏ½ππππΞ©π‘π‘
= 1
By construction, the shares of all inputs are added up to one. Notes: o Integrability is ensured: cf. Matsuyama-Ushchev (2017; Proposition 1). o Special cases: CES for π π (π§π§) = π§π§1βππ (ππ > 1), Translog for π π (π§π§) = max{β log(π§π§) , 0}
o lnππ(π©π©) = lnπ΄π΄(π©π©) β β« οΏ½β« π π (ππ)ππ dππβππ(ππ)π΄π΄(π©π©)
οΏ½ ππππΞ© + πππππππ π ππ
π΄π΄(π©π©) β ππππ(π©π©) (with the sole exception of CES), π΄π΄(π©π©) captures the cross-effects in the demand system ππ(π©π©) captures the productivity consequences of price changes
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 16 of 29
Pricing of Competitive Inputs: πππ‘π‘(ππ) = ππ = ππππ for all ππ β Ξ©π‘π‘ππ Pricing of Monopolistic Inputs: πππππ₯π₯ (πππ‘π‘(ππ) β ππ)π₯π₯π‘π‘(ππ) = οΏ½1 β
πππππ‘π‘(ππ)
οΏ½ π π οΏ½πππ‘π‘(ππ)π΄π΄(π©π©ππ)
οΏ½ πΏπΏ FOC: πππ‘π‘(ππ) οΏ½1 β
1ππ(πππ‘π‘(ππ) π΄π΄(π©π©ππ)β )
οΏ½ = ππ, where ππ(π§π§) is the price elasticity, given by
ππ(π§π§) β‘ 1 βπ§π§π π β²(π§π§)π π (π§π§)
> 1 βΊ π π (π§π§) = οΏ½ππ(ππ) β 1
ππππππ
β
π§π§> 0
(A1): π§π§ππ
β²(π§π§)ππ(π§π§)
> 1 β ππ(π§π§) for all π§π§ βΊ π§π§ οΏ½1 β 1ππ(π§π§ )
οΏ½ is increasing βΊ π π (π§π§)ππ( π§π§)
is decreasing (A1) guarantees the symmetric pricing: πππ‘π‘(ππ) = πππ‘π‘ππ for all ππ β Ξ©π‘π‘ππ Define π§π§π‘π‘ππ β‘ πππ‘π‘ππ π΄π΄(π©π©ππ)β ; π§π§π‘π‘ππ β‘ πππ‘π‘ππ π΄π΄(π©π©ππ)β . Then, Profit Maximization: π§π§π‘π‘ππ οΏ½1 β
1πποΏ½π§π§π‘π‘
πποΏ½οΏ½ = π§π§π‘π‘ππ π§π§π‘π‘ππ& π§π§π‘π‘ππ move together under (A1)
Maximized Profit: οΏ½1 β π§π§π‘π‘
ππ
π§π§π‘π‘πποΏ½ π π (π§π§π‘π‘ππ)πΏπΏ =
π π (π§π§π‘π‘ππ)πποΏ½π§π§π‘π‘
πποΏ½πΏπΏ decreasing in π§π§π‘π‘ππ under (A1)
Adding Up Constraint: πππ‘π‘πππ π (π§π§π‘π‘ππ) + πππ‘π‘πππ π (π§π§π‘π‘ππ) = 1.
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 17 of 29
Innovation (Free Entry): Complementarity Slackness Condition
πππ‘π‘ππ β₯ 0; π π (π§π§π‘π‘ππ)ππ(π§π§π‘π‘ππ)
πΏπΏ β€ πΉπΉ
For πππ‘π‘ππ > 0,
π§π§π‘π‘ππ = π§π§ππ; π§π§π‘π‘ππ = π§π§ππ β‘ π§π§ππ οΏ½1 β1
πποΏ½π§π§πποΏ½οΏ½
where π§π§ππ is defined by πποΏ½π§π§πποΏ½π π οΏ½π§π§πποΏ½
β‘πΏπΏπΉπΉ
π§π§ππ increasing in πΏπΏ/πΉπΉ under (A1). From πππ‘π‘πππ π οΏ½π§π§πποΏ½ + πππ‘π‘πππ π οΏ½π§π§πποΏ½ = 1,
πππ‘π‘ππ = ππ οΏ½1
π π οΏ½π§π§πποΏ½β πππ‘π‘πποΏ½ > 0,π€π€βππππππ ππ β‘
π π οΏ½π§π§πποΏ½π π οΏ½π§π§πποΏ½
> 1.
