Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based...

50
Economic Dynamics Stephen Turnbull Department of Policy and Planning Sciences Lecture 7: November 22, 2019 Abstract Discrete dynamics and market stability. Innovation dynamics. November 26, 2019 1 ohp7

Transcript of Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based...

Page 1: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Economic Dynamics

Stephen TurnbullDepartment of Policy and Planning Sciences

Lecture 7: November 22, 2019

AbstractDiscrete dynamics and market stability.

Innovation dynamics.

November 26, 2019 1 ohp7

Page 2: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Discrete dynamic systems

• A discrete dynamic system is a sequence of values (scalar or vector) {yt}defined recursively. (The braces indicate that it is a sequence of yτ forτ = 0, . . . , t.)

• Recursive means there is a functional relationship yt+1 = f({yt}) fort = 0, 1, 2, . . ..

• An alternative expression for a discrete dynamic system is a differenceequation ∆yt+1 ≡ yt+1 − yt = g({yt}).

• Clearly, we can always define a recursive equation from a given differenceequation by f({yt}) = g({yt}) + yt, and vice versa.

• Difference equations are closely related to differential equations, and are usedin computational simulation of differential equations.

November 26, 2019 2 discrete dynamics

Page 3: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Classifying Difference Equations

• As with differential equations, we can take differences of differences:

∆2yt+2 = ∆yt+2 −∆yt+1 = (yt+2 − yt+1)− (yt+1 − yt)

The exponent on ∆ is called the order of the difference equation.

• As with differential equations, a higher-order difference equation can bebroken down into a system of first-order difference equations.

• If f is linear, the difference equation is called linear.

• In general, f can depend on t. If it does, the dynamic system is notautonomous.

• We consider only the autonomous case where f is the same function of y forall t. This is more than a matter of simplifying computation and notation: itrarely makes sense to discuss steady states in non-autonomous systems.

• We classify recursive function systems in the same way.

November 26, 2019 3 difference equations

Page 4: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Difference Equations and Polynomials

• Consider a dynamic system where the position of a falling object is measuredwith respect to time. Suppose it started at rest at a height of c.

• Then its position at time t is xt = c− at2, where a = 12g ≈ 4.9m/s2.

• Suppose the position is measured at one second intervals.

• Then the first difference is∆xt = xt − xt−1 = (c− at2)− (c− a(t− 1)2) = a− 2at.

• The second difference is∆2xt = ∆xt −∆xt−1 = (a− 2at)− (a− 2a(t− 1)) = −2a.

• Thus a second-order difference equation based on a quadratic time path isalways a constant.

• Note that this analysis is based on the length of the interval analyzed beinga constant.

November 26, 2019 4 difference equations

Page 5: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

A Recursive Equation Example

• Consider an infinite sequence of binary choices. We represent one alternativeby “0” and the other by “1”.

• An example sequence could be written “z = .1101000110101 . . . ”, suggestingthe binary representation of real numbers between 0 and 1. In fact, this isthe case, and there is a continuum of such sequences (a “big” infinity). Note:z0 = 1, z1 = 1, z2 = 0, ….

• Consider the (non-autonomous!) dynamic system defined by a sequence offunctions ft such that xt = ft(x0, . . . , xt−1) for t = 1, 2, . . .. Evidently we canderive a new infinite sequence from any infinite sequence by applying thissystem to it. Call this sequence f(z) (no subscripts!)

• Question: how big is the set {f(z)|z ∈ [0, 1)}?

November 26, 2019 5 difference equations

Page 6: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

The Surprising (?) Answer

• You might think that with all the flexibility of an infinite sequence offunctions depending on everything in the past and a big infinity of z tochoose from, the answer would be something like “a lot.”

• In fact, the answer is two.

• The only part of z that affects the result sequence x is the first componentz0:

x0 = z0

x1 = f1(x0)

x2 = f2(x0, x1)

x3 = f3(x0, x1, x2)

...

November 26, 2019 6 difference equations

Page 7: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

The Surprising (?) Answer, cont.

• This sequence can be rewritten

x0 = z0

x1 = f1(z0)

x2 = f2(z0, f1(z0))

x3 = f3(z0, f1(z0), f2(z0, f1(z0))

...

• x = {x0, x1, x2, . . .} depends only on z0, which must be either 0 or 1!

• In fact, the answer is two.

• From now on, we consider only sequences of scalars, and the case where the{yt−1} are ignored. The general case isn’t harder in principle, but thecomputation and notation are tedious.

