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Chapter 21: Simultaneous Equation Models Identification

Chapter 21 Outline

Reviewo Demand and Supply Modelso Ordinary Least Squares (OLS) Estimation Procedureo Reduced Form (RF) Estimation Procedure

Two Stage Least Squares (TSLS): An Instrumental Variable Two Step Approach Comparison of Reduced Form (RF) and Two Stage Least Squares (TSLS) Estimates Statistical Software and Two Stage least Squares (TSLS) Identification of Simultaneous Equation Models

o Underidentificationo Overidentification

Summary of Identification Issues: Reduced Form and Two Stage Least SquaresEstimation Procedures

Chapter 21 Preview Questions

Beef Market Data:Monthly time series data relating to the market for beef from 1977 to 1986.

Qt Quantity of beef in month t(millions of pounds)

Pt Real price of beef in month t(1982-84 cents per pound)

FeedPt Real price of cattle feed in month t(1982-84 cents per pounds of corn cobs)

Inct Real disposable income in month t(thousands of chained 2005 dollars)

ChickPt Real rice of whole chickens in month t(1982-84 cents per pound)

Yeart Year

Consider the model for the beef market that we used in the last chapter:

Demand Model: QDt =

DConst +

DPPt +

DIInct + e

Dt

Supply Model: QSt =

SConst +

SPPt +

SFPFeedPt + e

St

Equilibrium: QDt = Q

St = Qt

Endogenous Variables: Qtand Pt Exogenous Variables: FeedPtand Inct

1. We shall now introduce another estimation procedure for simultaneous equation models, thetwo stage least squares (TSLS) estimation procedure:

1stStage: Estimate the variable that is creating the problem, the explanatory endogenous

variable:

Dependent variable:Original endogenous explanatory variable that creates thebias problem.

Explanatory variables:All exogenous variables.

2nd

Stage: Estimate the original models using the estimate of the problemexplanatory endogenous variable

Dependent variable:Original dependent variable. Explanatory variables:1st stage estimate of the problem explanatory

endogenous variable and any relevant exogenous explanatory variable.


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Naturally, begin by focusing on the first stage.1

stStage: Estimate the variable that is creating the problem, the explanatory endogenous

variable:

Dependent variable:Original endogenous explanatory variable that creates thebias problem. In this case, the price of beef, Pt, is the problem explanatory

variable.

Explanatory variables:All exogenous variables. In this case, the exogenous variablesare FeedPtand Inct.

Using the ordinary least squares (OLS) estimation procedure, what equation estimatesthe problem explanatory variable, the price of beef?

Click here to access data {EViewsLink}

EstP= ______________________________________________

Generate a new variable, EstP, that estimates the price of beef based on the 1ststage.

2. Next, we focus on the 2

nd

stage and consider the demand model:

Demand Model: QDt =

DConst +

DPPt +

DIInct + e

Dt

2nd

Stage: Estimate the original models using the estimate of the problemexplanatory endogenous variable

Dependent variable:Original dependent variable. In this case, the originalexplanatory variable is the quantity of beef, Qt.

Explanatory variables:1st stage estimate of the problem explanatoryendogenous variable and any relevant exogenous explanatory variable. In thiscase, the estimate of the price of beef and income, EstPtand Inct.

Beef Market Demand Model: Dependent Variable:QExplanatory Variables: EstPriceand Inc

a. Using the ordinary least squares (OLS) estimation procedure, estimate the EstPricecoefficent of the demand model.

b. Compare the two stage least squares coefficient estimate for the demand model withthe estimate computed using the reduced form estimation procedure in the previouschapter.
http://www3.amherst.edu/~fwesthoff/weblinks/55-Lec4-BeefMarket-1977-86.wf1http://www3.amherst.edu/~fwesthoff/weblinks/55-Lec4-BeefMarket-1977-86.wf1


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3. Now, consider the supply model:

Supply Model: QSt =

SConst +

SPPt +

SFPFeedPt + e

St

and the second stage of the two stage least squares estimation procedure.

2ndStage: Estimate the original models using the estimate of the problemexplanatory endogenous variable


Explanatory variables:1st stage estimate of the problem explanatoryendogenous variable and any relevant exogenous explanatory variable. In thiscase, the estimate of the price of beef and income, EstPtand PFeedt.

Beef Market Supply Model: Dependent Variable:QExplanatory Variables: EstPriceand FeedP

a. Using the ordinary least squares (OLS) estimation procedure, estimate the EstPricecoefficient of the supply model.

b. Compare the two stage least squares coefficient estimate for the supply model withthe estimate computed using the reduced form estimation procedure in the previouschapter.

4. Reconsider the following simultaneous equation model of the beef market and the reducedform estimates:

Demand and Supply Models:

Demand Model: QDt =

DConst +

DPPt +

DIInct + e

Dt

Supply Model: QSt =

SConst +

SPPt +

SFPFeedPt + e

St


St = Qt

Endogenous Variables: Qt

and Pt

Exogenous Variables:FeedPt

and Inct

Reduced Form (RF) Estimates

Quantity Reduced Form Estimates: EstQ = aQConst + a

QFPFeedPt + a

QIInct

Price Reduced Form Estimates: EstP = aPConst + a

PFPFeedPt + a

PIInct

a. Focus on the reduced form estimates for the income coefficients:

1) The reduced form income coefficient estimates, aQI and a

PI , allowed us to

estimate the slope of which curve? ___ Demand ___Supply2) If the reduced form income coefficient estimates were not available, would

we be able to estimate the slope of this curve? ___

b. Focus on the reduced form estimates for the feed price coefficients:1) The reduced form feed price coefficient estimates of these coefficients, a

QFP

and aPFP, allowed us to estimate the slope of which curve?

___Demand ___ Supply2) If the reduced form feed price coefficient estimates were not available, would

we be able to estimate the slope of this curve? ___


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Review: Demand and Supply Models

In simultaneous equation models the value of an explanatory variable is determined within themodel. For the economist, arguably the most important example of a simultaneous equationsmodel is the demand/supply model:

Demand Model: QDt = DConst + DPPt + Other Demand Factors + eDt

Supply Model: QSt =

SConst +

SPPt + Other Supply Factors + e

St


St = Qt

Endogenous Variables: Qtand Pt Exogenous Variables: Other Demand and Supply Factors

Project:Estimate the beef market demand and supply parameters

In a simultaneous equation model, it is important to emphasize the distinction betweenendogenous and exogenous variables. Endogenous variables are variables whose values aredetermined within the model. In the demand/supply example, both quantity and price aredetermined simultaneously within the model; the model is explaining both the equilibrium

quantity and the equilibrium price as depicted by the intersection of the supply and demandcurves. On the other hand, exogenous are determined outside the context of the model; thevalues of exogenous variables are taken as given. The model does not attempt to explain how thevalues of exogenous variables are determined.

