Chap2. Public Goods and Private Goods

 

Contents 2 Public Go ods and Private Go o ds One Public lic Good ood, One Private Good ood . . . . . . . . . . An Intr Intrig igui uing ng,, bu butt Gene Genera rall lly y Il Ille lega gall Dia Diagr gram am . . . . . A Fam Famil ily y of of Spe Speci cial al Case Casess – Qua Quasi sili line near ar Ut Util ilit ity y . . . . Example 2.1 . . . . . . . . . . . . . . . . . . . . . Generalizations . . . . . . . . . . . . . . . . . . . . . . Discrete Choice and Public Good oods . . . . . . . . . . . Appen Ap pendi dix: x: When When Sa Sam muels uelson on Cond Conditi ition onss are are Suffici Sufficien entt Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 4 7 9 10 11 11 13

 

Lecture 2

Public Goods and Private Goods One Public Good, One Private Good Cecil and Dorothy1 are roommate roommates, s, too. They They are not intere intereste sted d in card games or the temperature of their room. Each of them cares about the size of the flat that they share and the amount of money he or she has left for “private “priv ate goods”. Private Private goods, like chocolate chocolate or shoes, must be consumed consumed by one person person or the other, other, rather rather than being being joi joint ntly ly consum consumed ed like an apartment apartm ent or a game of cards. Cecil and Dorothy Dorothy do not work, but have have a fixed money income $W  $W .. This This money can be b e used in three three differen differentt ways. ways. It can be spent on private goods for Cecil, on private goods for Dorothy, or it can be spent on rent for the apartment. The rental cost of a flat is $c $ c  per  p er squaree foot. squar Let   X C  Let and   X D  be the amounts that Cecil and Dorothy, respectively, C   and  spend on private goods. Let Y  Let  Y  be  be the number of square feet of space in the flat. The set of possibl p ossiblee outcomes for Cecil and Dorothy Dorothy consists of all those triples, (X  (X C    ) that they can afford given their wealth of $W  $ W .. This is C , X D , Y  )  just the set: (X C  Y ) X C   W  C , X D , Y ) C   + X D  + cY 

{

|

 ≤ }

In general, Cecil’s utility function might depend on Dorothy’s private con1

When I first produced these notes, the protagonists were named Charles and Diana. Unfortunately many readers confused these characters with the Prince of Wales and his unhapp unh appy y spouse. spouse. Since Since it would would plainly plainly be in bad taste to sugges suggest, t, ho howe weve verr ina inadv dverertently, that members of the royal family might have quasi-linear preferences, two sturdy commoners, Cecil and Dorothy, have replaced Charles and Diana in our cast of characters.

 

2

Lecture 2. Public Go o ds and Private Goo ds

sumption as well as on his own and on the size of the apartment. He might, for example, like her to have more to spend on herself because he likes her to be happy. happy. Or he might be an envious envious lout who dislikes dislikes her having more to spend than he does. Thus, in general, we would want him to have a utility function of the form: U C  Y ). C (X C  C , X D , Y ) But for our first pass at the problem, let us simplify matters by assuming that both Cecil and Dorothy Dorothy are totally selfish about priv private ate goods. That is, neither neither cares cares how much much or little the other spends spends on pri priv vate goods. If  this is the case, then their utility functions would have the form: U C  Y ) C (X C  C , Y )

and   U D (X D , Y  ).  ).

With the examples of Anne and Bruce and of Cecil and Dorothy in mind, we are ready to present a general definition of public goods and of private goods. We define a public good to be a social decision variable variable that en enters ters simultaneously as an argument in more than one person’s utility function. In the tale of Anne and Bruce, both the room temperature and the number of games of cribbage cribbage were public goods. In the case of Cecil and Dorothy Dorothy, if both persons are selfish, the size of their flat is the only public good. But if, for example, both Cecil and Dorothy care about Dorothy’s consumption of chocolate, then Dorothy’s chocolate would by our definition have to be a public good. Perhaps surprisingly, the notion of a “private good” is a more complicated and special idea than that of a public good. In the standard economic models, private private goods have have two distinguishing distinguishing features. features. One is the distridistribution technolog technology y. For a good, say chocolate chocolate,, to be b e a priv private ate good it must be that the total supply of chocolate can be partitioned among the consumers in any way such that the sum of the amounts received by individuals adds to the total supply available.2 The second feature is selfishness. In the standard standa rd models of private private goods, goo ds, consumers care only about their own consumptions of any private good and not about the consumptions of private goods by others. In the story of Cecil Cecil and Doroth Dorothy y, we have have one public public good and one 3 private good. To fully describe an allocation of resources on the island we 2

It is possible possible to general generalize ize this model by addin adding g a more compl complicat icated ed distr distributi ibution on technology and still to have a model which is in most respects similar to the standard private goods model. 3 According to a standard result from microeconomic theory, we can treat “money for private goods” as a single private good for the purposes of our model so long as prices of  private goods relative to each other are held constant in the analysis.

