Chapter 7 – Empirical Demand Functions

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Chapter 7 – Empirical Demand Functions. Optimal pricing is critical to the success of any business. Given the stakes, it is frequently worth investing significant resources in determining the optimal price.

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Chapter 7 – Empirical Demand Functions
• Optimal pricing is critical to the success of any business.
• Given the stakes, it is frequently worth investing significant resources in determining the optimal price.
• Large firms have pioneered the use of econometrics in estimating demand functions to serve as the base for determining the best price.
Empirical Demand Functions
• Empirical demand functions are demand equations derived from actual market data.
• There are two basic methods for demand estimation
• Direct Methods
• Consumer Interviews
• Market Studies and Experiments
• Indirect Methods (econometrics/regression techniques)
Consumer Interviews
• Estimation of demand is based on asking individual consumers about their “potential” purchase decisions. Essentially playing the “what if” game – If the price is \$10, how many units would you purchase? However, if price is decreased to \$8, how many units would you now purchase?
• Sample must be representative of population.
• Random samples
• Response bias occurs when hypothetical answers are given to hypothetical questions.
• “People Lie”
• Sometimes unable to accurately answer the question
Market Studies and Experiments
• Market studies & experiments are the economists attempt to control or hold constant many of the factors affecting the amount purchased.
• All factors except price are held constant and the amount purchased is observed in response to variation in price.
Market Studies
• Use the market to directly react to changes in price
• M&M/Mars changed size of product and held price constant(effectively reduced price per oz). Response was positive and substantial
• Must be careful or may lose consumer goodwill especially if price was low during study and higher after research
Experiments
• Simulate market experience. Give income and make purchases(subject to price variation). Often known probabilities that you can keep your purchases - providing an incentive to make optimum choices.
Specification of the Empirical Demand Function
• Specification of the relationship includes two critical elements
• Choosing the correct set of independent variables
• Choosing appropriate functional form for the relationship
Choosing the Independent Variables

Best guide to choosing IVs is to turn to theory.

Price expectations and tastes and preferences are often

dropped due to the difficulty of measuring or estimating them.

Choosing the Appropriate Functional Form
• Choices we consider are
• Linear: Q=a+bP+cM+dPR+eN
• Log-linear: Q=aPbMc(PR)dNe
Linear Demand
• Q=a+bP+cM+dPR+eN
• Coefficient b yields the change in Q in response to a 1 unit increase in Price(b should be negative)
• Coefficient c can be positive(normal) or negative(inferior). It yields the change in Q in response to a 1 unit increase in incoMe.
• Coefficient d can be positive(substitutes) or negative(complements). It yields the change in Q in response to a 1 unit increase in the Price of a Related good.
• Coefficient e should be positive and yields the change in Q in response to a 1 unit increase in Number of buyers( proxy is often population)
Elasticities for Linear Specification

Own Price Elasticity

Income Elasticity

Cross Price Elasticity

Log-linear Demand
• Q=aPbMc(PR)dNe
• Converted into logs yields

lnQ = ln a + b ln P + c ln M + d ln PR + e ln N

• Elasticites are constant and equal to th estimated coefficients
Control of Price
• Market-determined price - For some products and firms, managers have no control over price. Price is set in the market and firms must accept it as a given.
• Manager-determined price – For other products and firms, mangers do have control over price and set price to achieve their goal.
• Estimating demand for these two cases is quite different, as we will see.
Endogenous versus Exogenous Variables
• An endogenous variable is a variable determined by a system of equations. If price is market-determined, it is an endogenous variable since it is determined simultaneously by S and D.
• An exogenous variable is a variable in a system of equations that is determined outside the system. If price is manager-determined, it is an exogenous variable since it is determined outside the S and D simultaneous equation model.
Estimating Demand for Price-Taking Firms
• The problem of estimating demand when price is market-determined is often referred to as the simultaneity problem.
• See Figure 7.1 as base for the simultaneity discussion.
Reduced Form Equations- Market for Gas
• Demand: Q = a + bP + cM + d
• Supply: Q = h + kP + lPc + s
• Endogenous variables are P and Q.
• Demand: Q = A + bP, where A=a+cM+d
• Supply: Q = H + kP, where H= lPc+s
• PE= f(M,Pc, d ,s) & QE = g(M,Pc, d ,s)
• The above are the reduced-form equations, which express the endogenous variables as a function of the exogenous variables and the random errors.

Pc is the price of crude oil

Reduced-Form Equations
• The reduced-form equations indicate
• The observed values of the endogenous variables, P & Q, are determined by all the exogenous variables and random errors in both the supply and the demand equations.
• The observed values of price are correlated with (depend on) the random errors in both S and D.
• Ordinary Least Squares(OLS) cannot be used if the observed values of price are correlated with the random errors.
Two-Stage Least Squares(2SLS)
• To estimate demand where price is endogenously determined by D & S, we must perform the following steps:
• Identification of demand – make sure the sample data will properly estimate the true demand relationship.
• Using data, estimate parameters(coefficients) of industry demand equation with 2SLS. 2SLS deals with simultaneous equation bias, in which endogenous variable is determined simultaneously by “two” relationships. If OLS had been used simultaneous equation bias would have resulted.
Identification of Demand
• A demand equation is identified when the true demand function can be estimated from sample data on equilibrium points.
• The above occurs when Supply includes at least one exogenous variable not in demand. In our example supply included Pc, the price of crude oil which was not in the demand function.
Estimating Demand for Price-Setting Firms
• When prices are manager-determined, the problem of simultaneity vanishes and single equation OLS can be used for estimating the parameters of the demand function without any regard for supply.