April 17-18, Stat Corner Deep Dive on Elasticities

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1 April 17-18, 2013 Sa Corner Deep Dive on Elasiciies

2 rice Response -- The Theoreical High Ground» Consumers maximize uiliy subjec o he prices of goods and services and heir budges. When he price of a normal good increases, here are wo effecs, an income effec and a subsiuion effec. The main effec is usually he subsiuion effec. Consumers subsiue away from he good ha has increased in price and owards oher goods.» roducers maximize profis and minimize coss. When he price of an inpu increases, producers subsiue away from ha inpu and oher complimenary inpus and oward oher inpus. Energy is complemenary wih equipmen. Energy and equipmen are ofen subsiues for labor.

3 rice Response The Energy Realiy» Consumers and producers do no consume energy direcly. Energy and equipmen are combined o provide services (hea, ligh, work).» Energy prices impac several ypes of decisions. Long erm responses include: - End-use equipmen acquisiion decisions, which deermine equipmen sauraion levels - Fuel choice decisions in new consrucion, replacemen and conversions (e.g. space heaing, cooking, waer heaing) - End-use efficiency decisions a he ime of equipmen purchase - Measure and device decisions ha impac efficiency and usage (e.g. se-back hermosas and occupancy sensors) Shor erm responses include: - Uilizaion levels (e.g. urning lighs off, hermosa seings) - Non-price conservaion (ofen in response o he news)

4 Saisical Models The mos general form of a saisical model of energy consumpion is: (,...) e Q F + Wihin his framework, price response can be measured by he derivaive of quaniy wih respec o price. riceslope dq d (,...) df d In his general form, he price slope is a funcion. I depends on he value of price and he value of oher facors. I also depends on he unis of price and quaniy. Q in GWh, in $/MWh: Slope -.5 GWh per $/MWh Q in KWh, in $/MWh: Slope -500,000 KWh per $/MWh Q in kwh, in cens/kwh: Slope -50,000 kwh per cens/kwh

5 General Definiion of rice Elasiciy In is mos general form, price elasiciy is compued as follows: (,... ) dq df Elas d Q d F(,... ) Like he derivaive, price elasiciy is a funcion. However, i is normalized o be uni free. Locally, i can be hough of as he percen change in Q caused by a one percen change in. Elas dq d Q dq / Q d / Q / Q / % Q %

6 rice Elasiciy - Wha Does i Mean? The price elasiciy gives he percen change in quaniy for a one percen change in price. For small price changes his is an accurae inerpreaion. For large price changes, his can be misleading. Elas dq d Q dq / Q d / Q / Q / % Q % Elasiciy -.15: a 1% price increase implies a.15% decline in quaniy. Elasiciy -.15: a 10% price increase implies abou a 1.5% decline in quaniy. Elasiciy -.15: a 100% price increase implies a?% decline in quaniy Elasiciy -.15, a 1000% price increase implies???% decline in quaniy

7 Esimaing Elasiciy wih Linear Models The mos common specificaion is linear. I looks like his. Q a + b e For his specificaion, he elasiciy is: dq d Elas b Q Q Example: Q Sidebar Economiss usually pu on he Y axis. Why is ha?

8 Linear Models A Deeper Look YQuaniy Q Elas Q-20 Elas Slope -20/ Xrice

9 Linear Models Arc Elasiciy YQuaniy Q ArcElas F ( 2) F( 1) ( 2 1) ( 2 + 1) 2 ( F( 2) + F( 1) ) 2 Elas ArcElas Q1200 Q2180 Q-20 Elas Slope -20/ Xrice

10 Esimaing Elasiciies wih Log/Log Models The geomeric or muliplicaive model is anoher common specificaion. The derivaive is: dq d Q a... Elas b So, he elasiciy is: dq d b b 1 ( a... ) Q For esimaion, aking logs of boh sides gives: ( Q ) ln( a) + b ln( ) +... e ln + b Q b Key Fac: ln(x) d ln Q d ln d ln(x) dx 1 X d ln Q dq dq d dq d Q b d d ln

11 Log/Log Models A Deeper Look YQuaniy Q Elas.179 Q1200 Q2180 Q-20 ArcElas.184 Elas.179 Elas Xrice

12 Wha MerixND rovides For a linear regression, he Elas ab provides he elasiciy a he mean of he hisorical price and quaniy values. X Mean b Y For linear regressions, he pracical significance of price variaions can be viewed on he BX ab. For neural neworks: F ab shows he derivaive of Y wr o each X for each obs. The Elas ab provides he corresponding elasiciies for each obs. dnn X (,..., X ) 1, n, Elas dxi, X Y i, i,

13 Making Elasiciies Dynamic Lagged Q One approach is o inroduce a lagged dependen variable. Q + e a + b r ice + ci XOheri, + d Q 1 i When price increases by 1, here are a series of impacs: Q drops by b p in he firs period. Q drops by c b p in he second period. Q drops by c 2 b p in he hird period This coninues wih increasingly smaller impacs So, he long run elasiciy is: 2 3 b ElasLR b ( 1+ d + d + d +...) Q 1 d Implicaions are: The bigges impac occurs in period 1 Impacs decline geomerically in subsequen periods The same lag srucure applies o oher X s (such as CDD), which does no make much sense in energy modeling. Q

14 Making Elasiciies Dynamic Disribued Lags Anoher approach is o inroduce disribued lags. Wih his approach, here many price coefficiens o esimae. One simplificaion is: Anoher simplificaion imposes a polynomial lag srucure. L L e... b... b b a Q ( ) L 1 e... DL a Q l l l L e... L b b a Q l l

15 Srucured rice Variables SAE Approach Wih his approach, srucured variables are consruced for each of he major uses (Heaing, Cooling, Oher). For example: XCool CoolIndex CoolUse CoolIndex Sa k Eff y CoolUse 98 ε c,p Income Income 98 ε c,inc HHSize HHSize 98 ε c,hhsize CDD CDD 98 ε c,p is he elasiciy of CoolUse wih respec o Similar equaions apply o XHea and XOher. Sauraion and efficiency can also be modeled o depend on price, bu he shor run effecs will all be in usage because equipmen socks change slowly over ime.

16 Srucured rice Variables SAE Approach Then we esimae he model (he saisical adjusmen sep): Q a + b XCool + b XHea + b XOher c h o dq d Q b c ε c,p XCool + b h ε h,p XHea + b o ε o,p XOher Q ε c,p bc XCool Q + ε h,p bh XHea Q + ε o,p bo XOher Q This says ha he overall price elasiciy is a weighed average of he hree inernal index price elasiciies (ε c,p, ε c,p, ε c,p ) where bx shares are he weighs. The weighs will sum o less han 1 when he consan erm (or oher variables in he model) have a posiive conribuion o he prediced value.

17 Complicaions Defining rice

18 Complicaions Defining rice Any nonlineariies in raes raise concerns abou how o compue price. Wih fixed charges or declining blocks, usage increases cause average prices o fall. This can cause srong biases in esimaed price elasiciies. Alernaives are: Use he marginal price. Use an index of ariffs (e.g., price ou a fixed consumpion paern) Use a 12 monh moving average of average prices Wih linear models, a price change will cause he same quaniy impac in all monhs (regardless of he level of consumpion). For gas, consumpion in summer is small, so price impac mus be small. To make price impacs seasonal, consruc an expeced bill using an average monhly consumpion paern. Or include separae price effecs in heaing, cooling, and oher componens. Finally, always use real (adjused for inflaion) price.

19 Discussion of racical roblems

20 THANK YOU