 

ONE PUBLIC GOOD, ONE PRIVATE GOOD

3

need to know not only the total output of private goods and of public goods, but also how the private private good is divided between between Cecil and Dorothy Dorothy. The allocation problem of Cecil and Dorothy is mathematically more complicated than that of Anne and Bruce. There are three decision variables instead of  two and there is a feasibility constraint as well as the two utility functions. Theref The refore ore it is more more difficult difficult to repres represen entt the whole whole sto story ry on a gra graph. ph. It is, however, quite easy to find Pareto optimality using Lagrangean Lagrangean methods. methods . Ininteresting fact, as weconditions will show,for these conditions conditi ons can also be deduced by a bit of careful “literary” reasoning. We begin begin with with th thee La Lagr gran ange gean an ap appr proa oacch. At a Paret aretoo op opti tim mum it should be impossible to find a feasible allocation that makes Cecil better off without without making Dorothy Dorothy wors worsee off. Theref Therefore ore,, Paret Paretoo opt optima imall allocaallocations can be found by setting Dorothy at an arbitrary (but possible) level ¯D  and maximizing Cecil’s utility subject to the constraint that of utility,  U  ¯D  and the feasibilit U D (X D , Y ) Y )  U  feasibility y constrain constraint. t. Formally ormally, we seek a solution to the constrained constrained maximization maximization problem: Choose  Choose   X C  and   Y   Y   to C ,   X D   and maximize U  maximize  U C  ) subje subject ct to: C  (X C  C , Y  )

 ≥

U D (X D , Y  )  )

 ≥  U ¯

D

  ≤ W.

  and   X C  C  + X D  + cY 

We define an  interior Pareto optimum  to be an allocation in the interior of the set of feasib feasible le allocat allocation ions. s. With With the techno technolog logy y discus discussed sed here, here, an interior Pareto optimum is a Pareto optimal allocation in which each consumer consumes a positive amount of public goods and where the amount of public goods is positive. The Lagrangean for this problem is: are U C Y ) C  (X C  C , Y )

− λ1



¯D U 



 − U  (X  , Y ) Y ) − λ2 (X   (X   + X   + cY  −  − W ) W ) D

D

C  C 

D

(2.1)

 ¯ C    ¯  ¯ A necessary condition for an allocation (X  Y ) to be an interior Pareto C , X D , Y ) optimum is that the partial derivatives of the Lagrangean are equal to zero. Thus we must have:  ¯ C ,  X  ¯D ) ∂U C  C (X C  λ2   = 0 (2.2) ∂X C  C 

−

λ1 ∂U D (X   ¯ C ,  X  ¯D ) λ2   = 0 ∂X D  ¯ C ,  X  ¯D )  ¯ C   ¯ ∂U C  ∂U D (X  C (X C  C , X D )   + λ1 λ2 c   = 0 ∂Y  ∂Y  From 2.2 it follows that:

−

  −

λ2  =

 ¯ C ,  X  ¯D )   ∂U C  C (X C  . ∂X C  C 

 

(2.3) (2.4)

(2.5)

 

4

Lecture 2. Public Go o ds and Private Goo ds

From 2.3 and 2.5 it follows that: λ1  =

 ¯ C ,  X  ¯D )   ∂U C  C (X C  ∂X C  C 

 ¯ C    ¯D C 

X  , X  ) ÷  ∂ U  (∂X  D

(2.6)

D

Use 2.6 and 2.5 to eliminate   λ1   and   λ2   fr from om Equati Equation on 2.4. Di Divi vide de the   ∂U C  resulting expression by ∂X C  and you will obtain:  ¯ C  ,X   ¯ D ) ∂U C C  (X C ∂Y    ¯ C  ,X   ¯ D ) ∂U C C  (X C ∂X C C  

+

 ¯ C   ¯ ∂U D (X  C ,X D ) ∂Y    ¯ C   ¯ ∂U D (X  C ,X D ) ∂X D

= c

 

(2.7)

This is the fundame fundament ntal al “Samu “Samuels elson on conditi condition” on” for efficien efficientt provis provision ion of  public goods. Stated in words, Equation 2.7 requires that the  sum  of   of Cecil’s and Dorothy’s marginal rates of substitution between flat size and private goods must equal the cost of an extra unit of flat relative to an extra unit of private goods. Let us now try to deduce this condition condition by literary literary methods. The rate at which either person is willing to exchange a marginal bit of private consumption for a marginal increase in the size of the flat is just his marginal rate of substitution. substitution. Thus Thus the left side of Equation Equation 2.7 represents represents the amount of private expenditure that Cecil would be willing to give up in return for an extra foot of space plus the amount that Dorothy is willing to forego for an extra foot of space. If the left side of 2.7 were greater than   c, then they could both be b e made better b etter off, since the total amount of private private expenditure expenditure that they are willing to give up for an extra square foot of space is greater than the total amount, amount,  c  c,, of private good that they would have spend to get an extra square square foot. Similarly Similarly if the left hand side of 2.7 were less than than   c, it would be possible to make both better off by renting a smaller flat and leaving lea ving each person more money to spend on private private goods. Therefore Therefore an allocation can be Pareto optimal only if Equation 2.7 holds.