Endogenous variables Variables determined within the model: Quantity and Price. Exogenous variables Variables determined outside the model.

Unlike single regression models, an endogenous variable can be anexplanatory variable in simultaneous equation models. In thedemand and supply models the price is such a variable. Both thequantity demanded and the quantity supplied depend on theprice; hence, the price is an explanatory variable. Furthermore, theprice is determined within the model; the price is an endogenous

variable. The price is determined by the intersection of the supplyand demand curves. The traditional demand/supply graph clearlyillustrates that both the quantity, Qt, and the price, Pt, are

endogenous, both are determined within the model.In our last lecture, we showed why simultaneous equations causea problem for the ordinary least squares (OLS) estimationprocedure:

Simultaneous Equations and Bias: Whenever an explanatory variable is also an endogenousvariable, the ordinary least squares (OLS) estimation procedure is biased.

In the demand/supply model, the price is an endogenous explanatory variable. When we used the

ordinary least squares (OLS) estimation procedure to estimate the value of the price coefficient inthe demand and supply models we observed that a problem emerged. In each model, price and theerror term were correlated; unfortunately, such correlation results in bias:

Figure 21.1: Demand/supply Model

S

D

Price

Quantity

P

Q

Q= Equlibrium Quantity

P= Equilibrium Price


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Demand Model: Supply Model:

QDt =

DConst+

DPPt + Other Demand Factors + e

Dt Q

St=

SConst +

SPPt + Other Supply Factors + e

St

S

D (eD

down) D

D (eD

up)

Price

Quantity

P

P(eD

down)

P(e

D

up)

Figure 21.2: Effect of Demand Error Term Figure 21.3: Effect of Supply Error Term

eDt up e

Dt down e

Stup e

St down

Ptup Ptdown Ptdown Ptup

Explanatory variable and Explanatory variable anderror term and positively correlated error term and negatively correlated

OLS estimation procedure OLS estimation procedure

for coefficient value for coefficient valuebiased upward biased downward

So, where did we go from here? We explored the possibility that the ordinary least squares (OLS)estimation procedure might be consistent. After all, is not half a loaf better than none? We tookadvantage of our Econometrics Lab to address this issue. Recall the distinction between anunbiased and consistent estimation procedure:

Unbiased: The estimation procedure does not systematically underestimate or overestimatethe actual value; that is, after many, many repetitions the average of the estimates equals theactual value.

Consistent but Biased:As consistent estimation procedure can be biased. But, as the samplesize, as the number of observations, grows:

The magnitude of the bias decreases. That is, the mean of the coefficient estimatesprobability distribution approaches the actual value.

The variance of the estimates probability distribution diminishes and approaches 0.Unfortunately, the Econometrics Lab illustrates the sad fact that the ordinary least squares (OLS)estimation procedure is neither unbiased nor consistent.

We then considered an alternative estimation procedure: the reduced form (RF) estimationprocedure. Our Econometrics Lab taught us that while the reduced form (RF) estimationprocedure is biased, it is consistent. That is, as the sample size grows, the average of thecoefficient estimates gets closer and closer to the actual value and the variance grew smallerand smaller after many, many repetitions. Arguably, when choosing between two biasedestimates, it is better to use the one that is consistent. This represents the econometricianspragmatic, half a loaf is better than none philosophy.

We shall now quickly review the reduced form (RF) estimation procedure.

SS (e

Sdown)

D

S (eSup)

Price

Quantity

P

P(eS

down)

P(eSup)


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Review:Reduced Form (RF) Estimation Procedure One Way to Cope withSimultaneous Equation Models

We begin with the simultaneous equation model and then constructed the reduced formequations:


Demand Model: QDt = DConst + DPPt + DIInct + eDt

Supply Model: QSt =

SConst +

SPPt +

SFPFeedPt + e

St


St = Qt


Reduced Form (RF) Estimates

Quantity Reduced Form Estimates: EstQ = aQConst + a

QFPFeedPt + a

QIInct

Price Reduced Form Estimates: EstP = aPConst + a

PFPFeedPt + a

PIInct

We can use the coefficient interpretation approach to estimate the slopes of the demand andsupply in terms of the reduced form estimates:

Suppose that FeedPincreases while Suppose that Inc increases whileIncremains constant: FeedPremains constant:

Q = aQFPFeedP Q = a

QIInc

P = aPFPFeedP P = a

PIInc

Figure 21.4: Reduced Form Summary and Coefficient Interpretation Approach

Q

P =

aQFPFeedP

aPFPFeedP

=a

QFP

aPFP

Q

P =

aQIInc

aPIInc

=a

QI

aPI

We are moving from one equilibrium to We are moving from one equilibrium toanother on the same demand curve. another on the same supply curve.This movement represents a change in This movement represents a change in

the quantity of beef demanded, QD

: the quantity of beef supplied, QS

:

bDP=Q

D

P =

aQFPFeedP

aPFPFeedP

=aQFP

aPFP

bSP=

QS

P=

aQIInc

aPIInc

=aQI

aPI

D

Price

Quantity

FeedPincreaes

S

SP=

F

Q

PFeedP

FQPFeedP

Q=

Incconstant S

Price

Quantity

FeedPconstant

D

D

P=

I

QInc

I

QInc

Q=

Incincreases

Initial

equilibrium


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Intuition: Critical Role of the Exogenous Variable Absent from the Model

Demand Model:QDt = DConst +

DPPt +

DIInct + e

Dt

Changes in the feed price, the exogenous variable absent from the demand model, allow usto estimate the slope of the demand curve. The supply curve shifts, but the demand curveremains stationary. Consequently, the equilbria trace out the stationary demand curve.

Supply Model: QS

t

= S

Const

+ S

P

Pt

+ S

FP

FeedPt

+ eS

t

Changes in income, the exogenous variable absent from the demand model, allow us toestimate the slope of the supply curve. The demand curve shifts, but the supply curveremains stationary. Consequently, the equilibria trace out the stationary supply curve.

Key Point:In each case, changes in the exogenous variable absent in the model allow us toestimate the parameters of the model.

Calculating the Reduced From Estimates

We use the ordinary least squares (OLS) estimation procedure to estimate the reduced formparameters and then use the ratio of the reduced form estimates to estimate the slopes of thedemand and supply curves:


Quantity Reduced Form Equation:Dependent Variable: QExplanatory Variables: FeedPand Inc

Dependent Variable: QMethod: Least SquaresSample: 1977M01 1986M12Included observations: 120

Coefficient Std. Error t-Statistic Prob.

