**Welfare Effects of Third-Degree Price Discrimination: Ippolito Meets Schmalensee and Varian**

Iñaki Aguirre *University of the Basque Country Spain* 

### **1. Introduction**

162 Social Welfare

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Price discrimination under imperfect competition is an important area of economic research,1 and third-degree price discrimination, the most prevalent form of price discrimination, is a major item in any standard treatment of monopoly theory covered in intermediate and advanced microeconomics courses (see, for instance, Pindyck and Rubinfeld, 2008, or Varian, 1992, 2006). Under third-degree price discrimination the seller can charge different prices to consumers belonging to different groups or submarkets. For example, the seller may charge different prices to customers who are separated geographically (the home and the export markets) or that are differentiated by age (senior citizen's discounts), occupation (student discounts), time of purchases (initial equipment and replacement purchases), or by end use (milk for liquid consumption or for further processing). Moving from non-discrimination to discrimination raises the firm's profits, harms consumers in markets where the prices increase and benefits the consumers who face lower prices. Consequently, the overall effect on welfare is undetermined.

Understanding the conditions under which the change in social welfare can be signed has concerned economists at least from the earlier work by Pigou (1920) and Robinson (1933). A move from uniform pricing to third-degree price discrimination generates, as will be shown below, two effects:2 firstly, price discrimination causes a misallocation of goods from high to low value users (that is, output is not efficiently distributed to the highest-value end); secondly, price discrimination affects total output. Therefore, since price discrimination is viewed as an inefficient way of distributing a given quantity of output between different consumers or submarkets, a necessary condition for price discrimination to increase social welfare is that it should increase total output.3 In consequence, in order for price

<sup>1</sup> See Stole (2007), Armstrong (2008) and Liu and Serfes (2010) for excellent theoretical surveys. See also McAfee (2008) for a modern view of price discrimination and for antitrust implications.

<sup>2</sup> McAfee (2008) provides a nice explanation of these effects.

<sup>3</sup> See, for example, Robinson (1933), Schmalensee (1981), Varian (1985), Schwartz (1990) and more recently Bertoletti (2004). However, when marginal cost varies across markets that result does not maintain (see, Bertoletti, 2009).

Welfare Effects of Third-Degree Price Discrimination: Ippolito Meets Schmalensee and Varian 165

It is illustrative to evaluate the marginal profit in market *i* ( 1,2 *i* ) at the optimal uniform

() ( ) ( ) ( )( )( ) ( ) . ( )

1 1 () () . () () *i i i*

The monopolist is willing to increase (decrease) the price in market *i* if the elasticity in that

<sup>1</sup> () () () () . () ()() ()()

*Dp Dp p pD p p p Dp p Dp*

market, <sup>0</sup> ( ) *<sup>i</sup> <sup>p</sup>* , is lower (higher) than the elasticity of the aggregate demand, <sup>0</sup> ( ) *<sup>p</sup>* .

0 0 ' 0 0 0 00 00 ()() ( ) ( ) ( ), ()() ()()

of demand and to reduce the price in the market with higher elasticity of demand.

monopolist can be expressed as ( ) / 1 / ( ), 1,2, *dd d*

Under price discrimination, the optimal policy for the monopolist is given by:

*<sup>p</sup> p p p Dp p Dp* 

where *ij j i* , 1,2, . Note that ' 0 ()0 *<sup>i</sup> p* iff 0 0 ( ) ( ), *j i p p ij j i* , 1,2, . Therefore, if possible the monopolist would want to increase the price in the market with lower elasticity

'

strictly concave in the relevant range). Under price discrimination, the optimal policy for the

*ii ii i ii p D p p D p* is the price-elasticity in market *i*. That is, the Lerner index in each market is inversely proportional to its elasticity of demand and the monopolist therefore sets a higher price in the market with the lower elasticity of demand. The quantity

*i ii i i q q Dp* Given the first order conditions in (6), total output can be

*<sup>i</sup> p* denotes the optimal price in market *i* (and profit functions are assumed to be

*i j i j i ii jj Dp Dp*

*Dp p c p D p p cD p pD p pD p p*

() ( ) () *i i*

(2)

0 0

(3)

(4)

(5)

*i i ii p c p pi* where

0' 0 0

*i*

0 0

0 00 00

*i ii j j*

*i j*

*p p* 

*i*

0 0

( ) ( ) ( ) 0, 1,2. *dd d D p p cD p i ii i ii* (6)

*i ii q Dp i* Total output under price discrimination is

2 2 0 0 0 '0 1 1

*i i q Dp p c Dp* 

' 0 0 0 ' 0 0' 0

' 0 0' 0

*p pD p*

*ii i i*

Therefore, the marginal profit becomes:

Condition (4) can be written, equivalently, as

' 0 0' 0

*i i*

pricing:

which leads to

where *<sup>d</sup>*

'() ()/() *d dd d*

2 2 1 1 ( ) *dd d*

expressed as:

sold in market *i* is ( ), 1,2. *d d*

discrimination to increase welfare a positive output effect must offset the negative effect of distributional inefficiency.

