Static ModelingSome Examples

U.S. Commercial Banking Model (Dickens, 1996):

Earningsi = ai + b1*(Ii ) + b2*(GDPi ) +b3*(PCi )+b4*(SCi ) + ui

where:a = the intercept termbi = regression coefficients

Ii = general level of interest rates GDPi = GDP used for general level of economic activity

(used only for loans)PCi = Processing Costs of issuing and maintaining

deposit and loan accounts SCi = Service Charge Income (used only for checkable

deposits)

“i” is the designation for group size and/or year.

Pharmaceutical Industry Model (Naylor, 1979):

Si = f(Mi , PEi , QPEi , DEi , QDEi , Pi )

where:Si = Sales volume of product “i”Mi = Market sales volume for the therapeutic market

class of product “i” PEi = Promotional expenditures for product “i” QPEi = Quality of promotional effect for product “i”

DEi = Detailing effort (sales calls on physicians) for product “i”

QDEi = Quality of detailing message for product “i” Pi = Relative price of product “i”

Therapeutic class markets “i” are: (1) Anti-infectives, (2) Anagesics and anti-inflammatory, (3) Psychopharmaceuticals,(4) Cough and cold preparations, (5) Cardiovasculars, (6) Nutritional sufficiency, (7) Oral contraceptives, (8) diabetic therapy, (9) Anticholinerics and antispasmodics, (10) Antiobesity.

Battery Industry Model (Naylor, 1979):

S = f(MV , PC , INV , TEMP , Q2 , Q3 , Q4 )

where:S = Sales volumeMV = Stock of vehicles two years old and older

PC = Personal consumption INV = Change in business inventories TEMP = Temperature variations from norm in Q1 and

Q4.Q2= Dummy variable for second quarter Q3 = Dummy variable for third quarter Q4 = Dummy variable for quarter quarter

• Change = yn = yn - yn-1 (Basic Form)

• If change is proportional to the current size of “y,” then the formula can be written as: yn = yn - yn-1 = k*yn-1

• If growth follows an “S” shaped pattern and approaches a capacity limit or maximum threshold (M), then the formula can be written as: yn = yn - yn-1 = k*(M - yn-1) yn-1

Modeling GrowthApproximating Data Patterns

The Modeling Paradigm: Future Value = Present Value + Change

Simple Example: Savings Certificate

Consider the case where you bought a savings certificate worth $1,000 that accumulates interest paid each month at 1% per month. Then the sequence representing accumulating value

over time is:

A = { 1000, 1010, 1020.10, 1030.30, . . . }

Then the first differences are defined as: a1 = a2 - a1 = 1010 - 1000 = 10

a2 = a3 - a2 = 1020.10 - 1010 = 10.10

and so on, and in general, an = an+1 - an

Then, this expression can be written:an = an+1 - a1 = k*an = 0.01*an

or, an+1 = an + 0.01*an

Example: Savings Certificate (Continued)

Assuming that you withdrew $50 from the account each month then, the model becomes

an = an+1 - a1 - 50 = k*an - 50 = 0.01*an - 50

or, an+1 = an + 0.01*an - 50

In general, the model is:

change = an = some function = f(terms in the sequence, external terms)

Example: Modeling Growth in Industry Sales

Period SalesFirst

Difference1 $9.602 $18.30 $8.703 $29.00 $10.704 $47.20 $18.205 $71.10 $23.906 $119.10 $48.007 $174.60 $55.508 $257.30 $82.709 $350.70 $93.4010 $441.00 $90.3011 $513.30 $72.3012 $559.70 $46.4013 $594.80 $35.1014 $629.40 $34.6015 $640.80 $11.4016 $651.10 $10.3017 $655.90 $4.8018 $659.60 $3.7019 $661.80 $2.20

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34

56

78

910

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19

Time

$0.00

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$700.00

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Sales

If you think that the maximum sales capacity is $675, then you can model the process as . . .

Since growth seems to be following an “S” shaped pattern and is approaching a capacity limit or maximum threshold (M) of $675, then the formula can be written as:

Salesn = Salesn+1 - Salesn

= k*(675 - Salesn) Salesn

To find the proportionality constant (k) estimate with OLS.