FEEDP -331.9966 121.6865 -2.728293 0.0073INC 17.34683 2.132027 8.136309 0.0000C 138725.5 13186.01 10.52066 0.0000

Table 21.1: EView Regression Results Quantity Reduced Form Equation

Price Reduced Form Equation: Dependent Variable: P

Explanatory Variables: FeedPand IncDependent Variable: PMethod: Least SquaresSample: 1977M01 1986M12Included observations: 120


FEEDP 1.056242 0.286474 3.687044 0.0003INC 0.018825 0.005019 3.750636 0.0003C 33.02715 31.04243 1.063936 0.2895

Table 21.2: EView Regression Results Price Reduced Form Equation

Estimated Slope Estimated Slopeof the Demand Curve of the Supply Curve

Ratio of Reduced Form Ratio of Reduced Form

Feed Price IncomeCoefficient Estimates Coefficient Estimates

Estimate of DP = bDP=

aQFP

aPFP Estimate of SP = b

SP=

aQI

aPI

=332.001.0562 = 314.3 =

17.347.018825= 921.5


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Two Stage Least Squares (TSLS): An Instrumental Variable Two StepApproach A Second Way to Cope with Simultaneous Equation Models

Another way to estimate simultaneous equation model is the two stage least squares (TSLS)estimation procedure. As the name suggests the procedure involves two steps. As we shall see,two stage least squares (TSLS) uses a strategy that is similar to the instrumental variable (IV)

approach.

1stStage: Estimate the variable that is creating the problem, the explanatory endogenous

variable:

Dependent variable:Original endogenous explanatory variable that creates the biasproblem.

Explanatory variables:All exogenous variables.

2nd

Stage: Estimate the original models using the estimate of the problem explanatoryendogenous variable

Dependent variable:Original dependent variable. Explanatory variables:1ststage estimate of the problem explanatory endogenous

variable and any relevant exogenous explanatory variables.

We shall now illustrate the two stage least squares (TSLS) approach by consider the beef market:


Qt Quantity of beef in month t(millions of pounds)

Pt Real price of beef in month t(1982-84 cents per pound)




Yeart Year

Consider the model for the beef market that we used in the last chapter:

Demand Model: QDt =

DConst +

DPPt +

DIInct + e

Dt

Supply Model: QSt =

SConst +

SPPt +

SFPFeedPt + e

St


St = Qt



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The strategy for the first stage is similar to the strategy used by instrumental variable (IV)approach. We use the ordinary least squares (OLS) regression to estimate an equation in whichthe problem endogenous explanatory variable becomes the dependent variable. Theexplanatory variables are all the exogenous variables. In our example, price is the problemexplanatory variable; consequently, price becomes the dependent variable in the first stage. Theexogenous variables, income and feed price, are the explanatory variables.

1stStage: Estimate the variable that is creating the problem, the explanatory endogenous variable:

Dependent variable:Original endogenous explanatory variable that creates the biasproblem. In this case, the price of beef, Pt, is the problem explanatory variable.

Explanatory variables:All exogenous variables. In this case, the exogenous variables areFeedPtand Inct.


1stStage: Dependent variable: PExplanatory variables: FeedP andInc

Dependent Variable: PMethod: Least SquaresSample: 1977M01 1986M12Included observations: 120


FEEDP 1.056242 0.286474 3.687044 0.0003INC 0.018825 0.005019 3.750636 0.0003C 33.02715 31.04243 1.063936 0.2895

Table 21.3: EViews Regression Results TSLS 1stStage

Estimated Equation:EstP= 33.027 + 1.0562FeedP+ .018825Inc

Using these regression results we can estimate the price of beef based on the exogenous variables,income and feed price.

The strategy for the second stage is also similar to the instrumental variable (IV) approach. Thedependent variable is the original dependent variable, quantity. The explanatory variables do notinclude the problem endogenous explanatory variable; instead, the estimate of the problemexplanatory variable based on the first stage is used. Instead of using the price as an explanatoryvariable, we use Stage 1s estimate of the price.

For the the demand model, the dependent variable is the quantity of beef, Q, and the explanatoryvariables EstPand Inc:

2nd



Explanatory variables:1st stage estimate of the problem explanatory endogenousvariable and any relevant exogenous explanatory variable. In this case, the estimatedof the price of beef and income, EstPt, and Inct.


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For the demand model, the dependent variable is the quantity of beef, Q, and the explanatoryvariables EstPand Inc:

2nd

Stage Beef Market Demand Model:Dependent variable: QExplanatory Variables: EstP and Inc

Dependent Variable: Q

Method: Least SquaresSample: 1977M01 1986M12Included observations: 120


ESTP -314.3312 115.2117 -2.728293 0.0073INC 23.26411 2.161914 10.76089 0.0000C 149106.9 16280.07 9.158860 0.0000

Table 21.4: EViews Regression Results TSLS 2ndStage Demand

Estimated Equation:EstQD= 149,107 314.3EstP+ 23.26Inc

We estimate the slope of the demand curve to be 314.3.

For the supply model, the dependent variable is the quantity of beef, Q, and the explanatoryvariables EstPand FeedP:

2nd



Explanatory variables:1st stage estimate of the problem explanatory endogenousvariable and any relevant exogenous explanatory variable. In this case, the estimatedof the price of beef and income, EstPt, and PFeedt.

2ndStage Beef Market Supply Model:Dependent variable: Q

Explanatory Variables: EstP and FeedPDependent Variable: QMethod: Least SquaresSample: 1977M01 1986M12Included observations: 120


ESTP 921.4783 113.2551 8.136309 0.0000FEEDP -1305.262 121.2969 -10.76089 0.0000

C 108291.8 16739.33 6.469303 0.0000

Table 21.5: EViews Regression Results TSLS 2ndStage Supply

Estimated Equation:EstQS= 108,292 + 921.5EstP 1,305.2FeedP

We estimate the slope of the demand curve to be 921.5.

Compare the estimates from the reduced form (RF) approach with the estimates from the twostage least squares (TSLS) approach:

Estimate of Reduced Form (RF) Two Stage Least Squares (TSLS)

DP 314.3 314.3

SP 921.5 921.5

The estimates are identical. In this case, the reduced form (RF) estimation procedure and the twostage least squares (TSLS) estimation procedure produce identical results.


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Software and Two Stage Least Squares (TSLS)

Many statistical packages provide an easy way to apply the two state least squares (TSLS)estimation procedure so that we do not need to generate the estimate of the problemexplanatory variable ourselves.

Getting Started in EViews

EViews makes it very easy for us to use the two stage least squares (TSLS) approach. EViewsdoes most of the work for us eliminating the need to generate a new variable:

In the Workfile window, highlight all relevant variables: q p feedp income Double click on one of the highlighted variables and click Open Equation. In the Equation Estimation window, click Options and then select TSLS Two-Stage Least

Squares (TSNLS and ARIMA).