An Intriguing, but Generally Illegal Diagram Economist Econom istss lo love ve diagrams diagrams.. Indeed Indeed,, if there there we were re an econom economist ists’ s’ coat coat of  4 arms, it would surely contain crossing supply and demand curves. When we discussed the special case where Cecil and Dorothy have quasilinear utility, we were able to use a diagram which is just as pretty as a supply and demand diagram, but tantalizingly different. 4

Perhaps also an Edgeworth box and an IS-LM diagram.

 

AN INTRIGUING INTRIGUING,, BUT GENERALLY GENERALLY ILLEGAL ILLEGAL DIAGRAM DIAGRAM 5 Recall that in the theory of private, competitive markets, we add individual demand curves “horizontally” to obtain the total quantity of a good demanded demand ed in the economy economy at any price. Equilibrium Equilibrium occurs at the price and quantity at which the aggregate demand curve crosses the aggregate supply curve. curv e. In Figure 2.1, the curves curves M   M 1 M 1  and  M 2 M 2  are the demand curves of  consumers 1 and 2 respectively and the horizontal sum is the thicker kinked cur curv ve. E Where Whe re the line line   SS  is SS  is the supply curve, equilibrium occurs at the point E  point . Figure 2.1:  With Private Goods, Demand Curves Add Horizontally

     S      R      M

M 2 M 1

S 

E 

S  M 1

M 2

Quantity

Howard Bowen [2] in a seminal article5 that was published in 1943 (ten years before Samuelson’s famous contribution) suggested a diagram in which one finds the efficient amount of public goods by summing marginal rate of substitution curves  vertically . A closely related related construction construction can also be found in a much much earlier paper by a Swedish Swedish economist, economist, Erik Lindahl Lindahl [3]. We will look at Lindahl’s diagram later. Lindahl and Bowen both realized that a Pareto efficient supply of public goods requires that the sum of marginal rates of substitution between the public good and the private good equals the marginal rate of transformation. Bowe Bo wen n propose proposed d the diagram diagram that that appears appears in Figure Figure 2.2, where where the amount of public good is represented on the horizontal axis and marginal ra rate tess of subs substi titu tuti tion on are are sh shoown on th thee verti ertica call ax axis is.. The curve curve   M i M i represents consumer   i’s marginal rate of substitution as a function of the 5

This article is best known for its pioneering contribution to the theory of voting and public choice.

 

6

Lecture 2. Public Go o ds and Private Goo ds

Figure 2.2:  With Public Goods, MRS Curves Add Vertically

     S      R      M

M 2 S  M 1

S 

E 

M 1

M 2 Quantity

amount of public good. These curves amount curves are added vertica vertically lly to find the sum of margin marginal al rates rates of substi substitut tution ion curve. curve. The interse intersecti ction on   E   E   between between the summed marginal rate of substitution curve and the marginal rate of transformation format ion curve curve   SS   SS   occurs occurs at the Pareto Pareto optima optimall quant quantit ity y for of public public goods. Now the problem with Figure 2.2 is that in general, we have no right to draw it, at least not without without further further assumpt assumption ions. s. The reason reason is that that a person’s marginal rate of substitution between public and private goods depends in general not only on the amount of public goods available but also on the amount of private goods he consumes. While in general we have no right to draw these curves, we showed in Lecture 2 that we are entitled to draw them if utility functions functions are quasilinear quasilinear.. We saw that when utility is quasilinear, an individual’s marginal rate of substitution between public and private goods depends only on the amount of public goods. The fact that we cannot, in general, draw marginal rate of substitution curves without making some assumption about the distribution of private goods was first brought clearly to the attention of the profession in Samuelson’s famous 1954 article. To be fair to Bowen, he seems to have been aware of this problem, since he remarks that these curves can be drawn “assuming a correct distribution of income”6 though he did not discuss this point as clearly as one might might wish. Even Even without quasilinear quasilinear utility utility, it would be 6

Notice the similarity to Musgrave’s later suggestion that each branch of government assume that the others are doing the right thing.

 

A FAMIL FAMILY Y OF SPECIAL CASES – QUASILINEAR QUASILINEAR UTILITY 7 possible to draw marginal rate of substitution curves if we made an explicit assumption about how private consumption will be allocated contingent on the level of public expenditure.

