Ch Rival Pure private goods (most retail goods) Non-Rival Impure public goods (internet service)

Transcription

1 h 7 1 Publc Goods o Rval goods: a good s rval f ts consumpton by one person precludes ts consumpton by another o Excludable goods: a good s excludable f you can reasonably prevent a person from consumng t Excludable Rval Pure prvate goods (most retal goods Non-Rval Impure publc goods (nternet servce Non-Excludable Impure publc goods (a parkng space at the mall Pure publc goods (natonal defense, lghthouse o Most of what we consder publc goods are mpure publc goods (eg hghways, parks We can thnk of publc goods as, to some degree, goods that have large postve externaltes o Optmal provson of prvate goods Suppose there are two ndvduals, artman and yle, choosng quanttes of two prvate goods, Salsbury steak and mashed potatoes (show sldes 1 & Lookng at the market for Salsbury steak, the ndvduals demand curves show the quantty they are wllng to purchase at a gven prce Addng together the quanttes gves us the total quantty demanded at that prce We can horzontally sum the demand curves to get the market demand curve (SMB curve Addng the supply curve (SM curve shows market equlbrum (socally effcent and the quanttes purchased by yle and Erc Mathematcally We allow mashed potatoes to be the numerare, and assume the prce of Salsbury steak s $ Each ndvdual chooses to consume where hs ndfference curve s tangent to hs budget constrant Ths s the pont where the (slope of I s equal to the prce rato (slope of budget constrant MU S PS o Thus, s, p s, p MU P From the supply perspectve, the frm chooses to produce where M = MB MB s prce o Then, at the socally optmal equlbrum, M S o Optmal provson of publc goods P P

2 h 7 Now suppose yle and artman are choosng between heesy Poofs, the numerare good, and Tjuana bottle rockets The dfference s that heesy Poofs are rval, but any Tjuana bottle rockets that one ndvdual buys are enjoyed by both chldren Thus yle and artman must consume the same amount of bottle rockets (look at slde, then show Now, lookng at the ndvdual demand curves, we can see each person s wllngness to pay for a gven level of the good (note ths s the reverse of what we dd before Summng the ndvduals wllngness to pay, we get the total wllngness to pay for the level of a good (SMB Now, snce any of the good provded wll be consumed by both ndvduals we sum the demand curves vertcally to get market demand (SMB Addng the supply (SM curve, we see the socally effcent outcome Note n the prevous case the effcent outcome was where the M was equal to each ndvdual s wllngness to pay Now the M s equal to the sum of the ndvduals wllngness to pay Mathematcally The margnal value to each ndvdual s ther Therefore the margnal value to socety s Thus the socally effcent outcome s where M o The bottom lne: for prvate goods, t s socally optmal to produce where M s equal to each consumer s margnal valuaton ( For publc goods t s socally optmal to produce where the M equals the sum of the consumer s margnal valuatons ( Ths s because prvate goods are rval, but publc goods are not If we produce one more unt of a prvate good, ts value to socety s ts value to the person who consumes t If we produce one more unt of a publc good, ts value to socety s the total value to all the people who consume t Prvate provson o Nash barganng: each person optmzes hs own utlty, gven the other s actons Then we look for a soluton where each person s pursung an optmal strategy at the same tme (Nash equlbrum Left enter Rght Left 5,5 4, 6,5 enter 6,5 8,8 6,6 Rght 7,5 6,4 5,5

3 h 7 o Suppose that heesy Poofs and Tjuana bottle rockets are a dollar each yle and artman each have utlty functons gven by: U log( log( F F and a budget constrant gven by F 100 Solvng for yle, we solve the budget constrant for and plug t nto the utlty functon: U log(100 F log( F F Settng the partal dervatve wth respect to F equal to zero gves: 1 0 (100 F F F ( 100 F Solvng for F results n F Ths s yle s best response functon That s, t shows hs optmal choce gven artman s level of freworks spendng Snce both boys have the same utlty functons and budget constrants, ( 100 F artman s best response functon s F So to fnd the Nash equlbrum we look for a soluton where each person s pursung hs optmal strategy That s, what values solve the system of two equatons? In ths case, solvng the equatons provdes the soluton F F 0 Thus both boys spend $0 each on bottle rockets and a total of $40 of Tjuana bottle rockets s purchased o Is ths optmal? Remember we found that the optmal level of publc goods s where M F U 1 MU F ( F F yle MU U Substtute 100 F from the budget constrant and you get 1 ( F F 100 F yle F (100 F 100 F 100 F Then the optmalty condton s M or F F 00 F 1 F Solvng ths gve us the optmal quantty s 6667 Thus the optmal quantty s hgher than what the prvate market would provde The publc good s underprovded o Show slde 4 Note that n the slde the utlty functons are not dentcal o What f one ndvdual s rcher than another? Assume the same stuaton as before, except that snce yle s dad s a lawyer, yle s allowance s 15, whle artman s s 75 Note that the total level of ncome s the same as before when t was 100 & 100