November 26, 2019 7 difference equations

Page 8: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Classifying Difference Equations

• As with differential equations, we can take differences of differences:

∆2yt+2 = ∆yt+2 −∆yt+1 = (yt+2 − yt+1)− (yt+1 − yt)

The exponent on ∆ is called the order of the difference equation.

• As with differential equations, a higher-order difference equation can bebroken down into a system of first-order difference equations.

• If f is linear, the difference equation is called linear.

• In general, f can depend on t. If it does, the dynamic system is notautonomous.

• We consider only the autonomous case where f is the same function of y forall t. This is more than a matter of simplifying computation and notation: itrarely makes sense to discuss steady states in non-autonomous systems.

• We classify recursive function systems in the same way.

November 26, 2019 8 difference equations

Page 9: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Difference Equations and Polynomials

• Consider a dynamic system where the position of a falling object is measuredwith respect to time. Suppose it started at rest at a height of c.

• Then its position at time t is xt = c− at2, where a = 12g ≈ 4.9m/s2.

• Suppose the position is measured at one second intervals.

• Then the first difference is∆xt = xt − xt−1 = (c− at2)− (c− a(t− 1)2) = a− 2at.

• The second difference is∆2xt = ∆xt −∆xt−1 = (a− 2at)− (a− 2a(t− 1)) = −2a.

• Thus a second-order difference equation based on a quadratic time path isalways a constant.

• Note that this analysis is based on the length of the interval analyzed beinga constant.

November 26, 2019 9 difference equations

Page 10: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Discrete Processes

Some interesting processes really are discrete. Examples:

• The cobweb, Hicks, and Marshall adjustment processes can be considered asthe result of (rather naive) behavior in a series of markets. For example,agricultural markets from one year to the next. (It is the inelasticity ofdemand and supply for food that made Marshall decide to use a modelwhere quantity adjusts and price responds to the quantity adjustment.)

• Human beings cannot be continuously active. They need to sleep, andfunction best on a consistent daily sleep cycle. Thus it makes sense tomeasure human activity per day.

• Most activities consist of a sequence of tasks. The tasks may take variableamount of clock time to complete, but it often makes sense to count “time”in terms of tasks completed rather than watching the clock.

• Newton’s method for solving equations.

November 26, 2019 10 discrete simulation

Page 11: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Discrete Simulations

But one of the most important uses of discrete processes is simulation ofcontinuous processes.

• You’ve probably used the Artisoc or Netlogo software in jisshuu to modelcrowd behavior in buildings or in disaster response.

• Weather simulations: global warming, general local weather, and specificphenomena like typhoons or El Niño.

• Numerical computation of processes such as price adjustment in a market.

November 26, 2019 11 discrete simulation

Page 12: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Example Simulations

• The Walras price adjustment process.

• The Solow growth model.

• The fishery.

November 26, 2019 12 discrete simulation

Page 13: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Stability

• The general idea of stability in dynamics is– A dynamic process is “stable” if it tends to an equilibrium.– An equilibrium is “stable under a process” if the process tends to the

equilibrium.

• The definitions for markets generally suppress the process, and distinguishbetween local and global stability.– A solution of a model is locally stable under a process if the process

converges to the equilibrium started at any point close enough to theequilibrium.

– A model is globally stable under a process if the process converges to somesolution no matter where it is started.

• In case of a market equilibrium, the definitions of solution (equilibrium) andprocess are independent of each other. In the case of a growth model,solution is defined as a steady state of the process.

November 26, 2019 13 market stability

Page 14: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Finding the Equilibrium

• Most microeconomic analysis simply assumes markets are in equilibrium.

• But in principle we need to worry about two issues:– After a major change in the environment (e.g., a new market), how can

the market find the equilibrium?– If the price is a little bit uncertain, will it tend to stay in the region of

equilibrium, or wander away?

• The first is determined by the condition called global stability, the second bylocal stability.

November 26, 2019 14 market stability

Page 15: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Why Call It Stability?

• The two tasks (finding equilibrium and staying in equilibrium) might seemto be different in principle.

• However, both are dynamic processes, and in the analysis of both,environmental conditions are assumed to be unchanging once the processstarts.

• Thus both dynamic processes are dependent on market price and quantity,and perhaps on the history of the process.

