Stat Corner

Deep Dive on Elasticities

April 17-18, 2013

Price Response -- The Theoretical High Ground

» Consumers maximize utility subject to the prices of goods and services and their budgets.

• When the price of a normal good increases, there are two effects, an income effect and a substitution effect.

• The main effect is usually the substitution effect. Consumers substitute away from the good that has increased in price and towards other goods.

» Producers maximize profits and minimize costs.

• When the price of an input increases, producers substitute away from that input and other complimentary inputs and toward other inputs.

• Energy is complementary with equipment. Energy and equipment are often substitutes for labor.

Price Response – The Energy Reality

» Consumers and producers do not consume energy directly. Energy and equipment are combined to provide services (heat, light, work).

» Energy prices impact several types of decisions.

• Long term responses include:

- End-use equipment acquisition decisions, which determine equipment saturation levels

- Fuel choice decisions in new construction, replacement and conversions (e.g. space heating, cooking, water heating)

- End-use efficiency decisions at the time of equipment purchase

- Measure and device decisions that impact efficiency and usage (e.g. set-back thermostats and occupancy sensors)

• Short term responses include:

- Utilization levels (e.g. turning lights off, thermostat settings)

- Non-price conservation (often in response to the news)

Statistical Models

( ) e...,PFQ +=

( )dP

...,PdF

dP

dQ==PriceSlope

• The most general form of a statistical model of energy consumption is:

• Within this framework, price response can be measured by the derivative of quantity with respect to price.

• In this general form, the price slope is a function. It depends on the value of price and the value of other factors. It also depends on the units of price and quantity.

– Q in GWh, P in $/MWh: Slope = -.5 GWh per $/MWh

– Q in KWh, P in $/MWh: Slope = -500,000 KWh per $/MWh

– Q in kWh, P in cents/kWh: Slope = -50,000 kWh per cents/KWh

General Definition of Price Elasticity

( )( )...,PF

P

dP

...,PdF

Q

P

dP

dQElast ×=×=

P%

Q%

P/P

Q/Q

P/dP

Q/dQ

Q

P

dP

dQElast

∆

∆=

∆

∆≈=×=

• In its most general form, price elasticity is computed as follows:

• Like the derivative, price elasticity is a function. However, it is normalized to be unit free. Locally, it can be thought of as the percent change in Q caused by a one percent change in P.

Price Elasticity - What Does it Mean?

P%

Q%

P/P

Q/Q

P/dP

Q/dQ

Q

P

dP

dQElast

∆

∆=

∆

∆≈=×=

• The price elasticity gives the percent change in quantity for a one percent change in price. For small price changes this is an accurate interpretation. For large price changes, this can be misleading.

– Elasticity = -.15: a 1% price increase implies a .15% decline in quantity.

– Elasticity = -.15: a 10% price increase implies about a 1.5% decline in quantity.

– Elasticity = -.15: a 100% price increase implies a ?% decline in quantity

– Elasticity = -.15, a 1000% price increase implies ???% decline in quantity

Estimating Elasticity with Linear Models

ttt e...PbaQ ++×+=

t

t

t

t

t

t

Q

Pb

Q

P

dP

dQElast ×=×=

• The most common specification is linear. It looks like this.

• For this specification, the elasticity is:

• Example:

Sidebar – Economists usually put P on the Y

axis. Why is that?

Q = 225 - .50×P

Linear Models – A Deeper Look

Y=Quantity

X=Price

Q = 225 - .50×P

9050

180

200

∆Q=-20

∆P=40Slope = -20/40 = -.50

180

9050.250.Elast ×−=−=

200

5050.125.Elast ×−=−=

Linear Models – Arc Elasticity

Y=Quantity

X=Price

Q = 225 - .50×P

P2=90P1=50

Q2=180

Q1=200

∆Q=-20

∆P=40Slope = -20/40 = -.50

190

70

40

20184.ArcElast ×−=−=

( ) ( )( )

( )( ) ( )( ) 21PF2PF

21P2P

1P2P

1PF2PFArcElast

+

+×

−

−=

Elast = -.125

Elast = -.250

Estimating Elasticities with Log/Log Models

• The geometric or multiplicative model is another common specification.

