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""CHAPTER 12 VALUING IMPACTS FROM OBSERVED BEHAVIOR: DIRECT ESTIMATION OF DEMAND CURVES
Purpose: The key concept for valuing policy impacts is change in social surplus. As discussed in Chapters 3 and 4, changes in social surplus are represented by areas, often as triangles or trapezoids, bounded by supply and demand curves. Measurement of changes in social surplus is relatively straightforward when we know the shape and positions of the supply and demand curves in the relevant primary market, before and after the policy change. In practice, however, these curves are usually not known. Analysts have to estimate them or find alternative ways to measure benefits and costs. In this chapter, we discuss direct estimation of these curves, focusing on estimating demand curves.
PROJECT REVENUES AS THE MEASURE OF (GROSS) BENEFITS
Revenues are a natural measure of benefits to firms in the private sector. But, in CBA as discussed in Chapter 4, revenues do not always equal (gross) consumer benefits.
Revenues may be used to measure benefits when there are no consumers with standing (and hence no CS) such as when all output is exported. They can also be used when the government sells goods in an undistorted market without affecting the market price unfortunately this usually is not the case in projects evaluated by CBA.
ESTIMATION KNOWING ONE POINT ON THE DEMAND CURVE AND ITS SLOPE OR ELASTICITY
Suppose we know only one point on the demand curve, but previous research provides an estimate of either the elasticity or slope of the demand curve. We first suppose the demand curve is linear and then suppose the demand curve has constant elasticity.
Linear Demand Curve
A linear demand curve assumes that the relationship between the quantity demanded and the price is linear; that is, the demand curve can be written as:
q = 0 + 1p (12.1)
where, q is the quantity demanded at price p, 0 is the quantity that would be demanded if price were zero (the intercept), and 1 indicates the change in the quantity demanded as a result of a one unit increase in price (the slope). If you know one point on the demand curve and its slope , then you can compute other points on the curve straightforwardly.
If the demand curve is linear and we have an estimate of its elasticity then we also need to know the price and quanity at which the elasticity was calculated. The price elasticity of demand, d, measures the responsiveness of the quantity demanded to changes in price -- the more it
responds, the higher the elasticity. For a linear demand curve, equation (12.1), the price elasticity of demand equals:
d = 1p/q (12.3)
Thus, the elasticity is non-constant -- it varies with both price and quantity. If we know the elasiticity and p and q, we can use equation (12.3) to estimate the slope of the demand curve and then it is straightforward to compute other points on the demand curve, as before.
Constant Elasticity Demand Curve
Economists have found that many goods have a constant elasticity demand curve, that is,
EMBED Equation.3 (12.4)
where, q denotes quantity demanded and p is price, as before, and 0 and 1 are parameters. In order to interpret 1 it is useful to take the natural logarithm, denoted by ln, of both sides of equation (12.4), which gives:
lnq = ln0 + 1lnp (12.5)
We see immediately that the constant elasticity demand curve is linear in logarithms. Furthermore, 1, the slope of this linear in logarithms demand curve, equals d, the price elasticity of demand. As d equals the slope of a linear curve, which is a constant, it follows that the price elasticity of demand is constant; hence the name of this demand curve. Again, given one point on the constant elasticity demand curve and an estimate of its elasticity, the whole curve can be plotted straightforwardly.
It is useful to note for the purpose of computing changes in social surplus that the area under a constant elasticity demand curve from quantity q0 to quantity q1 is given exactly by:
EMBED Equation.3 (12.6)
where = [1 + (1/1)].
Slope and elasticity estimates of demand curves can often be obtained from prior research. When this happens, you need to consider possible internal and external validity problems (i.e., how valid is the estimate [internal how was it measured and computed] and can it be used in this instance [external how similar is the case in question to the research case]).
EXTRAPOLATING FROM A FEW POINTS
If we know a few points on the demand curve, we can use them to predict another point of relevance to policy evaluation. There are two important considerations when extrapolating:
Different functional forms lead to different answers. Furthermore, the further we extrapolate from past experience, the more sensitive the predictions are to assumptions about the functional form.
The validity of attributing an outcome change to the policy change (i.e. other variables are assumed to remain constant) may be questionable.
More observations provide greater validity.
Econometric Estimation with Many Observations
If we have many different observations of prices and quantities, we can apply econometric methods to estimate the entire demand curve.
Model specification
The econometric model should include all explanatory (so-called independent) theoretically-relevant variables, even if one is not specifically interested in their effect. Excluding a theoretically important variable is one form of specification error. Using the incorrect functional form is another form of specification error.
Types of data
Sometimes you can generate your own data but, more often, limited resources require one to use data available at lower costs (previously published data, data originally collected for other purposes, and/or sampling administrative records or clients). Considerations in the choice of data are:
SEQ 1_0 \* Arabic \r 11) Level of aggregation, whether individual or group. Individual level data are preferred because most theory is based on individual utility maximization. Also, aggregate data can lead to less precise estimates.
SEQ 1_0 \* Arabic \n2) Choice of cross-sectional and time series data. Cross-sectional data generally provides estimates of long-run elasticities, while time series data usually provides estimates of short-run elasticities. Short-run elasticities are generally smaller than long-run elasticities (because there is less time to adjust to new prices). Cross-section data faces the possible problem of heteroskedasticity, which means the error terms have different variances. Time-series data may have problems of autocorrelation, which arises if the error terms are correlated over time. Both problems can be tested for and corrected using generalized least squares (GLS) instead of OLS. It is possible to have pooled cross-sectional and time-series data. This provides a rich source of information but requires more complicated econometric methods.
Identification
In a perfectly competitive market, price and quantity result from the simultaneous interaction of supply and demand. Changes in price and quantity can result from shifts of the supply curve, shifts of the demand curve, or both. In the absence of variables that affect only one side of the market (demand or supply, but not both), it may not be possible to estimate separate supply and demand curves. Indeed, if quantity supplied and quantity demanded depended only on price, then the equation for both the demand curve and the supply curve would look identical! How can we identify which is the demand curve and which is the supply curve? This is one example of the problem of identification. It occurs in multiple equation models in which some variables, such as price and quantity, are determined simultaneously. Such variables are called endogenous variables. In contrast, variables that are fixed or determined outside of the model are called exogenous variables.
To identify the demand curve, you need a variable that affects supply but not demand. This variable systematically shifts supply but not demand, thereby tracing out the demand curve. To identify the supply curve you require a variable that affects demand but not supply.
The identification problem does not arise when the government supplies a good or sets the price (there is no supply curve price is exogenous). Identification is only a problem if price is endogenous.
Confidence Intervals
The standard errors of the estimated coefficients can be used to construct confidence intervals. Confidence intervals provide some guidance for sensitivity analysis.
Prediction versus Hypothesis Testing
In cost-benefit analysis we often have to make a prediction. Thus, we are interested in all of the estimated coefficients, whether or not they are statistically different from zero.
Boardman, Greenberg, Vining, Weimer / Cost-Benefit Analysis, 3rd Edition
Instructor's Manual 12- PAGE 4
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