Schmalensee (1981)'s direct approach to the welfare effect and Varian's celebrated bounds on social welfare (1985, 1989,1992) have dominated both the research and the teaching of the welfare effects of third-degree price discrimination for the last twenty-five years. Our analysis, inspired by the pioneering work by Ippolito (1980) and its generalization to *n* markets by Aguirre (2008), offers some advantages over Schmalensee's and Varian's. Firstly, it focuses directly on the change in welfare (instead of on indirect Lagrangian techniques or on exogenous bounds) and allows the output effect (that is, the social valuation of the change in total output) to be distinguished neatly from the misallocation effect. In addition we show how it is possible to prove the theorem that "a necessary condition for third-degree price discrimination to increase social welfare is that total output increases" by using this decomposition. Our approach to the welfare effects of discrimination is also more intuitive and can be illustrated graphically.

#### **2. Analysis**

Consider a monopolist selling a good in two perfectly separated markets. The demand function in market *i* (*i*=1,2) is given by ( ) *D p i i* , where *<sup>i</sup> p* is the price charged in that market and the inverse demand function is ( ) *i i p q* , where *<sup>i</sup> q* is the quantity sold. Unit cost, *c*, is assumed to be constant. The price elasticity in market *i* is given by ' () ()/ () *ii iii ii p D p p D p* . The profit function in market *i*, ()( )() *ii i ii <sup>p</sup> p cD p* , is assumed to be strictly concave, ''0 *<sup>i</sup>* .

#### **2.1 Profit maximization**

Under simple monopoly pricing, profits are maximized by charging all consumers a common price <sup>0</sup> *p* such that:

$$\sum\_{i=1}^{2} D\_i(p^0) + (p^0 - c) \sum\_{i=1}^{2} D\_i^\dagger(p^0) = 0 \tag{1}$$

Therefore, under uniform pricing, the optimal policy is given by 00 0 ( )/ 1/ ( ) *p c p p* , where <sup>0</sup> *p* denotes the uniform price and <sup>0</sup> ( ) *p* is the elasticity of the aggregate demand at <sup>0</sup> *p* . If we let <sup>2</sup> <sup>1</sup> () () *<sup>i</sup> <sup>i</sup> Dp D p* denote the aggregate demand, then this elasticity is simply the weighted average elasticity: 0 00 <sup>2</sup> <sup>1</sup> ( ) ( )( ) *i i <sup>i</sup> <sup>p</sup> p p* , where the elasticity of market *i* is weighted by the "share" of that market at the optimal uniform price, 00 0 2 <sup>1</sup> ( ) ( )/ ( ) *ii i <sup>i</sup> <sup>p</sup> Dp Dp* . Let <sup>0</sup> *<sup>i</sup> q* denote the quantity sold in market *i*, 0 0 ( ) *i i q D p* ( *i* 1,2 ), and <sup>0</sup> *q* denote the total output, <sup>0</sup> <sup>2</sup> <sup>1</sup> ( ) *<sup>i</sup> <sup>i</sup> q Dp* , under uniform pricing which can be expressed as follows:

$$\log^{0} = \sum\_{i=1}^{2} D\_i(p^0) = -(p^0 - c) \sum\_{i=1}^{2} D\_i'(p^0) \tag{2}$$

It is illustrative to evaluate the marginal profit in market *i* ( 1,2 *i* ) at the optimal uniform pricing:

$$\pi\_i^!(p^0) = D\_i(p^0) + (p^0 - c)D\_i^!(p^0) = p^0 D\_i^!(p^0) \left| \frac{D\_i(p^0)}{p^0 D\_i^!(p^0)} + \frac{(p^0 - c)}{p^0} \right|. \tag{3}$$

Therefore, the marginal profit becomes:

$$
\pi\_i^{'}(p^0) = p^0 D\_i^{'}(p^0) \left[ -\frac{1}{\varepsilon\_i(p^0)} + \frac{1}{\varepsilon(p^0)} \right]. \tag{4}
$$

The monopolist is willing to increase (decrease) the price in market *i* if the elasticity in that market, <sup>0</sup> ( ) *<sup>i</sup> <sup>p</sup>* , is lower (higher) than the elasticity of the aggregate demand, <sup>0</sup> ( ) *<sup>p</sup>* .