For πππ‘π‘ππ = 0, π§π§π‘π‘ππ β₯ π§π§ππ & π§π§π‘π‘ππ β₯ π§π§ππ . Hence, πππ‘π‘ππ =
1π π οΏ½π§π§π‘π‘
πποΏ½β₯ 1
π π οΏ½π§π§πποΏ½ and,
πππ‘π‘ππ = πππππ₯π₯ οΏ½ππ οΏ½1
π π οΏ½π§π§πποΏ½β πππ‘π‘πποΏ½ , 0οΏ½ = πππππ₯π₯ οΏ½
πΏπΏπποΏ½π§π§πποΏ½πΉπΉ
β πππππ‘π‘ππ , 0οΏ½
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 18 of 29
Idiosyncratic Obsolescence Shock: πΏπΏ β (0,1), the survival rate
πππ‘π‘+1ππ = πΏπΏ(πππ‘π‘ππ + πππ‘π‘ππ) = πΏπΏπππππ₯π₯ οΏ½ππ
π π οΏ½π§π§πποΏ½+ (1 β ππ)πππ‘π‘ππ ,πππ‘π‘πποΏ½
Define πππ‘π‘ β‘ π π οΏ½π§π§πποΏ½πππ‘π‘ππ (the normalized measure of) competitive varieties (=market share of all competitive varieties, when πππ‘π‘ < 1) Dynamical System
πππ‘π‘+1 = ππ(πππ‘π‘) β‘ οΏ½πππΏπΏ(πππ‘π‘) β‘ πΏπΏ(ππ + (1 β ππ)πππ‘π‘) ππππππ πππ»π»(πππ‘π‘) β‘ πΏπΏπππ‘π‘ ππππππ
πππ‘π‘ < 1πππ‘π‘ > 1
where
ππ β‘π π οΏ½π§π§πποΏ½π π οΏ½π§π§πποΏ½
=π π οΏ½ π§π§ππ οΏ½1 β 1
πποΏ½ π§π§πποΏ½οΏ½οΏ½
π π οΏ½π§π§πποΏ½> 1;
πποΏ½π§π§πποΏ½π π οΏ½π§π§πποΏ½
β‘πΏπΏπΉπΉ
Notes: β’ Dynamics are still governed by the same 1D PWL (skewed V-shape) map, as before. β’ Its slopes still depend only on Ξ΄ and ππ, but ππ now depend on π π (β) and πΏπΏ/πΉπΉ. Note: We know that π§π§ππ and π§π§ππ are both increasing in πΏπΏ/πΉπΉ under (A1).
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 19 of 29
(A2): ππ(π§π§) is non-decreasing. β’ (A2) implies (A1). β’ As its relative price goes up, the demand curve for an input may become more price-
elastic, but never become less price-elastic. (Marshallβs 2nd Law of Demand) β’ By imposing (A2), we depart from CES only to allow for the Procompetitive Effect,
Strategic Complementarity in pricing, and (firm-level) Incomplete Pass-through, since A strictly increasing (decreasing) ππ(π§π§) implies i) A larger market size has a pro-competitive (anti-competitive) effect, since πποΏ½π§π§πποΏ½ is
strictly increasing (decreasing) in πΏπΏ/πΉπΉ. F.O.C. under heterogenous costs implies ii) Pricing of monopolistic varieties are strategic complements (strategic substitutes).
0 <dlogπππ‘π‘ππ(ππ)dlogπ΄π΄(pπ‘π‘)
=Ξ
1 + Ξ >()1,
where Ξ β‘ π§π§π‘π‘(ππ)ππβ²οΏ½π§π§π‘π‘(ππ)οΏ½
οΏ½πποΏ½π§π§π‘π‘(ππ)οΏ½β1οΏ½πποΏ½π§π§π‘π‘(ππ)οΏ½> ( 0.