A Family of Special Cases – Quasilinear Utility Suppose that Cecil and Dorothy have utility functions that have the special functional form: U C   ) = X C  (2.8) C (X C  C , Y  ) C   + f C  C (Y  ) U D (X D , Y  )  ) = X D  + f D (Y ) Y )

(2.9)

where the functions  functions   f C  and   f D   have positive first derivatives and negative C   and  7 second derivatives. derivatives. This type of utility function is said to be   quasilinear. ∂U C  Since   ∂X  marginal rate of substitutio substitution n C    = 1, it is easy to see that Cecil’s marginal   ∂U C  between the public and private goods is simply ∂Y     =   f C (Y  ). Likewi Likewise se Dorothy’s marginal rate of substitution is just f  just  f D (Y ). Y ). Therefore the necessary condition stated in equation 2.7 takes the special form 







f C (Y  )  ) + f D (Y ) Y ) =  c 

 

(2.10)



Since we have assumed that   f C   and  and   f D  are both negative, the left side of  (8) is a decreasing function of (Y  (Y  ).  ). There Therefore, fore, given given  c,  c , the there re can be at most one value of   Y   Y   that satisfies satisfies Equation 2.10. Equili Equilibrium brium is neatly depicted depicted in the Figure 2.3. The curves  curves   f C (Y  )   ) and  and   f D (Y  )   ) represent Cecil’s and Dorothy’s marginal rates of substitution between public and private goods. (In the special case treated here, marginal rates of substitution are independent of the other   ) is obtained by summing  ) + f D (Y  ) variables   X C variables and   X D .) The curve  curve   f C (Y  ) C    and  the individual individual m.r.s. curves curves  vertically  (rather   (rather than summing horizontally as one does with demand curves for private goods). The only value of   Y   Y   that satisfies condition 2.10 is   Y  , where the summed m.r.s. curve crosses the level   c. level As we will show show later, later, the result result that that the optimalit optimality y condit condition ion 2.10 2.10 uniquely determines the amount of public goods depends on the special 







∗

7

The assumptions about the derivatives of   f C C     and   f D  ensure that indifference curves slope down and are conve convex x toward the origin. The assumption that  U (X C  C , Y  ) is linear in X C C    implies that if the indifference curves are drawn with   X  on the horizontal axis and Y    on the vertical axis, each indifference curve is a horizontal translation of any other, making the indifference curves are parallel in the sense that for given   Y  , the slope of the indifference curve at (X, Y  ) is the same for all   X .

 

8

Lecture 2. Public Go o ds and Private Goo ds

Figure 2.3: Summing Marginal Rates of Substitution M RS C  C  + M RS D

M RS D

c M RS C  C 

Y 

∗

Y 

kind of utilit kind utility y functi function on that we chose. chose. The class class of models that has this this property is so special and so convenient that it has earned itself a special name. nam e. Models Models in in which which every everybody’s body’s utilit utility y functi function on is lin linear ear in some some commodity are said to have  quasilinear utility . As we have have see seen, n, when when there there is quasil quasiline inear ar utilit utility y, there there is a uni unique que Pareto optimal amount,   Y  of public good corresponding to allocations in which both consumers get positive amounts of the private good. This makes it very easy to calculate calculate the utility possibility possibility frontier. frontier. Since the supply of  public good is   Y  , we know that along this part of the utility possibility frontier, Cecil’s utility can be expressed as  U C  C (X C C  , Y  ) =  X C C   + f C C  (Y  ) and ∗

∗

∗

∗

∗

 ∗

is  U D (X D , Y  is U 

D + f D (Y  ). Therefore it must be that )=  X  +  ∗ ∗ frontier, frontier,    U  U D   =  X C  C  C  C  + X D + f C C  (Y  ) + f D (Y  ).

Dorothy’s utility on the utility possibility After the public good has been paid for, the amount of the family income that is left to be distributed between Cecil and Dorothy is W  is  W   pY  Y  . So it must be that U  that  U C  C  + U D   =  W   pY  Y  + f C C  (Y  ) + f D (Y  ). Since the right side of this expression is a constant, the part of the utility possibility frontier that is achievable with positive private consumptions for both persons will be a straight line with slope -1.

−

∗

∗

∗

 −  −

∗

 