Condition (4) can be written, equivalently, as

$$
\pi\_i^{\dot{}}(p^0) = p^0 D\_i^{\dot{}}(p^0) \left[ -\frac{1}{\varepsilon\_i(p^0)} + \frac{D\_i(p^0) + D\_j(p^0)}{\varepsilon\_i(p^0) D\_i(p^0) + \varepsilon\_j(p^0) D\_j(p^0)} \right].
$$

which leads to

164 Social Welfare

discrimination to increase welfare a positive output effect must offset the negative effect of

Schmalensee (1981)'s direct approach to the welfare effect and Varian's celebrated bounds on social welfare (1985, 1989,1992) have dominated both the research and the teaching of the welfare effects of third-degree price discrimination for the last twenty-five years. Our analysis, inspired by the pioneering work by Ippolito (1980) and its generalization to *n* markets by Aguirre (2008), offers some advantages over Schmalensee's and Varian's. Firstly, it focuses directly on the change in welfare (instead of on indirect Lagrangian techniques or on exogenous bounds) and allows the output effect (that is, the social valuation of the change in total output) to be distinguished neatly from the misallocation effect. In addition we show how it is possible to prove the theorem that "a necessary condition for third-degree price discrimination to increase social welfare is that total output increases" by using this decomposition. Our approach to the welfare effects of discrimination is also more intuitive

Consider a monopolist selling a good in two perfectly separated markets. The demand function in market *i* (*i*=1,2) is given by ( ) *D p i i* , where *<sup>i</sup> p* is the price charged in that market and the inverse demand function is ( ) *i i p q* , where *<sup>i</sup> q* is the quantity sold. Unit cost, *c*, is assumed to be constant. The price elasticity in market *i* is given by ' () ()/ () *ii iii ii p D p p D p* . The profit function in market *i*, ()( )() *ii i ii <sup>p</sup> p cD p* , is assumed to be strictly concave, ''

Under simple monopoly pricing, profits are maximized by charging all consumers a

0 0 '0

Therefore, under uniform pricing, the optimal policy is given by 00 0 ( )/ 1/ ( ) *p c p p* , where <sup>0</sup> *p* denotes the uniform price and <sup>0</sup> ( ) *p* is the elasticity of the aggregate demand at

<sup>1</sup> ( ) ( )( ) *i i <sup>i</sup>*

weighted by the "share" of that market at the optimal uniform price,

 

*Dp p c Dp* 

( )( ) ( ) 0 *i i*

<sup>1</sup> () () *<sup>i</sup> <sup>i</sup> Dp D p* denote the aggregate demand, then this elasticity is simply

(1)

*<sup>p</sup> p p* , where the elasticity of market *i* is

*<sup>i</sup> q* denote the quantity sold in market *i*, 0 0 ( ) *i i q D p*

<sup>1</sup> ( ) *<sup>i</sup> <sup>i</sup> q Dp* , under uniform pricing which can

2 2

1 1

*i i*

the weighted average elasticity: 0 00 <sup>2</sup>

( *i* 1,2 ), and <sup>0</sup> *q* denote the total output, <sup>0</sup> <sup>2</sup>

distributional inefficiency.

and can be illustrated graphically.

**2. Analysis** 

0 *<sup>i</sup>* .

**2.1 Profit maximization** 

<sup>0</sup> *p* . If we let <sup>2</sup>

be expressed as follows:

00 0 2 <sup>1</sup> ( ) ( )/ ( ) *ii i <sup>i</sup>*

*<sup>p</sup> Dp Dp* . Let <sup>0</sup>

common price <sup>0</sup> *p* such that:

$$\pi\_i^{\dot{\cdot}}(p^0) = \frac{D\_i(p^0)D\_j(p^0)}{\varepsilon\_i(p^0)D\_i(p^0) + \varepsilon\_j(p^0)D\_j(p^0)} \Big[\varepsilon\_j(p^0) - \varepsilon\_i(p^0)\Big],\tag{5}$$

where *ij j i* , 1,2, . Note that ' 0 ()0 *<sup>i</sup> p* iff 0 0 ( ) ( ), *j i p p ij j i* , 1,2, . Therefore, if possible the monopolist would want to increase the price in the market with lower elasticity of demand and to reduce the price in the market with higher elasticity of demand.

Under price discrimination, the optimal policy for the monopolist is given by:

$$D\_i(p\_i^d) + (p\_i^d - c)D\_i(p\_i^d) = 0, i = 1, 2. \tag{6}$$

where *<sup>d</sup> <sup>i</sup> p* denotes the optimal price in market *i* (and profit functions are assumed to be strictly concave in the relevant range). Under price discrimination, the optimal policy for the monopolist can be expressed as ( ) / 1 / ( ), 1,2, *dd d i i ii p c p pi* where ' () ()/() *d dd d ii ii i ii p D p p D p* is the price-elasticity in market *i*. That is, the Lerner index in each market is inversely proportional to its elasticity of demand and the monopolist therefore sets a higher price in the market with the lower elasticity of demand. The quantity sold in market *i* is ( ), 1,2. *d d i ii q Dp i* Total output under price discrimination is 2 2 1 1 ( ) *dd d i ii i i q q Dp* Given the first order conditions in (6), total output can be expressed as:

Welfare Effects of Third-Degree Price Discrimination: Ippolito Meets Schmalensee and Varian 167

market is strictly concave (strictly convex) and the demand in the higher elasticity market is strictly convex (strictly concave) then total output increases (decreases) with price

discrimination. When all demands are linear output remains unchanged.