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 20 of 29
The Main Propositions
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 21 of 29
Question #1: What is the range of ππ? For CES, ππ β (1, ππ), where ππ = 2.718 β¦ Proposition 1. Under (A2), ππ β (1, ππ), where ππ = 2.718 β¦
Thus, the types of asymptotic behaviors observable are the same with CES. Sketch of Proof: Under (A2), ππ(β ) is non-decreasing. Hence,
ππ β‘π π οΏ½π§π§πποΏ½π π οΏ½π§π§πποΏ½
= exp οΏ½οΏ½ππ(ππ) β 1
ππππππ
π§π§ππ
π§π§πποΏ½ β€ exp οΏ½οΏ½πποΏ½π§π§πποΏ½ β 1οΏ½οΏ½
ππππππ
π§π§ππ
π§π§πποΏ½ =
exp οΏ½οΏ½πποΏ½π§π§πποΏ½ β 1οΏ½ logοΏ½π§π§ππ
π§π§πποΏ½οΏ½ = exp οΏ½logοΏ½
π§π§ππ
π§π§πποΏ½1βπποΏ½π§π§πποΏ½
οΏ½ = οΏ½1 β1
πποΏ½π§π§πποΏ½οΏ½1βπποΏ½π§π§πποΏ½
< ππ
Intuition: Under (A2), price elasticities can become only smaller at lower prices. Hence, when a monopolistic variety becomes competitively priced, an increase in its market share caused by a drop in the price could only be smaller compared to the case of CES. Hence, it has the same upper bound, ππ < ππ. Remark: Without (A2), ππ could be arbitrarily large (see Appendix B), which could generate stable cycles of any period, or robust chaotic attractors with any number of intervals. By imposing (A2), we abstain ourselves from showing more exotic dynamic behaviors.
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 22 of 29
Question #2: How does ππ depend on πΏπΏ/πΉπΉ? For CES, it is independent of πΏπΏ/πΉπΉ. Proposition 2: If ππ(β ) β 1 is monotone and log-concave over an interval containing οΏ½π§π§ππ , π§π§πποΏ½, ππ is increasing in πΏπΏ/πΉπΉ. If one of the monotonicity and the log-concavity conditions is strict, ππ is strictly increasing in πΏπΏ/πΉπΉ. Corollary: Under (A2), the strict log-concavity of ππ(β ) β 1 is sufficient for ππ being strictly increasing in πΏπΏ/πΉπΉ.
Remark: The log-concavity of ππ(β ) β 1 is weaker than the concavity of ππ(β ) β 1, and hence the concavity of ππ(β ). In fact, if π π (β ) is thrice-continuously differentiable, ππ(β ) β 1
is strictly log-concave iff ππβ²β²(β ) <οΏ½ππβ²(β )οΏ½
2
ππ(β )β1, that is, when ππ(β ) is βnot too convex.β
Intuition: Under (A1), a higher πΏπΏ/πΉπΉ leads to an increase in both π§π§ππ and π§π§ππ . Under (A2), this leads to an increase in both πποΏ½π§π§πποΏ½ and πποΏ½π§π§πποΏ½. The former implies a lower markup rate, and hence the price drop due to the loss of monopoly is smaller, which contributes to a smaller ππ. The latter implies the market share responds more to the price drop, which contributes to a larger ππ. If ππ(β ) is βnot too convex,β the former effect could not dominate the latter, so that ππ is increasing in πΏπΏ/πΉπΉ.
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 23 of 29
Some Parametric Families within H.S.A.
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 24 of 29
Ex.1: Multiplicatively Perturbed CES with Linear Elasticity Function
π π (π§π§) = (π§π§ exp(πππ§π§))1βππ; (ππ > 1; ππ > 0)
βΉ ππ(π§π§) β 1 = (ππ β 1)(1 + πππ§π§) is strictly increasing and strictly log-concave. Hence, β’ From Proposition 1, ππ < ππ; β’ From Proposition 2, ππ is strictly increasing in πΏπΏ πΉπΉβ .