• Formally, the processes are indistinguishable, except that “non-wandering” isassumed to take place “near” the equilibrium (thus local stability, in theusual ϵ-δ sense of calculus), whereas “discovery” may have no historical priceor quantity information to work with (thus global stability).– Global stability requires convergence to some equilibrium (there may be

several) for any initial conditions of the process, while local stability mayrequire initial conditions to be “sufficiently near” a particular equilibrium.

November 26, 2019 15 market stability

Page 16: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Defining Stability

• There are multiple definitions of stability, each of which is characterized byan adjustment process which dynamically adjusts price and quantity, basedon current conditions.

• An equilibrium is stable if the adjustment process converges to theequilibrium.– It is globally stable if convergence occurs for all initial conditions in the

domain of the adjustment process.– It is locally stable if convergence occurs for all points in a region of the

domain around the equilibrium, where the equilibrium is not on theboundary of the region.

• There are many possible adjustment processes. Both local and globalstability are defined only for a particular adjustment process, which must bementioned when saying that an equilibrium is stable.

• An equilibrium that is globally stable necessarily is locally stable (with theregion being the whole domain!)

November 26, 2019 16 market stability

Page 17: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Defining an Adjustment Process

• An adjustment process is a function mapping information about the state ofthe process either to a new state (for discrete time) or to its derivative(continuous time).

• We need to pick the domain of the function …

• …and the function itself.

November 26, 2019 17 market stability

Page 18: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

A Simple Adjustment Process

• Assume we have a large number of buyers and sellers, each wishing to tradeone unit of a good if the price is advantageous. Denote the resulting demandand supply curves by D(p) and S(p), and define excess demand asZ(p) = D(p)− S(p).

• The domain is the price.

• Our adjustment process proceeds by picking prices as follows:1. Set ps = 0 and pd = ∞.2. The initial price p0 is picked at random.3. qt = min{D(pt), S(pt)}; all buyers with value v ≥ D−1(qt) and all sellers

with cost c ≤ S−1(qt) trade, and leave the market.4. If Z(pt) = 0, stop; pt is the equilibrium price.5. If Z(pt) < 0, set pd = pt, otherwise set ps = pt.6. If pd = ∞, set pt+1 = 2pt, otherwise set pt+1 = pd+ps

2 , and go to step 3.(pd = ∞ is a special case, and can only occur if D = 0.

November 26, 2019 18 market simple

Page 19: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Simple Process: Initial Condition

Linear demand andsupply curves withp∗ < p0 < a.

November 26, 2019 19 market simple

Page 20: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Simple Process: First Adjustment

Linear demand andsupply curves withp∗ < p0 < a.

November 26, 2019 20 market simple

Page 21: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Is This a “Good” Process?

• This process always converges to equilibrium. That’s good!

• But the assumption that trades take place at each price, and the extremevalue agents do all the trading is hard to justify.

• Especially so in a case of a good where agents trade multiple units. Why?Because the curves change due to income effects, so the computations areincorrect. (The correct computations would still converge to equilibrium, butare much more complicated.)

• Also, in either case, the traders on the long side of the market would benefitfrom waiting until the next adjustment (the terms of trade always improvefor them).

• Both problems can be avoided by use of a tâtonnement process (using“fictitious trading” as in the cobweb process).

November 26, 2019 21 market simple

Page 22: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Tâtonnement

• The term tâtonnement (“groping”) was introduced by the French economistLeon Walras (a contemporary of Karl Marx), generally considered theearliest general equilibrium theorist.

• Although there is some question about what Walras intended, today atâtonnement process is defined as one in which no trade is made atdisequilibrium prices.

• Both problems mentioned previously are obviously avoided, since theydepend on actual trades being made out of equilibrium.

• Tâtonnement is generally considered to be unrealistic, but (outside ofcontrolled experiments), not only do agents trade out of equilibrium,endogenously changing supply and demand, but exogenous parameters thataffect supply and demand are also changing.

• If our stability analysis abstracts from the “moving target” nature ofequilibrium, tâtonnement doesn’t seem too much harder to swallow.

November 26, 2019 22 market tatonnement

Page 23: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Adjustment processes for a market

• In the cobweb model, time is discrete, and supply responds to the past pricewhile demand responds to the current price.

• In Hicks and Marshall stability, the adjustment process is left abstract, andthe stability properties analyzed in terms of “arrow diagrams”.– This works for a single market because they imply a single differential

equation.

• Walrasian stability is like Hicks stability, except that the differentialequation is explicit, and worked out for more general cases.

November 26, 2019 23 market stability

Page 24: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

The Cobweb Model

• Partial equilibrium (a single market).