Fig. 1. Welfare effects of third-degree price discrimination.

*q*

1 *<sup>d</sup> q* <sup>0</sup>

<sup>2</sup> *D p*( ) <sup>1</sup>

consumption goods, 1,2 *<sup>i</sup>* . It is assumed that ' <sup>0</sup> *<sup>i</sup> <sup>u</sup>* and '' <sup>0</sup> *<sup>i</sup> <sup>u</sup>* , 1,2 *<sup>i</sup>* .

6 We consider the case of quasilinear-utility function with an aggregate utility function of the form

<sup>1</sup> *q* <sup>1</sup> *q q*

<sup>1</sup> *<sup>q</sup>* <sup>0</sup>

<sup>1</sup> ( ) *ii i <sup>i</sup> <sup>u</sup> q y* , where *<sup>i</sup> <sup>q</sup>* is consumption in submarket *i* and *<sup>i</sup> <sup>y</sup>* is the amount to be spent on other

<sup>2</sup>

A move from uniform pricing to price discrimination generates a welfare change of:

1 2

*d d q q*

0 0 1 2

*q q*

11 1 22 2 [() ] [() ] ,

that is, the change in welfare is the sum across markets of the cumulative difference between price and marginal cost for each market between the output under single pricing and the output under price discrimination.6 As output decreases in the market with lower elasticity

*W p q c dq p q c dq* (11)

*p* 

2 *d p*

> <sup>2</sup> *q* <sup>2</sup> *d q*

*q*

*c*

**2.3 Welfare effects** 

*p* 

*d p*

0 *p*

<sup>1</sup> *D p*( )

$$q^d = \sum\_{i=1}^{2} D\_i(p\_i^d) = -\sum\_{i=1}^{2} (p\_i^d - c)D\_i^\cdot(p\_i^d). \tag{7}$$

The change in the quantity sold in market *i* is given by <sup>0</sup> , 1,2. *<sup>d</sup> ii i q q qi* We assume with no loss of generality that market 1 is the market with the lower elasticity of demand (the strong market) and market 2 the market with the higher elasticity (the weak market). We have implicitly assumed that both markets are served under both price regimes so, given the strict concavity of the profit functions, then <sup>0</sup> 1 2 . *d d ppp* <sup>4</sup>

Therefore, price discrimination decreases the output in market 1 and increases output in market 2: 1 *q* 0 and 2 *q* 0 . The effect of third-degree price discrimination on social welfare depends crucially on the change in the total output given by <sup>0</sup> 1 2 *<sup>d</sup> qq q q q* . We next show that the demand curvature plays a relevant role in determining the effect on total output.

#### **2.2 The change in total output and demand curvature**

Given conditions (2) and (7), the change in total output, *<sup>d</sup>* <sup>0</sup> *qq q* , is given by:

$$
\Delta \boldsymbol{q} = \boldsymbol{q}^d - \boldsymbol{q}^0 = -\sum\_{i=1}^2 (p\_i^d - c) \boldsymbol{D}\_i^\cdot (p\_i^d) + \sum\_{i=1}^2 (p^0 - c) \boldsymbol{D}\_i^\cdot (p^0). \tag{8}
$$

We can write condition (8) as:

$$\Delta q = -\sum\_{i=1}^{2} \left[ \int\_{p^0}^{p\_i^d} d\left[ (p\_i - c) D\_i^\cdot (p\_i) \right] \right]. \tag{9}$$

Therefore, we get:

$$
\Delta q = -\sum\_{i=1}^{2} \left\{ \int\_{p^0}^{p^i} \left[ D\_i^\circ(p\_i) + (p\_i - c)D\_i^\circ(p\_i) \right] dp\_i \right\},
$$

$$
$$

$$
\tag{10}
$$

From (10) we obtain that the effect of third-degree price discrimination on total output depends on the demand curvature in each market.5 If the demand in the lower elasticity

<sup>4</sup> See in Nahata et al. (1990) the analysis when profit functions are not strictly concave.

<sup>5</sup> See, for example, Robinson (1933), Schmalensee (1981), Shih, Mai and Liu (1988), Cheung and Wang (1994), Cowan (2007) and Aguirre, Cowan and Vickers (2010).

market is strictly concave (strictly convex) and the demand in the higher elasticity market is strictly convex (strictly concave) then total output increases (decreases) with price discrimination. When all demands are linear output remains unchanged.