More explicitly,
ππ β‘π π οΏ½π§π§πποΏ½π π οΏ½π§π§πποΏ½
= οΏ½1 β1
πποΏ½π§π§πποΏ½οΏ½1βππ
expοΏ½1 βππ
πποΏ½π§π§πποΏ½οΏ½
As πΏπΏ πΉπΉβ β 0, π§π§ππ β 0, πποΏ½π§π§πποΏ½ β ππ, and ππ β οΏ½1 β 1πποΏ½1βππ
< ππ; As πΏπΏ πΉπΉβ β β, π§π§ππ β β, πποΏ½π§π§πποΏ½ β β, and ππ β ππ.
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 25 of 29
Ex.2; Multiplicatively Perturbed CES with Linear Fractional Elasticity Function
π π (π§π§) = π§π§1βππ οΏ½1 + π§π§
2οΏ½βππ
(ππ > 1; ππ > 0)
βΉ ππ(π§π§) = ππ +πππ§π§
1 + π§π§
is strictly increasing and strictly concave. Hence, β’ From Proposition 1, ππ < ππ. β’ From Proposition 2, ππ is strictly increasing in πΉπΉ/πΏπΏ.
More explicitly,
ππ β‘π π οΏ½π§π§πποΏ½π π οΏ½π§π§πποΏ½
= οΏ½1 β1
πποΏ½π§π§πποΏ½οΏ½1βππ
οΏ½1 β1
πποΏ½π§π§πποΏ½π§π§ππ
1 + π§π§πποΏ½βππ
As πΏπΏ πΉπΉβ β 0, π§π§ππ βΆ 0, πποΏ½π§π§πποΏ½ βΆ ππ, and ππ βΆ οΏ½1 β 1πποΏ½1βππ
< ππ;
As πΏπΏ πΉπΉβ β β, π§π§ππ βΆ β, πποΏ½π§π§πποΏ½ βΆ ππ + ππ, and ππ βΆ οΏ½1 β 1ππ+ππ
οΏ½1βππβππ
< ππ.
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 26 of 29
Ex.3: Generalized Translog:
π π (π§π§) = πΎπΎ οΏ½1 βππ β 1ππ
log οΏ½π§π§π½π½οΏ½οΏ½
ππ
= πΎπΎ οΏ½1 β ππππ
log οΏ½π§π§π§π§Μ οΏ½οΏ½ππ
for π§π§ < π§π§Μ β‘ π½π½ππππ
ππβ1,
( π½π½ > 0, ππ > 0; ππ > 1. )
ππ(π§π§) = 1 +ππ β 1
1 β ππ β 1ππ log οΏ½π§π§π½π½οΏ½
= 1 βππ
log οΏ½π§π§π§π§Μ οΏ½> 1, for π§π§ < π§π§Μ β‘ π½π½ππ
ππππβ1
is strictly increasing in π§π§ β (0, π§π§Μ ) with the range (1,β), and hence satisfying (A2).
β’ Translog is a special case, where ππ = 1. β’ CES is the limit case, where ππ β β, while holding π½π½ > 0 and ππ > 1 fixed, β’ πποΏ½π§π§πποΏ½ is strictly increasing in πΏπΏ πΉπΉβ β (0,β) with the range, (1,β).
ππ β‘π π οΏ½π§π§πποΏ½π π οΏ½π§π§πποΏ½
= οΏ½1 +1ππ log οΏ½1 β
1πποΏ½π§π§πποΏ½
οΏ½1βπποΏ½π§π§πποΏ½
οΏ½
ππ
< οΏ½1 +1πποΏ½
ππ
< ππ,
strictly increasing in πΏπΏ πΉπΉβ β (0,β) with the upper bound οΏ½1 + 1πποΏ½ππβ ππ, as ππ β β.
Notes: β’ ππ(π§π§) β 1 is not log-concave. Yet, ππ is strictly increasing in πΏπΏ πΉπΉβ . The log-concavity is
sufficient, but not necessary for ππ to be increasing in πΏπΏ πΉπΉβ . β’ With ππ > 1, ππ > 2 for a sufficiently large πΏπΏ πΉπΉβ and hence πΏπΏ(ππ β 1) > 1 for a
sufficiently large πΏπΏ, causing the instability of the steady state.