A FAMIL FAMILY Y OF SPECIAL CASES – QUASILINEAR QUASILINEAR UTILITY 9 Example 2.1

Suppose that Persons 1 and 2 have quasilinear utility functions  U i (X , Y ) Y ) = X i  + Y . Y . Public Public goods can be produce produced d from private private goods at a cost cost of  1 unit of private goods per unit of public goods and there are initially 3 units of private goods which can either be used to produce public goods or can be distribut distributed ed between between persons persons 1 and 2. Thus Thus the set of feasible feasible allocations is (X 1 , X 2 , Y ) Y )   0 X 1  +  + X   X 2  +  + Y   Y    3 . The sum sum of utilitie utilitiess is U 1 (X 1 , Y   )) + U 2 (X 2 , Y  )  ) =  X 1 + X 2 + 2 Y  which Y  which is equal to X  to  X + + 2 Y   Y   where X   = X 1 + X 2 . We will maximize the sum of utilities by maximizing X  maximizing  X +2 +2 Y  subject to   X  X    +   Y    3. The solution solution to this this constr constrain ained ed maximizat maximization ion problem is  is   Y   Y   = 1 and   X  X    = 3. Any Any alloca allocati tion on (X  (X 1 , X 2 , 1)   0 such that X 1  + X 2  = 2 is a Pareto optimum. In Figure 2, we draw the utility possibility possibility set. We start by finding the utility distributions that maximize the sum of utilities and in which   Y   Y   = 1 and   X 1  + X 2  = 2. At the allocation (2, and (2, 0, 1), where Person 1 gets all of the private goods, we have   U 1  = 2 + 1 = 3 and nd   U 2  = 0 + 1 = 1. Th This is is the the point   A. If Person 2 gets all of the private goods, then  U 1  = 0 + 1 = 1 and point U 2  = 2 + 1 = 3. This This is the the point point   B . Any Any point point on the line line   AB   can be achieved by supplying 1 unit of public goods and dividing 2 units of private goods between Persons 1 and 2 in some proportions.

√ 

 {

 ≥ |

 ≤ }

√ 

√ 

 ≤

√ 

 ≥

Figure 2.4:   A Utility Possibility Set U 2

D F 

B

A C  E 

U 1

Now let’s find the Pareto optimal points that do not maximize the sum of utilities utilities.. Consid Consider, er, for exampl example, e, the point that maximize maximizess Pe Perso rson n 1’s

 

10

Lecture 2. Public Go o ds and Private Goo ds

utility subject to the feasibility constraint  constraint   X 1  + X   +  X 2  + Y   +  Y    = 3. Since Perso Person n 1 has no interest in Person 2’s consumption, we will find this point by maximizing U  maximizing  U 1 (X 1 , Y  )  ) =  X 1 + Y  subject Y  subject to X  to  X 1 + Y  = Y  = 3. This is a standard consumer consu mer theory problem. problem. If you set the marginal marginal rate of substitution substitution equal to the relative prices, you will find that the solution is   Y   Y   = 1/4 and  and   X 1   = 3 1 2 4 . Wit With h this alloc allocati ation, on,   U 1   = 3 4   and   U 2   = 1/2. Th This is is the the poin point   C 

√ 

on Figure Figur e 2.hisNotic No tice tha that ty,when whe Person Pers onves 1s contro con trols allmbs ofs the reso max maximiz imizes es own ow ne utilit util ity hensti still ll leave lea som some e lscrumb cru for resourc Person Persurces ones2,and by providing public goods though he provides them from purely selfish motives. The curved line segment  segment   C A  comprises the utility distributions that result from allocations (W  (W  Y, 0, Y  )   ) where  where   Y  is Y  is varied over the interval [1/ [1 /2, 1]. An exactly symmetric argument will find the segment   DB   of the utility possibility frontier that corresponds to allocations in which Person 2 gets no private goods. The utility possibility frontier, which is the northeast boundary of the utility possibility set, is the curve  CABD  CABD.. There There are also also some boundary boundary points of the utility possibility possibility set that are not Pareto Pareto optimal. The curve curve segment   CE  consists segment CE  consists of the distributions of utility corresponding to allocations (W  (W   Y,  Y , 0, Y  )   ) where   Y   Y   is varied over the interval [0, [0, 1/2]. 2]. At thes thesee

 −  −

 −  −

allocations, Person 1 has all of the private goods and the amount of public goods is less than the amount that Person 1 would prefer to supply for himself. himsel f. Symmetrica Symmetrically lly,, there is the curve segment segment   DF  DF    in which Person 2 has all of the private goods and the amount of public goods is less than 1/2. Finally, every utility distribution in the interior of the region could be achieved by means of an allocation in which   X 1  + X 2  + Y <  3. In this example, we see that at every Pareto optimal allocation in which each consumer gets a positive amount of private goods the amount of public goods must be   Y   Y    = 1, which is the amount that maximizes the sum of  utilities.