#### **2.3 Welfare effects**

166 Social Welfare

2 2 '

*q D p p cD p* 

( ) ( ) ( ). *d d dd ii i ii*

no loss of generality that market 1 is the market with the lower elasticity of demand (the strong market) and market 2 the market with the higher elasticity (the weak market). We have implicitly assumed that both markets are served under both price regimes so, given the

Therefore, price discrimination decreases the output in market 1 and increases output in market 2: 1 *q* 0 and 2 *q* 0 . The effect of third-degree price discrimination on social

We next show that the demand curvature plays a relevant role in determining the effect on

2 2 0 ' 0 '0 1 1 ( ) ( ) ( ) ( ). *d dd i ii i*

(8)

(9)

(10)

( )().

()( ) () ,

( ) () ,

*ii i ii i*

*i ii*

*i i q q q p cD p p cD p* 

2 '

''

*i i ii i*

*i ii i*

*p c D p dp*

*q p c D p dp*

<sup>1</sup> ( ) () . <sup>2</sup>

From (10) we obtain that the effect of third-degree price discrimination on total output depends on the demand curvature in each market.5 If the demand in the lower elasticity

5 See, for example, Robinson (1933), Schmalensee (1981), Shih, Mai and Liu (1988), Cheung and Wang

*q D p p c D p dp*

*q d p cD p*

0

0 2 ''

2 ''

0

*d i p*

*d i p*

*d i p*

1

0 2 '

*i p*

1

*i p*

4 See in Nahata et al. (1990) the analysis when profit functions are not strictly concave.

(1994), Cowan (2007) and Aguirre, Cowan and Vickers (2010).

*d i p*

1

1

*i p*

*i p*

1 2 . *d d ppp* <sup>4</sup>

(7)

*ii i q q qi* We assume with

1 2

*<sup>d</sup> qq q q q* .

1 1

*i i*

welfare depends crucially on the change in the total output given by <sup>0</sup>

Given conditions (2) and (7), the change in total output, *<sup>d</sup>* <sup>0</sup> *qq q* , is given by:

The change in the quantity sold in market *i* is given by <sup>0</sup> , 1,2. *<sup>d</sup>*

strict concavity of the profit functions, then <sup>0</sup>

**2.2 The change in total output and demand curvature** 

total output.

We can write condition (8) as:

Therefore, we get:

A move from uniform pricing to price discrimination generates a welfare change of:

$$
\Delta \mathcal{W} = \bigcap\_{q\_1^0}^{q\_1^d} [p\_1(q\_1) - c] dq\_1 + \int\_{q\_2^0}^{q\_2^d} [p\_2(q\_2) - c] dq\_2. \tag{11}
$$

that is, the change in welfare is the sum across markets of the cumulative difference between price and marginal cost for each market between the output under single pricing and the output under price discrimination.6 As output decreases in the market with lower elasticity

Fig. 1. Welfare effects of third-degree price discrimination.

<sup>6</sup> We consider the case of quasilinear-utility function with an aggregate utility function of the form <sup>2</sup> <sup>1</sup> ( ) *ii i <sup>i</sup> <sup>u</sup> q y* , where *<sup>i</sup> <sup>q</sup>* is consumption in submarket *i* and *<sup>i</sup> <sup>y</sup>* is the amount to be spent on other consumption goods, 1,2 *<sup>i</sup>* . It is assumed that ' <sup>0</sup> *<sup>i</sup> <sup>u</sup>* and '' <sup>0</sup> *<sup>i</sup> <sup>u</sup>* , 1,2 *<sup>i</sup>* .

Welfare Effects of Third-Degree Price Discrimination: Ippolito Meets Schmalensee and Varian 169

*p*

<sup>1</sup> ( ) *p c q* .

<sup>1</sup> *q* <sup>1</sup> *q q*

<sup>1</sup> *<sup>q</sup>* <sup>0</sup>

where the misallocation effect, *ME*, and the output effect, *OE*, when total output increases

*ME u q u q dq u q u q dq*

*OE u q c dq* 

and may therefore be interpreted as the welfare loss due to the transfer of 1 *q* units of production from market 1 to market 2. The output effect (17), *OE*, can be interpreted as the effect of additional output on social welfare. It is positive because the social valuation of the increase in output exceeds the marginal social cost. Figure 3 illustrates the output effect (the

' '0 ' '0 11 11 1 22 22 2 [ ( ) ( )] [ ( ) ( )] ,

'

22 2 [() ] .