In the Instrument List box, enter the exogenous variables: feedp income In the Equation Specification box, enter the dependent variable followed by the

explanatory variables (both exogenous and endogenous) for each model:o To estimate the demand model enter q p incomeo

To estimate the supply model enter q p feedp


Beef Market Demand Model: Dependent variable: QExplanatory variables: Pand IncInstrument List: FeedP andInc

Dependent Variable: QMethod: Two-Stage Least SquaresSample: 1977M01 1986M12Included observations: 120Instrument list: FEEDP INC


P -314.3188 58.49828 -5.373129 0.0000INC 23.26395 1.097731 21.19276 0.0000C 149106.5 8266.413 18.03763 0.0000

Table 21.6: EViews Regression Results TSLS Demand

Beef Market Supply Model: Dependent variable: QExplanatory variables Pand FeedPInstrument List: FeedP andInc

Dependent Variable: QMethod: Two-Stage Least SquaresSample: 1977M01 1986M12Included observations: 120Instrument list: FEEDP INC


P 921.4678 348.8314 2.641585 0.0094FEEDP -1305.289 373.6098 -3.493723 0.0007

C 108292.0 51558.51 2.100372 0.0378

Table 21.7: EViews Regression Results TSLS Supply

Note that these are the same estimates that we obtained when we generate the estimate of theprice on our own.


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Taking Stock

Let us step back for a moment to review our beef market model:Demand and Supply Models:

Demand Model: QDt =

DConst +

DPPt +

DIInct + e

Dt

Supply Model: Q

S

t =

S

Const +

S

PPt +

S

FPFeedPt + e

S

tEquilibrium: Q

Dt = Q

St = Qt

Endogenous Variables: Qtand Pt Exogenous Variables:FeedPtand Inct

Reduced Form (RF) Equations

Quantity Reduced Form Equation: Qt = QConst +

QFPFeedPt +

QIInct +

Qt

Price Reduced Form Equation: Pt = PConst +

PFPFeedPt +

PIInct +

Pt

The reduced form estimation procedure uses the the reduced form estimates to estimate theslopes of the demand and supply curves:

In each model there is one exogenous variable absent and one endogenous explanatory variable.This one to one correspondence allows us to estimate the coefficient of the endogenousexplanatory variable.

Demand Model: The absent exogenous variable, FeedP, plays a critical role. Changes in FeedPshiftthe supply curve allowing us to estimate the slope of the demand curve, changes in FeedPallowus to estimate the coefficient of the demand models endogenous explanatory variable, P.


Changes in the feed price shift the Changes in income shift thesupply curve, but not the demand curve demand curve, but not the supply curve

bDP=

QD

P =

aQFPFeedP

aPFPFeedP =

aQFP

aPFP b

SP=

QS

P=

aQIInc

aPIInc =

aQI

aPI

The exogenous variable absent The exogenous variable absentin the demand model, FeedP, in the supply model, Inc,

allows us to estimate the coefficient allows us to estimate the coefficientof the endogenous explanatory of the endogenous explanatory

variable, P, in the demand model. variable, P, in the demand model.

D

Price

Quantity

FeedPincreaes

S

SP=

F

Q

PFeedP

F

Q

PFeedP

Q=

IncconstantS

Price

Quantity

FeedPconstant

D

D

P=

I

QInc

I

QInc

Q=

Incincreases

Initial

equilibrium

Estimated slope of demand curve: Estimated slope of supply curve:

bDP=QD

P

bSP=

QS

P


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Supply Model: The absent exogenous variable, Inc, plays a critical role. Changes in Incshift thedemand curve allowing us to estimate the slope of the supply curve, changes in Incallow us toestimate the coefficient of the supply models endogenous explanatory variable, P.

The Order Condition formalizes this relationship:

Number of Less Than Number ofexogenous variables Equal To endogenous explanatory

absent from the model Greater Than variables in the model

Model Model ModelUnderidentified Identified Overidentified

No Estimates Unique Estimates Multiple Estimates

Figure 21.6: Order Condition

UnderidentificationWe shall now illustrate the underidentificationproblem. Suppose that no income informationwas available. Obviously, if we have no income information, we cannot include Incas anexplanatory variable in either the original demand and supply models or the reduced formequations:


Demand Model: QDt =

DConst +

DPPt +

DIInct + e

Dt

Supply Model: QSt =

SConst +

SPPt +

SFPFeedPt + e

St


St = Qt




QFPFeedPt +

QIInct +

Qt


PFPFeedPt +

PIInct +

Pt

Let us now apply the order condition by counting the number of absent exogenous variables andendogenous explanatory variables in each model:

Demand Model Supply ModelExogenous Endogenous Exogenous Endogenousvariables explanatory variables explanatory

absent from variables in absent from variables inthe model the model the model the model

FeedP P None P

1 1 0 1


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The order condition suggests that we should

still be able to estimate the coefficient of the endogenous explanatory variable, P, in thedemand model.

not be able to estimate the coefficient of the endogenous explanatory variable, P, in thesupply model.

The coefficient interpretation approach explains why. We can still estimate the slope of thedemand curve, however, by calculating the ratio of the reduced form feed price coefficient

estimates, aQFPand a

PFP. We shall use the coefficient estimate approach to explain this phenomenon

to take advantage of the intuition it provides.

There is both good news and bad news when we have feed price information but no incomeinformation:

Good news:Since we still have feed price information, we still have information abouthow the supply curve shifts. The shifts in the supply curve cause the equilibriumquantity and price to move along the demand curve. In other words, shifts in the supplycurve trace out the demand curve; hence, we can still estimate the slope of thedemand curve.



bDP=

QD

P =

aQFPFeedP

aPFPFeedP =

aQFP

aPFP b

SP=

QS

P=

aQIInc

aPIInc =

aQI

aPI





bDP=

QD

P b

SP=

QS

P

D

Price

Quantity

FeedPincreaes

S

SP=

F

Q

PFeedP

FQ

PFeedP

Q=

Incconstant S

Price

Quantity

FeedPconstant

D

D

P=

I

QInc

I

QInc

Q=

Incincreases

Initial

equilibrium


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Bad news:On the other hand, since we have no income information, we have noinformation about how the demand curve shifts. Without knowing how the demandcurve shifts we have no idea how the equilibrium quantity and price move along thesupply curve. In other words, we cannot trace out the supply curve; hence, we cannotestimate the slope of the supply curve.