• Demand responds to the current price in the market.

qdt = D(pt)

• Supply in the current period is inelastic because of a production lag. It is setaccording to the previous period’s price. (Period is defined by theproduction lag.)

qst = S(pt−1)

• Equilibrium, as usual, clears the market:

qst = qdt

• This generates an iterated function system:

pt = D−1(S(pt−1))

November 26, 2019 24 market stability

Page 25: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Cobweb Example

• Take a simple linear supply and demand system:

D(p) = a− bp

S(p) = c+ dp

• Substituting in the iterated function system gives

pt =a− c

b− d

bpt−1

=a− c

b

t−1∑s=0

(−d

b

)s

+

(−d

b

)t

p0

Now limt→∞∑t−1

s=0

(−d

b

)s= b

b+d , so the first term in the expansion of pttends to a−c

b+d as t → ∞, and the second term converges to zero if and only if∣∣−db

∣∣ < 1, i.e., |d| < |b|. That is, supply is less elastic than demand.

November 26, 2019 25 market stability

Page 26: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Parametric stability conditions

Need a table here.

November 26, 2019 26 market stability

Page 27: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

The Cobweb Dynamic

1. This adjustment process starts by assuming that firms set a common price.

2. The consumers respond by announcing demand quantity.

3. Then the firms respond by setting price again, to elicit the quantity theywant to supply at the current price.

4. Repeat steps 2 and 3 until convergence.

5. Because we start and end with price in each cycle, the domain of the processis the price axis. Recall that the inverse supply function isS−1(P ) = {Q | Q = S(P )}, where Q = S(P ) is the supply function. Then wecan write the equation

Pt+1 = S−1(D(Pt)),

for the cobweb adjustment process, where Q = D(P ) is the demand function.

November 26, 2019 27 market stability

Page 28: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

The Cobweb Dynamic

At each price, the supplyside determines the quantity.The excess demand causesfierce competition among themembers of the long side ofthe market, inducing a newprice.“Fictitious trading:” notrades actually occur untilequilibrium price is found.Requires a slope condition orthere is no convergence.

-

6

P

Q

@@

@@@

@@

@@@

S

D

r?6?

November 26, 2019 28 market stability

Page 29: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Problems of the Cobweb Dynamic

• Firm behavior is very unrealistic. To compute the next price, the firm needsto know not only the market supply function (perhaps reasonable, since otherfirms are likely to have similar marginal cost), but also the market demandfunction. If it knows all that, why doesn’t it go directly to the equilibrium?

• The cobweb adjustment process fails to converge when 0 < −D′(P ) ≤ S′(P ).(Notice that this means that supply is upward-sloping and demand isdownward-sloping as usual, and that demand is steeper than supply on theusual graph.)

November 26, 2019 29 market stability

Page 30: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Nonconvergence in the Cobweb Dynamic

-

6

P

Q

@@

@@@

@@@

@@�

��

��

���

���

S

D

r?6?

6

?

• Occurs when supply is more elastic (flatter on the conventional graph) thandemand.

November 26, 2019 30 market stability

Page 31: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Marshall and Hicks Stability

• Marshall and Hicks did not fully define adjustment processes.

• Similar to the cobweb process in that current price and quantity are used tocompute the next approximation on one of the market-defining curves.

• They differ in whether they are quantity adjustment processes or priceadjustment processes.– In Marshallian stability, if previous period’s quantity induced excess

demand (the demand price is higher than the supply price) the quantityis increased, while quantity is reduced when there is excess supply.

– In Hicksian stability, if there was excess demand at the previous price(quantity demand greater than quantity supplied), the price is increased,while it is decreased for excess supply.

• It is assumed that the step is small enough that eventually the processconverges even if it overshoots sometimes.

• Both processes are stable with upward-sloping supply and downward-slopingdemand, but can be unstable with “perverse” demand and/or supply.

November 26, 2019 31 market stability

Page 32: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Walras’ Adjustment Process

• The Walras adjustment process is a continuous time process that is morerealistic than the discrete-time processes described above:– It approximates small adjustments leading to equilibrium.– The cobweb process requires the firm to know the whole demand curve,

Walras only the “local” condition of the market at the current price.– It “aims at” equilibrium, while the “short side adjusts” processes assume

current conditions will persist despite planned changes.

• It’s very simple to express as a mathematical formula. Define the excessdemand at price p as z(p) = D(p)− S(p). (Note that excess supply isrepresented as negative excess demand.) Then the continuous-time Walrasadjustment process is just

p = kz(p).

k > 0 is the “speed of adjustment” parameter, usually taken as k = 1.