0 0 0 0

11 11 1 22 1 22 *ME u q u q q u q q u q* [ ( ) ( )] [ ( ) ( )], (18)

(16)

2 *d p*

(17)

<sup>2</sup> *q* <sup>2</sup> *d q* *q*

*c*

0 0 1 1 2 1

*q q q q*

0 0 1 2

*q q*

0 2 1

> 0 2 1

*q q*

*qqq*

Fig. 2. The addition and subtraction of <sup>0</sup>

1 *<sup>d</sup> q* <sup>0</sup>

*q*

<sup>2</sup> *D p*( ) <sup>1</sup>

The misallocation effect (16) can be written as:

green area) and the misallocation effect (the red area).

*q* 0 are given by:

*p*

*d p*

0 *p*

<sup>1</sup> *D p*( )

of demand and increases in the market with higher elasticity of demand, the first term in (11) is the welfare loss in market 1, whereas the second term is the welfare gain in market 2.7

Figure 1 illustrates how the welfare effect of third-degree price discrimination is measured as the addition of the (negative) change in total surplus in market 1 and the (positive) change in total surplus in market 2.8

We want to break down the effect on social welfare into two effects: a misallocation effect, which can be interpreted as the welfare loss due to the transfer of q units of production from market 1 (the market with lower elasticity) to the market 2 (the market with higher elasticity), and an output effect, which can be interpreted as the effect of the change in total output on social welfare. Obviously, the effect of total output on social welfare crucially depends on whether third-degree price discrimination increases total output or not. Then we decompose the change in welfare into the two effects for the case where price discrimination increases total output.9

We assume that *q* 0 and since the change in total output is given by 1 2 *qq q* , the change in output in market 2 is 2 1 *q qq* <sup>1</sup> *q q* 0 . We can express the change in welfare as:

$$
\Delta V = \prod\_{q\_1^0}^{q\_1^d} p\_1(q\_1) dq\_1 + \int\_{q\_2^0}^{q\_2^0 - \Delta q\_1} p\_2(q\_2) dq\_2 + \int\_{q\_2^0 - \Delta q\_1}^{q\_2^d} [p\_2(q\_2) - c] dq\_2. \tag{12}
$$

Given that <sup>0</sup> , 1,2, *<sup>d</sup> ii i q q qi* we have:

$$
\Delta \mathcal{W} = \int\_{q\_1^0}^{q\_1^0 + \Delta q\_1} p\_1(q\_1) dq\_1 + \int\_{q\_2^0}^{q\_2^0 - \Delta q\_1} p\_2(q\_2) dq\_2 + \int\_{q\_2^0 - \Delta q\_1}^{q\_2^0 - \Delta q\_1 + \Delta q} [p\_2(q\_2) - c] dq\_2. \tag{13}
$$

which under quasilinear utility ' ( ) ( ), 1,2, *ii ii pq uq i* becomes:

$$
\Delta \mathcal{W} = \int\_{q\_1^0}^{q\_1^0 + \Delta q\_1} u\_1^\dagger(q\_1) dq\_1 + \int\_{q\_2^0}^{q\_2^0 - \Delta q\_1} u\_2^\dagger(q\_2) dq\_2 + \int\_{q\_2^0 - \Delta q\_1}^{q\_2^0 - \Delta q\_1 + \Delta q} [u\_2^\dagger(q\_2) - c] dq\_2. \tag{14}$$

Taking into account that the optimal uniform price satisfies 0 '0 '0 11 22 *p uq uq* () () and by adding and subtracting <sup>0</sup> <sup>1</sup> ( ) *p c q* , see Figure 2, we can express the change in welfare as:

$$
\Delta W = \text{ME} + \text{OE} \tag{15}
$$

<sup>7</sup> The overall effect on welfare may be positive or negative. See Aguirre, Cowan and Vickers (2010) for sufficient conditions based on the shape of the demand and inverse demand functions to determine the sign of the welfare effect.

<sup>8</sup> Ippolito (1980) and more recently Cowan (2011) analyze the effect of third-degree price discrimination on consumer surplus and find reasonable settings where the effect is positive. In a related paper Leeson and Sobel (2008) consider costly price discrimination. Note that if consumer surplus increases then social welfare would increase even though price discrimination costs offset the private incentive to price discriminate.

<sup>9</sup> The Appendix considers the case where price discrimination reduces total output.

Fig. 2. The addition and subtraction of <sup>0</sup> <sup>1</sup> ( ) *p c q* .

of demand and increases in the market with higher elasticity of demand, the first term in (11) is the welfare loss in market 1, whereas the second term is the welfare gain in market 2.7 Figure 1 illustrates how the welfare effect of third-degree price discrimination is measured as the addition of the (negative) change in total surplus in market 1 and the (positive)

We want to break down the effect on social welfare into two effects: a misallocation effect, which can be interpreted as the welfare loss due to the transfer of q units of production from market 1 (the market with lower elasticity) to the market 2 (the market with higher elasticity), and an output effect, which can be interpreted as the effect of the change in total output on social welfare. Obviously, the effect of total output on social welfare crucially depends on whether third-degree price discrimination increases total output or not. Then we decompose the change in welfare into the two effects for the case where price