To use the reduced form (RF) approach to estimate the slope of the demand curve, we first useordinary least squares (OLS) to estimate the parameters of the reduced form (RF) equations:


Quantity Reduced Form Equation:Dependent Variable: QExplanatory Variable: FeedP



FEEDP -821.8494 131.7644 -6.237266 0.0000C 239158.3 5777.771 41.39283 0.0000

Table 21.8: EViews Regression Results RF Quantity

Price Reduced Form Equation: Dependent Variable: QExplanatory Variable: FeedP



FEEDP 0.524641 0.262377 1.999571 0.0478C 142.0193 11.50503 12.34411 0.0000

Table 21.9: EViews Regression Results RF Price

Then, we can estimate the slope of the demand curve by calculating the ratio of the feed priceestimates:

Estimated slope of the demand curve = bSP=aQI

aPI=

821.85.52464 = 1,566.5

Now, let us use the two stage least squares (TSLS) estimation procedure to estimate the slope ofthe demand curve:

Beef Market Demand Model: Dependent variable: QExplanatory variable: PInstrument List: FeedP

Dependent Variable: QMethod: Two-Stage Least SquaresSample: 1977M01 1986M12Included observations: 120Instrument list: FEEDP


P -1566.499 703.8335 -2.225667 0.0279C 461631.4 115943.8 3.981510 0.0001


In both cases, the estimated slope of the demand curve is 1,566.5.


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Similarly, an underidentification problem would exist if income information was available, butfeed price information was not.


Demand Model: QDt =

DConst +

DPPt +

DIInct + e

Dt

Supply Model: QS

t

= S

Const

+ S

P

Pt

+ S

FP

FeedPt

+ eS

t


St = Qt




QFPFeedPt +

QIInct +

Qt


PFPFeedPt +

PIInct +

Pt

Again, let us now apply the order condition by counting the number of absent exogenousvariables and endogenous explanatory variables in each model:

Number of Less Than Number of

exogenous variables Equal To endogenous explanatoryabsent from the model Greater Than variables in the model




Demand Model Supply Model

Exogenous Endogenous Exogenous Endogenousvariables explanatory variables explanatoryabsent from variables in absent from variables inthe model the model the model the model

None P Inc P

0 1 1 1

The order condition suggests that we should

still be able to estimate the coefficient of the endogenous explanatory variable, P, in thesupply model.

not be able to estimate the coefficient of the endogenous explanatory variable, P, in the

demand model.


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17

The coefficient interpretation approach explains why.

Again, there is both good news and bad news when we have income information, but no feedprice information:

Good news:Since we have income information, we still have information about how thedemand curve shifts. The shifts in the demand curve cause the equilibrium quantity andprice to move along the supply curve. In other words, shifts in the demand curve traceout the supply curve; hence, we can still estimate the slope of the supply curve.

Bad news:On the other hand, since we have no feed price information, we have noinformation about how the supply curve shifts. Without knowing how the supply curveshifts we have no idea how the equilibrium quantity and price move along the demandcurve. In other words, we cannot trace out the demand curve; hence, we cannotestimate the slope of the demand curve.



bDP=

QD

P =

aQFPFeedP

aPFPFeedP

=a

QFP

aPFP

bSP=

QS

P=

aQIInc

aPIInc

=a

QI

aPI




D

Price

Quantity

FeedPincreaes

S

SP=

F

Q

PFeedP

FQ

PFeedP

Q=

Incconstant S

Price

Quantity

FeedPconstant

D

D

P=

I

QInc

I

QInc

Q=

Incincreases

Initial

equilibrium


bDP=Q

D

P b

SP=

QS

P


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18

To use the reduced form (RF) approach to estimate the slope of the supply curve, we first useordinary least squares (OLS) to estimate the parameters of the reduced form (RF) equations:


Quantity Reduced Form Equation:Dependent Variable: QExplanatory Variable: Inc



INC 20.22475 1.902708 10.62946 0.0000C 111231.3 8733.000 12.73690 0.0000


Price Reduced Form Equation: Dependent Variable: PExplanatory Variable: Inc



INC 0.009669 0.004589 2.107161 0.0372C 120.4994 21.06113 5.721413 0.0000


Then, we can estimate the slope of the supply curve by calculating the ratio of the incomeestimates:

Estimated slope of the supply curve = bDP=aQFP

aPFP=

20.225.009669= 2,091.7

Once again, two stage least squares (TSLS) provide the same estimate:

Beef Market Supply Model:Dependent variable: QExplanatory Variable: PInstrument List:Inc

Dependent Variable: QMethod: Two-Stage Least SquaresSample: 1977M01 1986M12Included observations: 120Instrument list: INC


P 2091.679 1169.349 1.788756 0.0762C -140814.8 192634.8 -0.730994 0.4662


Conclusion: When a simultaneous equations model is underidentified, we cannot estimate all itsparameters. For those parameters we can estimate, however, the reduced form estimation procedureand the two stage least squares (TSLS) estimation procedures are equivalent.


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19

Overidentification

While an underidentification problem arises when too little information is available, anoveridentificationproblem arises when too much information is available. To illustrate thissuppose that in addition to the feed price and income information, the price of chicken is alsoavailable. The simultaneous equation model and the reduced form estimates would become:


Demand Model: QDt =

DConst +

DPPt +

DIInct +

DCPChickPt + e

Dt

Supply Model: QSt =

SConst +

SPPt +

SFPFeedPt + e

St


St = Qt

Endogenous Variables: Qtand Pt Exogenous Variables: FeedPt, Inct, and ChickPt



QFPFeedPt +

QIInct +

QCPChickPt +

Qt


PFPFeedPt +

PIInct +

PCPChickPt +

Pt

Let us now apply the order condition by counting the number of absent exogenous variables andendogenous explanatory variables in each model:






Demand Model Supply ModelExogenous Endogenous Exogenous Endogenousvariables explanatory variables explanatory

absent from variables in absent from variables inthe model the model the model the model

FeedP P Inc andChickP P

1 1 2 1

The order condition suggests that we should still be able to estimate the coefficient of the endogenous explanatory variable, P, in the

demand model.

should encounter some difficulties when estimating the coefficient of the endogenousexplanatory variable, P, in the supply model.

We shall now explain these difficulties.