PeriodFirst

DifferenceSales* (675

-Sales)1 --- $6,3882 $8.70 $12,0183 $10.70 $18,7344 $18.20 $29,6325 $23.90 $42,9376 $48.00 $66,2087 $55.50 $87,3708 $82.70 $107,4749 $93.40 $113,73210 $90.30 $103,19411 $72.30 $83,00112 $46.40 $64,53313 $35.10 $47,70314 $34.60 $28,70115 $11.40 $21,91516 $10.30 $15,56117 $4.80 $12,52818 $3.70 $10,15819 $2.20 $8,736

Regression Output:Constant 0Std Err of Y Est 6.6405643R Squared 0.9549745No. of Observations 18Degrees of Freedom 17

X Coefficient(s) 0.000776Std Err of Coef. 0.0000259

$0.0

0

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$0

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Thousands

Sales*(665-Sales)

Sales Growth Constrained by Capacity

“k” = proportionality constant

Period ActualEstimated/Predicted

1 $9.602 $18.30 $14.563 $29.00 $22.034 $47.20 $33.195 $71.10 $49.736 $119.10 $73.877 $174.60 $108.358 $257.30 $156.029 $350.70 $218.8810 $441.00 $296.3911 $513.30 $383.5212 $559.70 $470.3113 $594.80 $545.0514 $629.40 $600.0415 $640.80 $634.9616 $651.10 $654.7017 $655.90 $665.0218 $659.60 $670.1719 $661.80 $672.6820 $673.8921 $674.4722 $674.7523 $674.8824 $674.9425 $674.97

12

34

56

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910

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1920

2122

2324

X-Axis

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Sales

Actual Predicted

Model Prediction and Observation

In imposing a certain functional form, you are imposing an assumption on the process you

are modeling.

Although that example using a “sales ceiling” may be unrealistic, it may be a very realistic way of modeling production in the face of a short-run capacity constraint. In this case, the maximum value would represent “maximum feasible capacity.” This formulation also is used to estimate technological diffusion.

Moreover, the OLS estimation of the “k” in a nonlinear functional form is consistent with the logic of a “scanning method” suggested in Dhrymes, P. J., “A Model of Short Run Labor Adjustment,” The Brookings Model: Some Further Results, ed. by James S. Duessenberry, et. al., Rand-McNally & Co, Chicago, 1969.

Consider Four Models of Growth

• Unconstrained Growth Model

• Constrained Growth Model

• Mutual Growth Model

• Competitive Growth Model

A. Unconstrained Growth Model

• Assuming that the industry population changes by new business formations and business failures.

• Assuming that during each time period the number of new business formations is a percentage of the current business population (number of existing establishments denoted by “p”), then it can be modeled as, b*pn , where “b” is a non-negative constant.

A. Unconstrained Growth Model

• Similarly, assuming that during each time period the number of business failures is a percentage of the current population (number of existing establishments denoted by “p”), then it can be modeled as, f*pn , where “f” is a non-negative constant.

• Then the unconstrained growth model will take the form: pn = pn - pn-1 = b*pn-1 - f*pn-1 = k*pn-1

where k = b-f represents the growth constant.

B. Constrained Growth Model

• Assuming that the industry population changes by new business formations and business failures. Further assuming that overall market demand will support a business population of the size M, where M represents the capacity (maximum demand) of the market.

• Assuming that the growth rate of the industry population will slow as “p” approaches “M.”

B. Constrained Growth Model

• One model that would capture these assumptions

of constrained growth could be formulated as:

pn = pn - pn-1 = k*(M-pn-1)*pn-1 where k

represents the growth constant (positive).

C. Mutual Growth Model

• Now suppose that a second industry (designated as “c”) exists and competes for a portion of consumer or business spending, but that the industries are not competitors.

• Assume further that the industries are characterized by unconstrained growth, such that each can be model as: (1) pn = pn - pn-1 = b1*pn-1 - f1*pn-1 = k1* pn-1

(2) cn = cn - cn-1 = b2*cn-1 - f2*cn-1 = k2*cn-1

where ki are positive growth rates (for i = 1, 2).