We assume that *q* 0 and since the change in total output is given by 1 2 *qq q* , the change in output in market 2 is 2 1 *q qq* <sup>1</sup> *q q* 0 . We can express the change in

*W p q dq p q dq p q c dq*

*W p q dq p q dq p q c dq* 

'' '

7 The overall effect on welfare may be positive or negative. See Aguirre, Cowan and Vickers (2010) for sufficient conditions based on the shape of the demand and inverse demand functions to determine the

8 Ippolito (1980) and more recently Cowan (2011) analyze the effect of third-degree price discrimination on consumer surplus and find reasonable settings where the effect is positive. In a related paper Leeson and Sobel (2008) consider costly price discrimination. Note that if consumer surplus increases then social welfare would increase even though price discrimination costs offset the private incentive to price

*W u q dq u q dq u q c dq* 

11 1 22 2 22 2 () () [() ] ,

11 1 22 2 22 2 () () [() ] ,

11 1 22 2 22 2 () () [() ] ,

(14)

(13)

 (12)

<sup>1</sup> ( ) *p c q* , see Figure 2, we can express the change in welfare as:

*W ME OE* (15)

11 22 *p uq uq* () () and by

0 1 2 1 2

00 0 1 2 2 1

00 0 1 1 2 1 2 1

00 0 1 1 2 1 2 1

000 1 2 2 1

*q q q q*

Taking into account that the optimal uniform price satisfies 0 '0 '0

9 The Appendix considers the case where price discrimination reduces total output.

000 1 2 2 1

*q q q q*

*q q q q qqq*

*q q q q qqq*

*q q q q*

*d d q q q q*

change in total surplus in market 2.8

discrimination increases total output.9

Given that <sup>0</sup> , 1,2, *<sup>d</sup>*

adding and subtracting <sup>0</sup>

sign of the welfare effect.

discriminate.

*ii i q q qi* we have:

which under quasilinear utility ' ( ) ( ), 1,2, *ii ii pq uq i* becomes:

welfare as:

where the misallocation effect, *ME*, and the output effect, *OE*, when total output increases *q* 0 are given by:

$$ME = \int\_{q\_1^0}^{q\_1^0 + \Delta q\_1} [\dot{u\_1}(q\_1) - \dot{u\_1}(q\_1^0)] dq\_1 + \int\_{q\_2^0}^{q\_2^0 - \Delta q\_1} [\dot{u\_2}(q\_2) - \dot{u\_2}(q\_2^0)] dq\_2. \tag{16}$$

$$OE = \int\_{q\_2^0 - \Delta q\_1}^{q\_2^0 - \Delta q\_1 + \Delta q} [\mu\_2'(q\_2) - c] dq\_2. \tag{17}$$

The misallocation effect (16) can be written as:

$$ME = -[\mu\_1(q\_1^0) - \mu\_1(q\_1^0 - |\Delta q\_1|)] + [\mu\_2(q\_2^0 + |\Delta q\_1|) - \mu\_2(q\_2^0)].\tag{18}$$

and may therefore be interpreted as the welfare loss due to the transfer of 1 *q* units of production from market 1 to market 2. The output effect (17), *OE*, can be interpreted as the effect of additional output on social welfare. It is positive because the social valuation of the increase in output exceeds the marginal social cost. Figure 3 illustrates the output effect (the green area) and the misallocation effect (the red area).

Welfare Effects of Third-Degree Price Discrimination: Ippolito Meets Schmalensee and Varian 171

Aguirre, Cowan and Vickers (2010) find sufficient conditions, based on the curvatures of direct and inverse demand functions for third-degree price discrimination to increase (or decrease) social welfare. Their main results are that the output effect is stronger than the misallocation effect (that is price discrimination increases social welfare) when inverse demand in the weak market is more convex than that in the strong market and the price difference with discrimination is small, and discrimination reduces welfare when the direct

Cowan (2011) shows that aggregate consumer surplus is higher with discrimination if the ratio of pass-through to the price elasticity (at the uniform price) is the same or larger in the weak market.10 As an application he shows that discrimination always increases surplus for logit demand functions whose pass-through rates exceed 0.5 (so demand is convex). Note that an increase in the consumer surplus ensures an increase in social welfare given that price discrimination increases profits (at least for a monopolist). Therefore, with this demand family results are just contrary to those under linear demand: the output effect always dominates the misallocation effect for logit demand functions (with pass-through

The constant elasticity demand family is very appropriate for illustrating the tradeoff between the two effects given that as total output increases with discrimination (see, Ippolito, 1980, Aguirre, 2006, and Aguirre, Cowan and Vickers, 2010) output effect is positive. If both the share of the strong market under uniform pricing and the elasticity difference between markets are big enough then the output effect dominates to the

iv. Third-degree price discrimination is a topic covered by any microeconomics text book. However, there is a gap in the literature with respect to an appropriate graphical analysis of the effects on social welfare.11 The above analysis fills this gap and provides a graphic treatment that is accessible for most readers and highlights the welfare effects

Schmalensee (1981) in his graphical analysis decomposed social welfare into two effects:12

concept which generalizes and simplifies the analysis of many industrial organization models.