4 h 7 4 Then yle s budget constrant s F 15 and artman s s 75 F yle maxmzes U log(15 F log( F F to get a best ( 15 F response functon: F artman maxmzes U log( 75 F log( F F to get a best ( 75 F response functon: F Solvng ths system of equatons provdes a Nash equlbrum where F 45 and F 5 Thus yle provdes all of the bottle rockets Really t s 4167 and 0 Note that now the prvately provded equlbrum quantty s hgher than n the prevous example Ths llustrates the result that greater ncome nequalty leads to hgher provson of publc goods Also of nterest s that more dssmlar preferences lead to hgher provson of publc goods However, n each case the publc good s stll underprovded unless one ndvdual has all of the ncome or s the only one to get utlty out of the good o Why do we get ths outcome? Because ndvduals are only concerned about ther own utlty It s smlar to the externaltes problem n that people choose the optmal level of the publc good for themselves wthout consderng the postve externalty that ther purchase has for others In the second example artman s free rdng off of the purchases of yle If yle purchased no freworks (F =0, artman would purchase ( 75 F F 5 Snce yle does purchase freworks, artman chooses to spend all of hs ncome on heesy Poofs and watch the Tjuana bottle rockets that yle sends up o Possble solutons: The ncome/preference nequalty we just dscussed Altrusm: ndvduals care about the total welfare of others rather than just maxmzng ther own Warm Glow models: people get some utlty from beng chartable hanges the utlty functon from U f ( c, F to U f, F, F ( Problems wth government provson of publc goods o Determnng the costs and benefts In theory the government can provde the optmal level of a publc good through taxaton

5 h 7 5 In practce t s hard to determne the optmal level of the good People often do not know ther own valuaton of publc goods such as defense and sports stada Even f a person has a good dea of ther personal value, they may not accurately report t from fear that the government wll tax them at that level (free rder problem osts of government projects can be very hard to estmate o A recent study showed that the last ten sports stada bult had an average cost overrun of % o On the $450m stadum the Marlns are lookng for that s ~$150m over budget What often happens s the government provdes some of the publc good and lets prvate nvestors provde some of t In the Mam stadum case, The cty, county, and state would cover roughly half the cost and the Marlns would cover the other half o rowdng out Returnng to the orgnal Tjuana bottle rocket example, suppose the mayor of South Park decdes to tax yle and artman $1 each to provde an addtonal $667 to make up the dfference between the prvate equlbrum and the socally optmal level Now Erc and yle are maxmzng utlty functons that take nto account the publc fundng: U log( log( F F 667 The budget constrant becomes: F U log(8667 F log( F F Frst dervatve: 0 (8667 F F F 667 ( Fj Best response functon: F Result: F F 6 67 Thus the prvately provded amount of the good s 1 The publcly provded amount s 667, thus the total amount of freworks produced s $4000, just as before The government spendng completely crowds out the prvate provson o Erc and yle are taxed $1 and they reduce ther prvate spendng by $1 o Note that the boys are smply makng the adjustment to get back to the prvate equlbrum To them, they are stll spendng the same amount on freworks Now consder the case where yle has more ncome than artman yle maxmzes U log(15 15 F log( F F 5

6 h 7 6 ( 65 F Hs best response functon s F artman maxmzes U log( F log( F F 5 ( 15 F Hs best response functon s F Solvng the system of equatons provdes us wth the Nash equlbrum outcome: F 9 167, F 17 5 o Actually, the equlbrum s artman provdes zero and yle provdes the total prvate spendng of $08 o The total publc and prvate spendng s $08 + $667 = $4750 In ths case, publc spendng does not completely crowd out prvate spendng Why not? o The government effectvely forced artman to become a contrbutor o When we tax non-contrbutors to pay for a publc good, we get only partal crowdng We also get less than complete crowdng f ndvduals are altrustc or f we subscrbe to the warm glow model In the real world, crowdng out s not complete, however, some crowdng out does occur There s no consensus n the emprcal lterature, and estmates of the level of crowdng out vary sgnfcantly