• The derivative is:

• For estimation, taking logs of both sides gives:

...PaQb

tt ××=

( ) ( ) ( ) tt e...PlnbalnQln ++×+=

bQ

P

dP

dQElast

t

t

t

t =×=

Key Fact:X

1

dX

)Xln(d=

-3.00

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

1 2 3 4 5

ln(X

)

( )t

t1b

t

t

t

P

Qb...Pab

dP

dQ×=×××=

−

• So, the elasticity is:

bQ

P

dP

dQ

Plnd

dP

dP

dQ

dQ

Qlnd

Plnd

Qlnd

=×=

××=

Log/Log Models – A Deeper Look

Y=Quantity

Q2=180

Q1=200

∆Q=-20

∆P=40

179.Elast −=

184.ArcElast −=

P2=90P1=50

Elast = -.179

X=Price

179.P2.403Q −×=

179.Elast −=

What MetrixND Provides

• For a linear regression, the Elas tab provides the elasticity at the mean of the historical price and quantity values.

• For linear regressions, the practical significance of price variations can be viewed on the BX tab.

• For neural networks:

• F’ tab shows the derivative of Y wrt to each X for each obs.

• The Elas tab provides the corresponding elasticities for each obs.

Y

XbMean@Elast ×=

( )

t,i

t,i

t,i

t,nt,1

tY

X

dX

X,...,XdNNElast ×=

Making Elasticities Dynamic – Lagged Q

• One approach is to introduce a lagged dependent variable.

• When price increases by 1, there are a series of impacts:

– Q drops by bp in the first period.

– Q drops by c×××× bp in the second period.

– Q drops by c2×××× bp in the third period

– This continues with increasingly smaller impacts

• Implications are:

– The biggest impact occurs in period 1

– Impacts decline geometrically in subsequent periods

– The same lag structure applies to other X’s (such as CDD), which does not make much sense in energy modeling.

• So, the long run elasticity is:

t1t

i

t,iitt eQdXOthercicePrbaQ +×+×+×+= −∑

( )t

t

t

t32

Q

P

d1

b

Q

P...ddd1bElastLR ×

−=×++++×=

Making Elasticities Dynamic – Distributed Lags

• Another approach is to introduce distributed lags.

• With this approach, there many price coefficients to estimate.

• One simplification is:

• Another “simplification” imposes a polynomial lag structure.

tLtL1t1t0t e...Pb...PbPbaQ ++×+×+×+= −−

( ) t

L

1

tt e...PPDLaQ ++×+= ∑=

−

l

ll

t

L

1

t

1t0t e...L

P

bPbaQ ++×+×+=

∑=

−

l

l

Structured Price Variables – SAE Approach

ttt CoolUseCoolIndexXCool ×=

×

ε

×

ε

×

ε

=

98

thhsize,c

98

tinc,c

98

tp,c

98

tt

CDD

CDD

HHSize

HHSize

Income

Income

P

PCoolUse

×=

y

tt Eff

SatkCoolIndex

• εεεεc,p is the elasticity of CoolUse with respect to P

• Similar equations apply to XHeat and XOther.

• Saturation and efficiency can also be modeled to depend on price, but the short run effects will all be in usage because equipment stocks change slowly over time.

• With this approach, structured variables are constructed for each of the major uses (Heating, Cooling, Other). For example:

Structured Price Variables – SAE Approach

tothtct XOtherbXHeatbXCoolbaQ ×+×+×+=

××ε+

××ε+

××ε=

×

×ε×+

×ε×+

×ε×=×

t

top,o

t

thp,h

t

tcp,c

t

t

t

tp,oo

t

tp,hh

t

tp,cc

t

t

t

t

Q

XOtherb

Q

XHeatb

Q

XCoolb

Q

P

P

XOtherb

P

XHeatb

P

XCoolb

Q

P

dP

dQ

• This says that the overall price elasticity is a weighted average of the three internal index price elasticities (εεεεc,p, εεεεc,p, εεεεc,p ) where bX shares are the weights. The weights will sum to less than 1 when the constant term (or other variables in the model) have a positive contribution to the predicted value.

• Then we estimate the model (the statistical adjustment step):

Complications Defining Price

Complications Defining Price

• Any nonlinearities in rates raise concerns about how to compute price.

– With fixed charges or declining blocks, usage increases cause average prices to fall.

– This can cause strong biases in estimated price elasticities.

• Alternatives are:

– Use the marginal price.

– Use an index of tariffs (e.g., price out a fixed consumption pattern)

– Use a 12 month moving average of average prices

• With linear models, a price change will cause the same quantity impact in all months (regardless of the level of consumption).

– For gas, consumption in summer is small, so price impact must be small.

– To make price impacts seasonal, construct an “expected” bill using an average monthly consumption pattern. Or include separate price effects in heating, cooling, and other components.

• Finally, always use real (adjusted for inflation) price.

Discussion of Practical Problems

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