Generalizations For clarity of exposition we usually discuss the case where a single public good is produced from private goods at a constant unit cost   c. In th this case, the set of feasible allocations consists of all (X  ( X 1 , . . . , Xn  , Y ) Y )  0 such that i X i  +  cY   cY    =   W   W   where   W   W   is the initial allocation of public goods. Mostt of our results Mos results extend extend quite quite easily easily to a more more gen genera erall techno technolog logy y in which the cost of production of public goods is not constant but where there is a conve convex x production production possibility possibility set. For this more general general technolog technology y,

 

 ≥

 

DISCRETE CHOICE AND PUBLIC GOODS

11

 ≥

the set of feasible allocations consists of all (X  ( X 1 , . . . , Xn  , Y ) Y )  0 such that ( i X i , Y ) Y )  T    T   where T  where  T  is  is a convex set, specifying all feasible combinations of gotal output of private goods and total amount of public goods produced. Almost all of our results also extend in a pretty straightforward way to cases where there are many public goods, many private goods, and many people. peop le. I don’t don’t need to presen presentt the story story here here because because the details details of how



 ∈

this works are nicely sketched out Samuelson’s classic paper ”The list. Pure Theory of Public Expenditure” [4],in which is on your required reading

Discrete Choice and Public Goods Some decisions decisions about public goods are essentially essentially discrete. discrete. For example, Cecil and Dorothy may choose whether or not to buy a car or whether or not to mov move fro from m London London to Chicago Chicago.. For this kind of choice choice,, the public public goods setup is useful even though Samuelsonian calculations of marginal rates of substitution are not relevant. Let X  Let  X C  and  X D  denote private consumption for Cecil and Dorothy, and C   and X  let let Y   Y  =  = 0 if they don’t have a car and  Y   Y  =  = 1. Assume that in initially, they 0 0 have no car and their private consumptions are X  are  X C  X D . Let us define Cecil’s willingness to pay for a car as the greatest reduction in private consumption thatt he would tha would accept accept in return return for ha havin vingg a car. car. Similar Similarly ly for Doroth Dorothy y. In general, their willingnesses to pay for a car will depend on their initial consumptions consum ptions if they don’t buy a car. In particular particular Person i Person i’s ’s willingness to pay for a car is the function V  function  V i ( ) is defined defined to be the solution solution to the equati equation on 0 0 0 U i (X i  V i (X i ),  ), 1) =   U i (X i , 0). Suppose Suppose that that the cost of the car is  is   c. It will be possible for Cecil and Dorothy to achieve a Pareto improvement by buying the car if and only if the sum of their willingnesses to pay for the car exceed exceedss the cost of the car. If this is the case, case, then then each each could could pay for the car with each of them paying less than his or her willingness to pay. Therefore both would be better off.

 −

·

Appendix Appen dix:: Whe When n Sam Samuel uelson son Cond Conditi itions ons are Sufficient So far, we have shown only that the Samuelson conditions are necessary for an interior Pareto optimum. The outcome is exactly analogous to the usual maximizatio maximi zation n story in calculus. calculus. For continu continuously ously differentiable differentiable functions, functions, the first-order conditions are necessary for either a local or global maximum, but they are not in general sufficient. sufficient. As you know, in consumer theory, theory, if 

 

12

Lecture 2. Public Go o ds and Private Goo ds

utility functions are quasi-concave and continuously differentiable, then a consumption bundle that satisfies an individual’s first-order conditions for constrained maximization will actually be a maximum for the constrained optimization problem. For the Samuelson conditions, we have the following result.   If all individu-

Proposition 1 (Sufficiency of Samuelson conditions) als have continuously differentiable, quasi-concave utility functions  U (  U (X i , Y  )  ) which are strictly increasing in   X i  and if the set of possible allocations consists of all   (X 1 , . . . , Xn  , Y )  +  cY    =   W , then the feasibility  Y )   such that  X i  + cY    ¯ ¯ condition  X i  + cY   Y   = W   together with the Samuelson condition 





 ¯ i ,Y   ¯) ∂U i (X  ∂Y    ¯ i ,Y   ¯) ∂U i (X  i=1 ∂X i n



= c

 

(2.11)

 ¯ 1 , . . . ,  X  ¯n ,  ¯ is sufficient as well as necessary for an interior allocation  (  ( X  Y ) Y )  >> 0   to be Pareto optimal. To prove Proposition 1, we need the following lemma, which is in general astraightforward us usef eful ul thing thing for eco nomi mist s to know kno about about.. inABlume pr proof oof and of th the e le lemm mma a is andecono can be sts found, for w example, Simon’s text Mathematics for Economists  Economists   [1]. A function with the properties properties described in Lemma 1 is called a  pseudo-concave   function. Lemma 1   If the functi function  on   U  U ((X, Y  )  )   is continuous ontinuously ly differe differentia ntiable, ble, quasiquasiconcave, and defined on a convex set and if at least one of the partial derivatives of   U  is non-zero at every point in the domain, then if   (X, Y  )  )  is in the    interior of the domain of   U , then for any   (X  , Y  ), 