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20

Now we have two exogenous factors that shift the demand curve: income and the price ofchicken. Consequently, there are two ways to trace out the supply curve. There are now twodifferent ways to use the reduced form (RF) estimates to estimate the slope of the supply curve:

Ratio of the reduced form Ratio of the reduced formincome coefficients chicken feed coefficients

Estimated slope Estimated slopeof supply curve: of supply curve:

bSP=a

Q

Ia

PI

bSP=a

Q

CPa

PCP


Changes in income shift the Changes in the chicken price shift thesupply curve, but not the demand curve demand curve, but not the supply curve

bSP=

QS

P=

aQIInc

aPIInc =

aQI

aPI b

SP=

QS

P=

aQIChickP

aPIChickP =

aQI

aPI




Estimated slope of demand curve:

bSP=

QS

P

S

Price

Quantity

FeedPconstant

D

D

P=

I

QInc

I

QInc

Q=

Incincreases

Initial

equilibrium

ChickPconstantS

Price

Quantity

FeedPconstant

D

D

P=

F

Q

PChickP

C

QPChickP

Q=

Incconstant

Initial

equilibrium

ChickPincreases


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21

We shall now go through the mechanics of the reduced form (RF) estimation procedures toillustrate the overidentification problem. First, we use the ordinary least squares (OLS) toestimate the reduced form (RF) parameters:


Quantity Reduced Form Equation:Dependent Variable: Q

Explanatory Variables: FeedP,Inc, andChickP



FEEDP -349.5411 135.3993 -2.581558 0.0111INC 16.86458 2.675264 6.303894 0.0000

CHICKP 47.59963 158.4147 0.300475 0.7644C 138194.2 13355.13 10.34765 0.0000


Price Reduced Form Equation:Dependent Variable: PExplanatory Variables: FeedP,Inc, andChickP



FEEDP 0.955012 0.318135 3.001912 0.0033INC 0.016043 0.006286 2.552210 0.0120

CHICKP 0.274644 0.372212 0.737870 0.4621C 29.96187 31.37924 0.954831 0.3416


Estimated slope Estimated slope Estimated slope

of demand curve of the supply curve of the supply curve

Ratio of reduced form Ratio of reduced form Ratio of reduced formfeed price income chicken price

coefficient estimates coefficient estimates coefficient estimates

bDP=aQFP

aPFP=

349.54.95501 = 366.0 b

SP=

aQI

aPI=

16.865.016043= 1051.2 b

SP=

aQCP

aPCP

=47.600.27464= 173.3

The reduced form (RF) estimation procedure produces two different estimates for the slope forthe supply curve. The slope of the supply curve is overidentified.


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22

Two-Stage Least Squares (TSLS)

While reduced form (RF) estimation procedure cannot resolve the overidentification problem,two stage least squares (TSLS) approach can. The two squares least squares estimation procedureprovides a single estimate of the slope of the supply curve. The following regression printoutreveals this:


Beef Market Demand Model:Dependent variable: QExplanatory Variables: P,Inc, and ChickPInstrument List:FeedP,Inc, andChickP

Dependent Variable: QMethod: Two-Stage Least SquaresSample: 1977M01 1986M12Included observations: 120Instrument list: FEEDP INC CHICKP


P -366.0071 68.47718 -5.344950 0.0000INC 22.73632 1.062099 21.40697 0.0000

CHICKP 148.1212 86.30740 1.716205 0.0888

C 149160.5 7899.140 18.88313 0.0000


The estimated slope of the demand curve is 366.0. This is the same estimate ascomputed by the reduced form (RF) estimation procedure.

Beef Market Supply Model:Dependent variable: QExplanatory Variables: Pand FeedPInstrument List:FeedP,Inc, andChickP

Dependent Variable: QMethod: Two-Stage Least SquaresSample: 1977M01 1986M12Included observations: 120

Instrument list: FEEDP INC CHICKPCoefficient Std. Error t-Statistic Prob.

P 893.4857 335.0311 2.666874 0.0087FEEDP -1290.609 364.0891 -3.544761 0.0006

C 112266.0 49592.54 2.263769 0.0254


Two stage least squares (TSLS) provides a single estimate for the slope of the supply curve:

bSP= 893.5


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Overidentification: Comparison of Reduced Form and Two Stage Least Squares Estimates

Table 21.18 compares the estimates that result when using the two different estimation procedures:

Estimated slope Estimated Slopeof demand curve of supply curve

Reduced Form 366.0Based on Income Coefficients 1051.2Based on Chicken Price Coefficients 173.3

Two Stage Least Squares 366.0 893.5Table 21.18: Comparison of RF and TSLS Estimates

Note that the slope of the demand curve is not overidentified; furthermore, both the reducedform (RF) estimation procedure and the two stage least squares (TSLS) estimation procedureprovide the same estimate. On the other hand, the slope of the supply curve is overidentified.The reduced form (RF) estimation procedure provides two estimates; the two stage least squares(TSLS) estimation procedure provides only one.

Summary of Identification Issues: Reduced Form and Two Stage Least SquaresEstimation Procedures






Reduced Form and Two Stage Least Squares Estimation Procedures: A Comparison

Identified: The procedures are equilivalent. Underidentified: The procedures are equilivalent. Overidentified: The reduced form estimation procedure produces multiple estimates

while two stage least squares produces a single estimate.

Chapter 21 Review Questions

1. What does it mean for a simultaneous equation model to be underidentified?

2. What does it mean for a simultaneous equation model to be overidentified?

3. Compare the reduced form (RF) estimation procedure and the two stage least squares (TSLS)estimation procedure:

a. When will the two procedures produce identical results?b. When will the two procedures produce different results? How do the results differ?


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24

Chapter 21 Exercises

The following workfile contains the data we used in class to analyze the beef market:


Qt Quantity of beef in month t(millions of pounds)Pt Real price of beef in month t(1982-84 cents per pound)




Yeart Year

Consider the following constant elasticity model describing the beef market:

Demand Model: log(QDt ) =

DConst +

DPlog(Pt) +

DIlog(Inct) +

DCPlog(ChickPt) + e

Dt

Supply Model: log(QSt) =

SConst +

SPlog(Pt) +

SFPlog(FeedPt) + e

St

Equilibrium: log(QDt ) = log(QSt) = log(Qt)

1. Suppose that there were no data for the price of chicken and income; that is, while you caninclude the variable FeedP in your analysis, you cannot use the variables Incand ChickP.


a. Consider the reduced form (RF) estimation procedure:

1) Can we estimate the own price elasticity of demand, DP? If not, explain why

not. If so, does the reduced form estimation procedure provide a singleestimate? What is (are) the estimate (estimates)?

2) Can we estimate the own price elasticity of supply, SP? If not, explain why


b. Consider the two stage least squares (TSLS) estimation procedure:

1) Can we estimate the own price elasticity of demand, DP? If so, what is (are)

the estimate (estimates)?

2) Can we estimate the own price elasticity of supply, SP? If so, what is (are) the

estimate (estimates)?


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2. On the other hand, suppose that there were no data for the price of feed; that is, while youcan include the variables Incand ChickP in your analysis, you cannot use the variable FeedP.












3. Last, suppose that you can use all the variables in your analysis.











http://www3.amherst.edu/~fwesthoff/weblinks/55-Lec4-BeefMarket-1977-86.wf1http://www3.amherst.edu/~fwesthoff/weblinks/55-Lec4-BeefMarket-1977-86.wf1http://www3.amherst.edu/~fwesthoff/weblinks/55-Lec4-BeefMarket-1977-86.wf1http://www3.amherst.edu/~fwesthoff/weblinks/55-Lec4-BeefMarket-1977-86.wf1


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26

Chicken Market Data:Monthly time series data relating to the market for chicken from 1980 to1985.