C. Mutual Growth Model

• If we assume that the mutually detrimental

interaction of the two industries is proportional

then the model can be formulated as:

(1) pn = pn - pn-1 = k1*pn-1 - k4* pn-1*cn-1

(2) cn = cn - cn-1 = k2*cn-1 - k3* pn-1*cn-1

where k3 and k4 represent the relative intensities

of the competitive interactions.

D. Competitive Growth Model• Assume that a new industry develops (“c”) and its

primary source of demand comes at the expense of an existing industry (“e”).

• We can model the mutually detrimental interaction of the two industries as in the previous case: cn = cn - cn-1 = k1*cn-1 - k2* en-1*cn-1

where k1 and k2 are positive constants.

• But if the new industry is taking demand from the old industry, then the larger the market size of the existing industry “e,” that will mean the larger the potential size of “c.” This is capture by the term: + k4* en-1*cn-1 where k4 is positive.

D. Competitive Growth Model

• If the existing market size is zero, then the new industry will be zero and similarly the larger “e” is the larger “c” could be. Assume further that “c” is inversely related to “e.” Alternatively, c could be negatively related to “e,” which also captures the concept that as one goes up, the other goes down. For example, that model aspect can be indicated as -k3*cn-1 where k3 is positive. This could be formulated as: cn = cn - cn-1 = -k3*cn-1 + k4* en-1*cn-1

where k1 and k2 are positive constants.

D. Competitive Growth Model• Then the complete model becomes:

cn = cn - cn-1 = -k3*cn-1 + k4* en-1*cn-1

en = en - en-1 = k1*en-1 - k2* en-1*cn-1

This model is known also as the “predator-prey” model where, in our case, the predator is the new industry and the prey is the existing industry.

Consider One of the Classic Models of Industry Growth

The Cobweb Growth Model

Cobweb Growth Model

A basic supply-demand model characterized as a “recursive model”

Pt = Price in time period “t”Dt = Quantity Demanded in period “t” St = Quantity Supplied in period “t”

Dt = a + b* Pt demand equation

St = c + d* Pt-1 supply equation

St = Dt market clearing identity

Cobweb Model with Convergent Cycle

The simplest cobweb model can be solved as follows (order and simulate):

St = a + b* Pt-1

Dt = St

Pt = c - d* Dt

Types of Cycle Results:

|-d| > |1/b| => divergent cycle

|-d| < |1/b| => convergent cycle

|-d| = |1/b| => continuous cycle

Models of Technological Diffusion

Most studies of the spread or

diffusion of a new technology

employ an S-shaped (sigmoid)

curve -- as represented here -- to model the

process.

Reference

Paul Stoneman, The Economic Analysis of Technological

Changed, Oxford University Press, 1983.

Approximating the Rate of Diffusion

Basic Model

g(t) is also the speed of diffusion.

Continuous vs. Discrete Model

How to Interpret g(t):

1. g(t) can be a function of time;

2. g(t) can be a function of the number of firms adopting the technology. (More common, but can be identical to approach 1, if you assume that population changes, then g(t) also is a function of time.)

In Time-Dependent Formulation

First Approach:

• N-bar can be considered as an “equilibrium” point, which is time dependent.

where i = number of firms. N-bar can be estimated as a function of relative prices (p), income (Y) and maximum number of firms (N). N-bar = f(p, Y, N)

If Adopter-Model AssumedSecond Approach:

• N-bar can be considered as a “satiation” or “saturation” point, which is not time dependent (unless you assume that the point changes over time, which is often done).

Types of Diffusion Models

• External-Influence Model

• Internal-Influence Model

• Mixed-Influence Model

External-Influence Model

Internal-Influence Model

Mixed-Influence Model

This model has been used to forecast long-term sales of consumer durable goods (televisions, dishwashers, close dryers, etc.)

This Mixed Influence Diffusion Model is estimated using nonlinear methods or

rewriting the equation as below allows for OLS estimation.