•   if   U (X  , Y  ) > U  U ((X, Y  )  )  then  ∂U (X, Y  ) ∂U (  )   (X  ∂X 





∂U ((X, Y  )  ) − X ) +   ∂U    (Y  − Y ) Y )  > 0  >  0.. ∂Y  



•   if   U (X  , Y  ) ≥ U (  U (X, Y  )  )  then  ∂U (X, Y  ) ∂U (  )   (X  ∂X 



U (X, Y  )  ) − X ) +  ∂ U (   (Y  − X ) ≥ 0.  0 . ∂Y   

Proof of Proposition 1.   Suppose that the allocation allocation (X  (X 1 , . . . , Xn  , Y ) Y ) satis





fies the Samuelson Samuelson conditions and and suppose that the allocation (X  (X 1 , . . . , Xn  , Y  )

 

EXERCISES

13 

 

is Pareto superior to (X  (X 1 , . . . , Xn  , Y  ).  ). Then ffor or some some i, U i (X i , Y  )  > U i (X i , Y ) Y ) and for all i all  i U i (X i , Y  )  U i (X i , Y  ).   ). From Lemma 1 it follows that for all  i,  i , 



 ≥

∂U i (X i , Y ) Y ) (X i ∂X i

X  , Y ) Y )  − X  ) +  ∂ U  (∂Y    (Y  − Y ) Y ) ≥  0 i



i

 

i



(2.12)



with a strict equality for all   i  such that  that   U i (X i , Y  )   > U (X i , Y ). Y ). Since, Since, by by assumption,   ∂U i (X i , Y  )/∂X  assumption,  )/∂X i   > 0, it follows that 

∂U i (X i ,Y  ) ∂Y   ∂U i (X i ,Y  ) ∂X i

 − X   +

X i

i

 

(Y 

− Y ) Y ) ≥   0

(2.13)

for all  all   i  with strict inequality for some   i. Adding these inequalities over all consumers, we find that n

n

 i=1



 − X  ) +

(X i

i

 i=1

∂U i (X i ,Y  ) ∂Y   ∂U i (X i ,Y  ) ∂X i



 

(Y 

− Y ) Y ) ≥  0.  0 .

 

(2.14)

Using the Samuelson condition as stated in Equation 2.7, this expression simplifies simpli fies to n





  i=1

(X i

 −

X i ) +

c(Y 

− Y ) Y ) ≥  0.  0 .

 

(2.15)

But Expression 2.15 implies that n

n





X i  + cY  >

i=1



X i  + cY  cY    = W.

 

(2.16)

i=1







From Expression 2.16, we see that the allocation (X  ( X 1 , . . . , Xn  , Y  ) is not feasible. Therefore there can be no feasible allocation that is Pareto superior to (X 1 , . . . , Xn  , Y  ).  ).

Exercises 2.1   Muskrat, Muskrat, Ontario, Ontario, has 1,000 people p eople.. Citizens Citizens of Muskrat consume only one private private good, Labatt’s Labatt’s ale. There is one public good, the town skating skating rink. Although Although they may differ in other respects, respects, inhabitants inhabitants have have the same utility utilit y function. This function function is  is   U i (X i , Y ) Y ) =  X i 100 100/Y  /Y  ,   , where  where   X i  is the number of bottles of Labatt’s consumed by citizen   i   and   Y   Y   is the size of  the town skating rink, measured in square meters. The price of Labatt’s ale is $1 per bottle and the price of the skating rink is $10 per square meter. Everyone who lives in Muskrat has an income of $1,000 per year.

 −

 

14

Lecture 2. Public Go o ds and Private Goo ds

a). Write out the equation implied by the Samuelson Samuelson conditions. conditions. b). Show Show that this equation equation uniquely determines determines the efficient efficient rink size for Muskrat. What is that size? 2.2   Cowflop, Wisconsin, has 1,100 people. Every year year they have have a fireworks show sho w on the fourth of July. July. The citize citizens ns are interes interested ted in only two two things – – drinking drinking milk and watching watching fireworks. fireworks. Firew Fireworks orks cost 1 gallon of milk per unit. Every Everybody body in to town wn is named named Johnson Johnson,, so in order to be able able to identify iden tify each other, the citizens citizens have taken taken numbers numbers 1 through through 1,100. The utility function of citizen  citizen   i   is

√ 

  ai y U i (xi , y) =  x i  + 1000 where  xi  is the number of gallons of milk per year consumed by citizen i where x citizen  i and  and y  is the number of units of fireworks exploded in the town’s Fourth of July extrav extra vaganza. aganza. (Private (Private use of fireworks fireworks is outlawed). outlawed). Although Although Cowflop is quite unremarkable in most ways, there is one remarkable feature. For each Johnsons ns i  from 1 to 1,000, Johnson number   i  has parameter   ai   =   i/10. i/10. Johnso with numbers bigger than 1,000 have   ai   = 0. For each each Johnso Johnson n in to town, wn, Johnson  i  has income of 10 + i/ Johnson i i/10 10 units of milk. a). Find the Pareto optimal amount amount of fireworks fireworks for Cowflop. Hint:   It is true that you hav have e to sum a series series of nu numb mbers ers.. But  this is a series that Karl Friedrich Gauss is said to have solved  when he was in second grade.