Qt Quantity of chicken in month t(millions of pounds)

Pt Real price of whole chickens in month t(1982-84 cents per pound)

FeedPt Real price chicken formula feed in month t(1982-84 cents per pound)


PorkPt Real price of pork in month t(1982-84 cents per pound)

Yeart Year

Consider the following constant elasticity model describing the beef market:

Demand Model: log(QDt ) =

DConst +

DPlog(Pt) +

DIlog(Inct) +

DPPlog(PorkPt) + e

Dt

Supply Model: log(QSt) =

SConst +

SPlog(Pt) +

SFPlog(FeedPt) + e

St

Equilibrium: log(QDt ) = log(Q

St) = log(Qt)

4. Suppose that there were no data for the price of pork and income; that is, while you caninclude the variable FeedPin your analysis, you cannot use the variables Incand PorkP.








1) Can we estimate the own price elasticity of demand,D

P? If so, what is (are)the estimate (estimates)?


http://www3.amherst.edu/~fwesthoff/weblinks/55-Lec4-ChickenMarket-1980-85.wf1http://www3.amherst.edu/~fwesthoff/weblinks/55-Lec4-ChickenMarket-1980-85.wf1


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27

5. On the other hand, suppose that there were no data for the price of feed; that is, while youcan include the variables Incand PorkPin your analysis, you cannot use the variable FeedP.












6. Last, suppose that you use all the variables in your analysis.












In general, compare the reduced form (RF) estimation procedure and the two stage least squares(TSLS) estimation procedure.

7. When the reduced form estimation procedure (RF) provides no estimates for a coefficient,how many estimates does the (TSLS) estimation procedure provide?

8. When the reduced form estimation procedure (RF) provides a single estimate for acoefficient, how many estimates does the (TSLS) estimation procedure provide? How are theestimates related?

9. When the reduced form estimation procedure (RF) provides multiple estimates for acoefficient, how many estimates does the (TSLS) estimation procedure provide?
http://www3.amherst.edu/~fwesthoff/weblinks/55-Lec4-ChickenMarket-1980-85.wf1http://www3.amherst.edu/~fwesthoff/weblinks/55-Lec4-ChickenMarket-1980-85.wf1http://www3.amherst.edu/~fwesthoff/weblinks/55-Lec4-ChickenMarket-1980-85.wf1http://www3.amherst.edu/~fwesthoff/weblinks/55-Lec4-ChickenMarket-1980-85.wf1


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Appendix 21.1 Algebraic Derivation of the Reduced From Equations -Underidentification

Demand Model: QDt =

DConst +

DPPt + e

Dt

Supply Model: QSt =

SConst +

SPPt +

SFPFeedPt + e

St

Equilibrium: QDt = QSt = Qt

There are 5 parameters in the demand and supply models: DConst, DP,

SConst,

SP, and

SFP. Ideally,

we would like to estimate them all. Unfortunately, we will not be able to do so. To show thisformally we shall algebraically derive the reduced form equations.

Strategy to derive the reduced form equation for Pt:

Substitute Qtfor QDt and Q

St.

Subtract the equation for the supply model from the equation for the demand model. Solve for Pt.

QDt =

DConst +

DPPt + e

Dt

QSt =

SConst +

SPPt +

SFPFeedPt + e

St

Substitute

Qt = DConst +

DPPt + e

Dt

Qt = SConst +

SPPt +

SFPFeedPt + e

St

Subtract

0 = DConstSConst +

DPPt

SPPt

SFPFeedPt + e

Dt e

St

Solve

SPPt DPPt =

DConst

SConst

SFPFeedPt + e

Dt e

St

(S

P

D

P)Pt =D

Const

S

Const

S

FPFeedPt + e

D

t

e

S

t

Pt =DConst

SConst

SPDP

SFP

SPDP

FeedPt +eDt e

St

SPDP

Strategy to derive the reduced form equation for Qt:


St.

Multiply the equation for the demand model by SPand the equation for the supply

model by DP.

Subtract the equation for the supply model from the equation for the demand model.

Solve for Qt.


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29

QDt =

DConst +

DPPt + e

Dt

QSt =

SConst +

SPPt +

SFPFeedPt + e

S

Substitute.

Qt = DConst +

DPPt + e

Dt

Qt = SConst +

SPPt +

SFPFeedPt + e

S

Multiply.

SPQt = SP

DConst +

SP

DPPt +

SPe

Dt

DPQt = DP

SConst +

DP

SPPt +

DP

SFPFeedPt +

DPe

S

Subtract.

SPQt DPQt =

SP

DConst

DP

SConst + 0

DP

SFPFeedPt +

SPe

Dt

DPe

S

Solve.

(SPDP)Qt =

SP

DConst

DP

SConst

DP

SFPFeedPt +

SPe

Dt

DPe

S

Qt =SP

DConst

DP

SConst

SPDP

DP

SFP

SPDP

FeedPt +SPe

Dt

DPe

S

SPDP

Compare the reduced form equations for Qtand Pt:

Qt =SP

DConst

DP

SConst

SPDP

DP

SFP

SPDP

FeedPt +SPe

Dt

DPe

S

SPDP

Pt =DConst

SConst

SPDP

SFP

SPDP

FeedPt +eDt e

St

SPDP

Next, let the s represent the constants and coefficients of the reduced form (RF) equations:

Qt = QConst +

QFPFeedPt +

Qt

Pt = PConst +

PFPFeedPt +

Pt

where the following equations specify the 4 s:

QConst =SP

DConst

DP

SConst

SPDP

QFP = DP

SFP

SPDP

PConst =DConst

SConst

SPDP

PFP = SFP

SPDP

There are 5 parameters in the original demand/supply model, 5 unknown s, and only 4equations specifying the s. We cannot solve for all 5 unknowns with only 4 equations. Theoriginal demand/supply model is underidentified. More specifically, we cannot solve for the

slope of the supply curve, SP; on the other hand, we can solve for the slope of the demand

curve, DP:

Ratio of FeedPtcoefficients:QFP

PFP =

DP

SFP

SPDP

SFP

SPDP

= DP= Slope of the demand curve


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30

Appendix 21.2 Algebraic Derivation of the Reduced From Equations -Underidentification

Demand Model: QDt =

DConst +

DPPt +

DIInct + e

Dt

Supply Model: QS

t =

S

Const +

S

PPt + eS

t


St = Qt


DI,

SConst, and

SP. Ideally, we

would like to estimate them all. Unfortunately, we will not be able to do so. To show thisformally we shall algebraically derive the reduced form equations.