November 26, 2019 32 market stability

Page 33: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Properties of Walras’ Adjustment Process

• For z′(p) < 0 (globally downward-sloping excess demand), we have a negativefeedback loop and the market is globally stable with a unique equilibriumunder Walras’ process.– If demand is downward-sloping and supply upward-sloping in the usual

way, excess demand will be downward-sloping.

• In a multimarket model (including general equilibrium), Walras’ processagain gives stability if all markets have downward-sloping excess demand andsatisfy a condition called weak gross substitutes on cross-price elasticities.

November 26, 2019 33 market stability

Page 34: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

The Theory of the “Second Best”

• There are a large number of circumstances where the First Welfare Theorem,i.e., “market outcomes are Pareto efficient,” does not hold.

• These are called market failures.

• The “Theorem of the Second Best” says that in cases of market failure, thereis often no policy that achieves the first best. In these cases, the mosteffective policy often involves introducing additional market failures.

November 26, 2019 34 Second Best

Page 35: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Intellectual Property as a Second Best Policy

• It is claimed that there is a market failure in the R&D industry, that thereare unpriced external benefits: the “spillovers.”

• In his famous article “The Problem of Social Cost” (Journal of Law andEconomics, October 1960, pp. 1-44), Ronald Coase showed that manyproblems of externalities can be solved simply by allocating a privateproperty right, either to conduct an activity or to prohibit it.

• “Intellectual property” are those rights created by law to conduct activitiesinvolving certain ideas.

November 26, 2019 35 Second Best

Page 36: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Effects of Intellectual Property

• An intellectual property right is the exclusive right to conduct an activityinvolving an idea.

• Copyright is the exclusive right to copy the expression of an idea, as fixed ina medium.– Independent development of the idea is not restricted.– Copyright leads to monopolistic competition.

• A patent is the exclusive right to practice an idea, which is to construct adevice based on the idea, to be used for a specified purpose.– Independent development and practice is restricted.– Patent leads to (pure) monopoly or “tight” oligopoly.

• In each case, an additional market failure is introduced: creation of amonopoly.

November 26, 2019 36 Second Best

Page 37: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Motivation

• Monopoly has known effects of creating inefficiency; economists recommendreducing monopoly whenever possible.

• Monopolistic competition may result in “second best”, where P = ATC.

• Boldrin and Levine argue that intellectual property protection isunnecessary, because the technology of the innovation process need notinvolve increasing returns to scale, merely indivisibilities.

• They argue that intellectual property protection is harmful, because whereeach innovation depends on previous innovations (an innovation chain), themonopoly will cause cumulative damage to the economy by inhibiting futureinnovations.

November 26, 2019 37 Boldrin-Levine Motivation

Page 38: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Historical Innovation

• There is no definitive history of innovation, but we can point out someinteresting facts.

• Charles Dickens (author of A Tale of Two Cities, Oliver Twist, and otherfamous novels in English) complained that in the U.S. in the 1800s anyonewas free to reprint foreign publications without payment to the author.Nevertheless, testimony to a commission on publishing showed that Englishauthors would sometimes receive more from American publishers than fromEnglish ones.

• The mail-order business was invented by Montgomery Ward and Sears in the1870s and 1880s, and despite a lack of price-fixing power were able to makevery large profits from their innovation in business practice.

November 26, 2019 38 Boldrin-Levine Motivation

Page 39: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Spontaneous Innovation

• Innovation clearly occurs, and economic rewards to innovation can be great,without government-enforced monopoly.

• Boldrin and Levine conclude that “the issue of whether government grants ofmonopolies over ideas is second best is an empirical rather than theoreticalissue.”

November 26, 2019 39 Boldrin-Levine Motivation

Page 40: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Creative Destruction

• Creative destruction was proposed by Joseph Schumpeter as a way to viewthe dynamics of industry.– When a new technology is created, older ones are made obsolete, and

often disappear (i.e., are destroyed).

• Modern endogenous growth theory focuses on creative destruction as amechanism for propagating new technology.

• Economists such as Paul Romer argue that a technology is a fixed input, sothe R&D expense to create it is a fixed cost, and therefore there areautomatically increasing returns to scale (decreasing average cost) related totechnological progress.– These returns to scale are what makes per capita growth in the steady

state possible, it is claimed.