2.3   Some Some miles west west of Cowflop, Cowflop, Wisc. is the town town of Heifer Heifer’s ’s Breath. Breath. Heifer’s Heif er’s Breath, Breath, like like Cowflop has 1000 people. As in Cowflop, the citizens are interested interested only in drinking drinking milk and watching watching fireworks. fireworks. Firewor Fireworks ks cost 1 gallon gallon of milk per unit. unit. Heifer Heifer’s ’s Breath Breath has tw twoo kinds kinds of people, people, Larsons and Olsens. The Larsons are numbered 1 through 500 and the Olsen’s are numbere numbered d 1 throug through h 500. 500. The Larsons Larsons have have all CobbCobb-Dou Dougla glass utilutilα 1−α

ity (functions  functions  β U L1(−xβ i., y) U  O xi , y ) =  x i  y

=   xi  y

and the Olsons all have utility functions

a). Write an expression expression for the optimal amount amount of public goods as a function of the parameters of the problem. b). If   α   =   β , show that the Pareto optimal amount of public goods depends on the aggregate income in the community, but not on how that income is distributed.

 

EXERCISES

15

√ 

2.4   Let  Let   U C  Y ) =   X C  Y   and  and   U D (X D , Y  )  ) =   X D  + C (X C  C , Y ) C   + 2 Y   that c that  c =  = 1.

√ Y . Y .

Suppo Suppose se

a). Determine Determine the amount amount of   Y  that Y  that must be produced if the output is to be Pareto optimal and if both persons are to have positive consumption of private goods. b). Find and describe describe the set of Pareto Pareto optimal allocations allocations in which one or the other other person consume consumess no private private goods. Show Show that that at these Pareto optima, the Samuelson conditions do not necessarily apply. c). Draw Draw the utility possibility possibility frontier. frontier. d). Writ ritee do down wn an explic explicit it formu formula la for the straig straight ht lin linee portion portion of the utility possibility frontier. e). Write down explicit expressions expressions for the curved curved portion p ortionss of the utility utility possibility frontier. 2.5  Cecil and Dorothy found a place to live rent-free. Now they are deciding whethe whe therr to buy a car. car. They They have have a total total income income of $1000 $1000 and a car would would cost $400. Cecil’s Cecil’s utility is given given by X  by  X C (1 + Y ) Y ) and Dorothy’s by X  by  X D (3 + Y ) Y ) where Y  where  Y  =  = 1 if they buy a car and Y  and  Y  =  = 0 if they do not, and where X  where  X C   and C   X D  are the amounts that Cecil and Dorothy spend on private goods. 1. Writ ritee an equati equation on for the utilit utility y possibi possibilit lity y front frontier ier if they they are not allowed to buy a car and another equation for the utility possibility frontier if they must buy a car. 2. Graph these two two utility utility possibility possibility frontiers frontiers and shade in the utility possibility set if they are free to decide whether or not to buy a car. 3. Suppose that if they don’t purchase purchase the car, Cecil and Dorothy split their money equally equally.. Could they achiev achievee a Pareto impro improvem vement ent by buying the car? Show Show your answer answer in two two ways (i) Compare the sum of willingne willingnesse ssess to pa pay y to the cost cost of the car. (ii (ii)) Show Show whethe whetherr or not the outcome where each has $500 and they have no car produces a point on the overall utility possibility frontier. 4. Suppose that if they don’t purchase purchase the car, income wou would ld be divided divided so that   X C  350. 0. Coul Could d th they ey achi achiev evee a Paret aretoo C    = 650 and   X D   = 35 improvem impro vement ent by buying the car? Show your your answer in two ways ways (i) Compar Com paree the sum of willingn willingness esses es to pa pay y to the cost of the car. (ii)

 

16

Lecture 2. Public Go o ds and Private Goo ds Show whether or not the situation where  X C  and   X D  = $350 C  = $650 and  and they have no car is on the overall utility possibility frontier.

 

Bibliography [1] Lawrence Lawrence Blume Blume and Carl Simon.  Mathematics for Economists . Norton, New York, 1994. [2] H. Bowe Bowen. n. The interpr interpreta etatio tion n of vo votin tingg in the allocatio allocation n of resource resources. s. Quarterly Journal of Economics , 58(1):27–48, November 1943. [3] Erik Erik Lindah Lindahl. l. Just Just taxati taxationon-aa positiv positivee sol soluti ution. on. In Richard Richard Musgra Musgrave ve Classicss in the The Theory ory of Public Public Financ Finance , and Alan Peacock Peacock,, editors, editors,   Classic pages 98–123. Macmillan, London, 1958. [4] Pa Paul ul Samuel Samuelson son.Statistics  . The pure theory theory of public expendit diture ures. s.   Review of  Economics and , 36(4):387–389, 1954.expen