St.

Subtract the equation for the supply model from the equation for the demand model. Solve for Pt.

QDt =

DConst +

DPPt +

DIInct + e

Dt

QSt =

SConst +

SPPt + e

St

Substitute

Qt = DConst +

DPPt +

DIInct + e

Dt

Qt = SConst +

SPPt + e

St

Subtract

0 = DConstSConst +

DPPt

SPPt +

DIInct + e

Dt e

St

Solve

SPPt DPPt =

DConst

SConst +

DIInct + e

Dt e

St

(SPD

P)P

t = D

ConstS

Const+ D

IInc

t + e

D

teS

t

Pt =DConst

SConst

SPDP

+DI

SPDP

Inct +e

Dt e

St

SPDP



St.


model by DP.

Subtract the equation for the supply model from the equation for the demand model. Solve for Qt.


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31

QDt =

DConst +

DPPt +

DIInct + e

Dt

QSt =

SConst +

SPPt + e

S

Substitute.

Qt = DConst +

DPPt +

DIInct + e

Dt

Qt = SConst +

SPPt + e

S

Multiply.

SPQt = SP

DConst +

SP

DPPt +

SP

DIInct +

SPe

Dt

DPQt = DP

SConst +

DP

SPPt +

DPe

S

Subtract.

SPQt DPQt =

SP

DConst

DP

SConst + 0 +

SP

DIInct +

SPe

Dt

DPe

S

Solve.

(SPDP)Qt =

SP

DConst

DP

SConst +

SP

DIInct +

SPe

Dt

DPe

S

Qt =SP

DConst

DP

SConst

SPDP

+SP

DI

SPDP

Inct +SPe

Dt

DPe

S

SPDP

Compare the reduced form equations for Qtand Pt:

Qt =SP

DConst

DP

SConst

SPDP

+SP

DI

SPDP

Inct +SPe

Dt

DPe

S

SPDP

Pt =DConst

SConst

SPDP

+DI

SPDP

Inct +e

Dt e

St

SPDP


Qt = QConst +

QI Inct +

Qt

Pt = PConst +

PI Inct +

Pt


QConst =SP

DConst

DP

SConst

SPDP

QI =SP

DI

SPDP

PConst =DConst

SConst

SPDP

PI =DI

SPDP

There are 5 parameters in the original demand/supply model, 5 unknown s, and only 4equations specifying the s. We cannot solve for all 5 unknowns with only 4 equations. Theoriginal demand/supply model is underidentified.

More specifically, we cannot solve for the slope of the supply curve, DP; on the other hand, wecan solve for the slope of the demand curve, SP:

Ratio of Inctcoefficients:QI

PI =

SPDI

SPDP

DI

SPDP

= SP= Slope of the supply curve


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Appendix 21.3 Algebraic Derivation of the Reduced From Equations -Overidentification

Demand Model: QDt =

DConst +

DPPt +

DIInct +

DCPChickPt + e

Dt

Supply Model: QS

t =

S

Const +

S

PPt +

S

FPFeedPt + eS

tEquilibrium: Q

Dt = Q

St = Qt


DI,

DCP,

SConst,

SP, and

SFP.

These models are overidentified meaning that (at least) one of the parameters can be estimated intwo different ways. To show this formally we shall algebraically derive the reduced formequations.



St.


Solve for Pt.

QDt = DConst +

DPPt +

DIInct +

DCPChickPt + e

QSt = SConst +

SPPt +

SFPFeedPt + e

Substitute

Qt = DConst +

DPPt +

DIInct +

DCPChickPt + e

Qt = SConst +

SPPt +

SFPFeedPt + e

Subtract

0 = DConstSConst +

DPPt

SPPt

SFPFeedPt +

DIInct +

DCPChickPt + e

Dt

Solve

SPPt DPPt = DConstSConst SFPFeedPt + DIInct + DCPChickPt + eDt

(SPDP)Pt =

DConst

SConst

SFPFeedPt +

DIInct +

DCPChickPt + e

Dt

Pt =DConst

SConst

SPDP

SFP

SPDP

FeedPt +DI

SPDP

Inct +DCP

SPDP

ChickPt +e

Dt

SP


33/34

33



St.


model by DP.


Solve for Qt.

QDt =

DConst +

DPPt +

DIInct +

DCPChickPt +

QSt =

SConst +

SPPt +

SFPFeedPt +

Substitute.

Qt = DConst +

DPPt +

DIInct +

DCPChickPt +

Qt = SConst +

SPPt +

SFPFeedPt +

Multiply.

SPQt = SP

DConst +

SP

DPPt +

SP

DIInct +

SP

DCPChickPt +

DPQt = DP

SConst +

DP

SPPt +

DP

SFPFeedPt +

Subtract.SPQt

DPQt=

SP

DConst

DP

SConst + 0

DP

SFPFeedPt +

SP

DIInct +

SP

DCPChickPt +

SPe

Dt

Solve.

(SPDP)Qt =

SP

DConst

DP

SConst

DP

SFPFeedPt +

SP

DIInct +

SP

DCPChickPt +

SPe

Dt

Divide.

Qt =SP

DConst

DP

SConst

SPDP

DP

SFP

SPDP

FeedPt +SP

DI

SPDP

Inct +SP

DCP

SPDP

ChickPt +SPe

Dt

SPCompare the reduced form equations for Qtand Pt:

Qt =

SPDConst

DP

SConst

SPDP

DPSFP

SPDPFeedPt +

SPDI

SPDPInct +

SPDCP

SPDPChickPt +

SPeDt

SP

Pt =DConst

SConst

SPDP

SFP

SPDP

FeedPt +DI

SPDP

Inct +DCP

SPDP

ChickPt +e

Dt

SP


Qt = QConst

QFPFeedPt +

QI Inct +

QCPChickPt +

Pt = PConst

PFPFeedPt +

PIInct +

PCPChickPt +


QConst=SP

DConst

DP

SConst

SPDP

QFP= DP

SFP

SPDP

QI =SP

DI

SPDP

QCP=SP

DCP

SPDP

PConst=DConst

SConst

SPDP

PFP= SFP

SPDP

PI =DI

SPDP

PCP=DCP

SPDP


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34

There are 7 parameters in the original demand/supply model, 7 unknown s, and 8 equationsspecifying the s. We have more equations than unknowns. The original demand/supply model

is overidentified. More specifically, we can solve for the slope of the supply curve, SP, in two

ways:

Ratio of Inctcoefficients:QI

PI =

S

P

D

ISP

DP

DI

SPDP


Ratio of ChickPtcoefficients:QCP

PCP =

SPDCP

SPDP

DCP

SPDP


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