Where

Dynamic System Models

Functional Forms

Competitive Markets without Inventories

• ct = f( pt, yt , zt, ut ) Demand

• qt = f( pt-1, yt , zt, ut ) Production

• pt = f-1(ct, yt , zt, ut ) Price (inverted

consumption/demand function)

• qt = ct Identity

c = demand (sales), q = production, p = price, z = exogenous variables, u = disturbance term

Competitive Markets with Inventories

• ct = f( pt-1, yt , zt, ut ) Demand

• qt = f( pt-1, yt , zt, ut ) Production

• pt = f(st, zt, ut ) Price

• st = qt - ct Stock-Adjustment Identity

c = demand (sales), q = production, y = income, p = price,

z = exogenous variables, u = disturbance term

Competitive Markets with Inventories

Flow Adjustment Specification:

pt = f(ct-qt, zt, ut ) Flow Version

Price equation embodies conventional equilibrium theory that excess demand or excess supply leads to an increase or decrease in prices and that price differences will be reduced as consumers and producers react to the new market situation, prices eventually return to their equilibrium.

Competitive Markets with Inventories

Stock Adjustment Specification:

ct = f( pt-1, yt , zt, ut ) Demand

qt = f( pt-1, yt , zt, ut ) Production

pt = f(st, zt, ut ) Stock Version

st= st-1 + qt - ct Inventory Identity

c = demand (sales), q = production, y = income, p = price,

z = exogenous variables, u = disturbance term

Competitive Markets with Inventories

Stock-Flow Adjustment Specification: Combination of stock and flow adjustment. Flow adjustment is captured pressure placed by consumption on production. However, adjustment is stock formulated since it represents pressure of consumption or production on inventories.

Competitive Markets with Inventories

Stock-Flow Adjustment Specification: This can be specified in terms of ratios -- such as, the consumption/ production to inventory ratios, which inverted represent “inventory coverage” measures. The price equation might be specified as:

pt = f(st /ct, st /qt zt, ut ) Stock-Flow Version

Competitive Markets with Inventories

Stock-Flow Adjustment Specification:

ct = f( pt-1, yt , zt, ut ) Demand

qt = f( pt-1, yt , zt, ut ) Production

pt = f(st /ct, st /qt zt, ut ) Stock-Flow Version

st= st-1 + qt - ct Stock-Adjustment Identity

c = demand (sales), q = production, y = income, p = price,

z = exogenous variables, u = disturbance term

Noncompetitive Markets(Monopoly, Duopoly, or Oligopoly)

Main Difference from Competitive Markets Case is that Price is Considered from the Point of View of the Actions of the Individual Market Participants Instead of Total Market.

Noncompetitive Markets(Monopoly, Duopoly, or Oligopoly)

Example: J. Burrows, Cobalt: An Industry Analysis, D.C. Heath & Co., Lexington, MA, 1971.

Noncompetitive Markets(Monopoly, Duopoly, or Oligopoly)

• ct = f( pt , yt, ut ) Demand

• qt = f( Cost Function* ) Production

• ct = qt Identity

c = demand (sales), q = production, y = income, p = price, z = exogenous variables, u = disturbance term * For a monopoly, it is possible to assume that the monopolist maximizes profits,

Profit(t) = p(t)*q(t) - Cost(t) and you can solve that for the price function.

Exploring the Numerical Solutions of Dynamic Systems is Important to Determine How Well the System

Traces out the History and Provides a Prediction.

Exploring Long-Term Behavior of Systems

A Few Possible Patterns

Nelson & Winter Model: Structure of a System-Dynamics Type of Modeling (Simulation)

Background Reading:

Richard R. Nelson and Sidney G. Winter, “An Evolutionary Model of Economic Growth,” in An

Evolutionary Theory of Economic Change, The Belknap Press of Harvard University Press,

Cambridge, MA., 1982, pp.206-233.

http://www.business.auc.dk/evolution/evolecon/nelwin/Druidmar96/Druidwp.html

To Explore the Nelson and Winter Models, See:

“The Nelson and Winter Models Revisited”Prototypes for Computer-Based Reconstruction of

Schumpeterian CompetitionBy Esben S. Andersen, Anne K. Jensen, Lars Madsen and

Martin Jørgensen

What we have done so far is to sketch out some basic model structures. The selection of the model form will depend on the descriptive characteristics of the industry and the pattern of the data. So look at the data you have!!! The empirical discussion of the industry you have given will provide the set of stylized facts that you need to capture in your model.