November 26, 2019 40 Creative Destruction

Page 41: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Spillovers

• Technology also generates positive externalities called spillovers.

• A spillover occurs when a new technology is developed, and the innovationbecomes known to others. They automatically become able to use theinnovation (or some part of it).

• Spillovers result in social benefit that cannot be captured by the investors innew technology.– Inefficiently low incentive for R&D investment.– Free riding on the technology by rivals can lead to “dog eat dog

competition” and a strong second-mover advantage, so nobody wants todeveloper new technology.

– To address this market failure, intellectual property is created by law.

November 26, 2019 41 Creative Destruction

Page 42: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

The Process of Innovation

• The innovation process is two-stage.

• First, a new idea is invented, and developed to some target level of quality.– Boldrin and Levine describe this as production of a prototype. The

prototype must be complete, and thus involves an indivisibility.

• Once the invention is completed, it is replicated. Replication means– low (or zero) fixed cost– constant marginal cost– large scale

November 26, 2019 42 Spillovers

Page 43: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Taking It to the Market

• Another common terminology is where replication is called innovation ineconomics.

• Usually it’s not important to determine whether “innovation” means“invention and replication” or just “replication”.

• “Innovation” does imply “taking a product to market,” which is theimportant part in economics.

November 26, 2019 43 Spillovers

Page 44: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Sources of Market Failure

• Benefits may not be appropriable. Because of spillovers, the total socialbenefit is not received by the innovator, who lacks incentive to produce theinnovation.

• The product may involve increasing returns to scale. Typically this is due tothe presence of a fixed cost.– The microeconomic problem is that ATC > MC, so the efficient Q∗

where P (Q∗) = MC(Q∗) cannot be produced at a profit.

• The product may be indivisible.– The microeconomic problem is that marginal analysis fails, because there

is no “margin”. Production must make big jumps, each of whose costsmust be covered in total.

November 26, 2019 44 Spillovers

Page 45: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Embodiment of Ideas

• Ideas are non-rival: two people may use the same idea at the same time.E.g., all students taking a test may know the correct answer.

• Boldrin and Levine claim that this is not interesting for economics. To eachstudent, mere existence of the correct answer does not ensure high grades;the student himself must know it.

• Boldrin and Levine would say that the correct answer is embodied instudents who know it. The process of embodiment is costly: the studentmust attend class or read a book to know the answer, and in many casesmust practice or repeatedly study it.

November 26, 2019 45 Spillovers

Page 46: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Nonexistence of Unpriced Spillovers

• Note that in class the idea is “embodied” in the lecturer, while the bookitself is an “embodiment” of the idea.

• Thus, transmission of ideas requires communication from one embodiment ofthe idea to another.

• If the source of the idea is priced, then even if the idea cannot be priceddirectly, its value will increase the value of the source embodiment, and thusthe idea can be priced.

• Only the original invention is different; we can trace the source of allembodied ideas to their prototype, and the prototype can thereforeaccumulate the total value.

• There is no failure of appropriability.

November 26, 2019 46 Spillovers

Page 47: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Increasing Returns vs. Indivisibility

• Once the innovation is produced, the costs of R&D are sunk (they cannot berecovered by abandoning the product). Sunk costs are hard to distinguishfrom fixed costs in reality.

• Thus for the long run increasing returns (fixed cost giving decreasing averagetotal cost) and indivisibility (presence of sunk cost) are hard to tell apart.

• Boldrin and Levine assume indivisibility, but not increasing returns, andremind us that in this case competition can fail to provide incentive forinnovation, accounting for observed market failures in innovation.

November 26, 2019 47 Spillovers

Page 48: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Production with Increasing Returns

TC

Q

FC

-

6

��

���

����TC

AC, MC

Q-

6

AC

MC

Figure 1: Cost with increasing returns

November 26, 2019 48 Spillovers

Page 49: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Market with Constant Returns

TC

Q-

6

���

���

��

���TC

AC, MC

Q-

6

AC = MC

Figure 2: Cost with constant returns

November 26, 2019 49 Spillovers

Page 50: Economic Dynamics - turnbull.sk.tsukuba.ac.jp · • Thus a second-order difference equation based on a quadratic time path is always a constant. ... • The cobweb, Hicks, and Marshall

Market with Indivisibility

TC

Q

TCmin

Qmin

-

6

r����

����TC

AC, MC

Q-

6

rQmin

AC = MC

Figure 3: Cost with nonconvexity

November 26, 2019 50 Spillovers