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 27 of 29
Ex.4: Constant Pass-Through: For a positive constant, Ξ> 0,
π π (π§π§) = πΎπΎ οΏ½ππ β (ππ β 1) οΏ½π§π§π½π½οΏ½ΞοΏ½1 Ξβ
= πΎπΎππ1 Ξβ οΏ½1 β οΏ½π§π§π§π§Μ οΏ½
ΞοΏ½1 Ξβ
, with π§π§Μ β‘ π½π½ οΏ½ππ
ππ β 1οΏ½1 Ξβ
βΉ ππ(π§π§) =1
1 β οΏ½1 β 1πποΏ½ οΏ½π§π§π½π½οΏ½
Ξ =1
1 β οΏ½π§π§π§π§Μ οΏ½Ξ > 1
strictly increasing with limπ§π§βοΏ½Μ οΏ½π§
ππ(π§π§) = β and limππβ0
ππ(ππ) = 1.
β’ The pass-through rate, ππ lnππππ
ππ lnππ= 1
1+Ξ< 1, is constant in this example.
β’ CES is the limit case, where Ξ β 0, while holding π½π½ > 0 and ππ > 1 fixed, β’ π§π§ππ and π§π§ππ both strictly increasing in πΏπΏ πΉπΉβ , with π§π§ππ β π§π§Μ and π§π§ππ β π§π§Μ , as πΏπΏ πΉπΉβ β β.
ππ β‘π π οΏ½π§π§πποΏ½π π οΏ½π§π§πποΏ½
=
β£β’β’β’β‘1 β οΏ½
π§π§πππ§π§Μ οΏ½
Ξ
1 β οΏ½π§π§πππ§π§Μ οΏ½
Ξ
β¦β₯β₯β₯β€1 Ξβ
=
β£β’β’β’β‘1 β οΏ½
π§π§πππ§π§Μ οΏ½
Ξ(1+Ξ)
1 β οΏ½π§π§πππ§π§Μ οΏ½
Ξ
β¦β₯β₯β₯β€1 Ξβ
< [1 + Ξ]1 Ξβ < ππ
strictly increasing in πΏπΏ πΉπΉβ , with the upper bound, [1 + Ξ]1 Ξβ β ππ, as Ξ β 0. β’ ππ(π§π§) β 1 is not log-concave. Yet, ππ is strictly increasing in πΏπΏ πΉπΉβ . The log-concavity is
sufficient, but not necessary for ππ to be increasing in πΏπΏ πΉπΉβ . β’ With 0 < Ξ < 1, ππ > 2 for a sufficiently large πΏπΏ πΉπΉβ and hence πΏπΏ(ππ β 1) > 1 for a
sufficiently large πΏπΏ, causing the instability of the steady state.
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 28 of 29
Concluding Remarks
-
Destabilizing Effects of Market Size in the Dynamics of Innovation
Page 29 of 29
β’ In existing models of Endogenous Innovation Cycles, market size alters the amplitude of fluctuations without affecting the patterns of fluctuations.
β’ Due to the CES homothetic demand system, monopolistically competitive firms sell their products at an exogenous markup rate o Procompetitive effect of market size is missing
β’ We extended the Judd model of endogenous innovation cycles, using H.S.A., a more
general homothetic demand system to allow for the procompetitive effect. o Under H.S.A., the price elasticity of demand for each product, ππ(β), depends solely
on its βown relative price.β If increasing, the procompetitive effect. o Dynamics, still generated by the skewed-V shape map, but its parameter, the
delayed impact of innovation, depends on the market size/innovation cost ratio, πΏπΏ/πΉπΉ.
β’ A larger π³π³ ππβ has destabilizing effects through the procompetitive effect. o Under the log-concavity of ππ(β) β 1, i.e., ππ(β) is βnot too convex.β o In two parametric families with choke prices, βGeneralized Translogβ and βConstant
Pass-Throughβ β’ Monopolistic competition under H.S.A. should be valuable extensions to CES o Tractable and flexible o Four parametric families we introduced should also be useful for many applications.
,π-πΏ.,,π-π‘..β‘πΏπβπΏ,πβ1.,π-π‘.45ΒΊ,π-π‘+1.
πΏπ,π-β.,π-π».,,π-π‘..=πΏ,π-π‘.πΏ,π-π‘.
,π-πΏ.,πΏ.,π+,1βπ.,π-π‘..O1Active InnovationNo Innovation