10 Pass-through is extensively analyzed by Weyl and Fabinger (2011) and shown to be a unifying

11 Graphical presentations of third-degree price discrimination typically focus on the comparison of the corresponding profit maximization problems. See for example Round and McIver (2006) and Weber and

12 In order to facilitate comparison we have rewritten his graphical analysis in terms of our notation.

**4. Advantages over the Schmalensee's and Varian's analyses** 

We next compare our analysis with Schmalensee's and Varian's.

increases social welfare but both markets are served.

demand function is more convex in the high-price market.

rates exceeding 0.5).

misallocation effect (see, Aguirre, 2011).

of third-degree price discrimination.

Pasche (2008) for recent analysis.

the only way for third-degree price discrimination to increase welfare is by opening markets. However, the change in welfare depends on two effects: a misallocation effect and an output effect. It is easy to construct examples where price discrimination

Fig. 3. Output and Misallocation Effects.

#### **3. Some lessons on the welfare effects of price discrimination**

Some important lessons can be drawn from the above analysis:


*p* 

2 *d p*

Fig. 3. Output and Misallocation Effects.

1 *<sup>d</sup> q* <sup>0</sup> *q*

*p* 

*d p*

0 *p*

<sup>1</sup> *D* ( ) *p*

increases.

**3. Some lessons on the welfare effects of price discrimination** 

<sup>1</sup> *<sup>q</sup>* <sup>0</sup>

<sup>1</sup> *q* <sup>1</sup> *q q*

Misallocation effect

<sup>2</sup> *<sup>D</sup>* ( ) *<sup>p</sup>* <sup>1</sup>

remains constant, social welfare is reduced by price discrimination.

degree price discrimination. Note that in this case <sup>0</sup>

i. *An increase in total output is a necessary* (but of course not sufficient) *condition for thirddegree price discrimination to increase social welfare*. This conclusion is not based on exogenous bounds. Since the misallocation effect, (16), is always non-positive then a positive output effect (based on an increase in output) is needed to increase social welfare. In fact, that argument represents an earlier, easier and more intuitive demonstration of the theorem that an increase in output is a necessary condition for discrimination raises social welfare. Under linear demand, given that total output

ii. *Market Opening*. In the above analysis we assume that both markets are served under both price regimes. We now analyze the case in which third-degree price discrimination serves to open markets; that is, we assume that market 2 is only served under third-

and 2 2 <sup>0</sup> *<sup>d</sup> q q* . Therefore, in this case price discrimination not only increases social welfare but also implies a Pareto improvement. Notice that the misallocation effect would be zero and the output effect would obviously be positive given that total output

iii. The use of linear demands is not appropriate for illustrating the welfare effects of thirddegree price discrimination. Non-specialized readers might reach the conclusion that

1 2

<sup>2</sup> *q* <sup>2</sup> *d q*

*d d p p p* and therefore 1 *<sup>q</sup>* <sup>0</sup>

Output effect

*q*

*c*

Some important lessons can be drawn from the above analysis:

the only way for third-degree price discrimination to increase welfare is by opening markets. However, the change in welfare depends on two effects: a misallocation effect and an output effect. It is easy to construct examples where price discrimination increases social welfare but both markets are served.

Aguirre, Cowan and Vickers (2010) find sufficient conditions, based on the curvatures of direct and inverse demand functions for third-degree price discrimination to increase (or decrease) social welfare. Their main results are that the output effect is stronger than the misallocation effect (that is price discrimination increases social welfare) when inverse demand in the weak market is more convex than that in the strong market and the price difference with discrimination is small, and discrimination reduces welfare when the direct demand function is more convex in the high-price market.

Cowan (2011) shows that aggregate consumer surplus is higher with discrimination if the ratio of pass-through to the price elasticity (at the uniform price) is the same or larger in the weak market.10 As an application he shows that discrimination always increases surplus for logit demand functions whose pass-through rates exceed 0.5 (so demand is convex). Note that an increase in the consumer surplus ensures an increase in social welfare given that price discrimination increases profits (at least for a monopolist). Therefore, with this demand family results are just contrary to those under linear demand: the output effect always dominates the misallocation effect for logit demand functions (with pass-through rates exceeding 0.5).

The constant elasticity demand family is very appropriate for illustrating the tradeoff between the two effects given that as total output increases with discrimination (see, Ippolito, 1980, Aguirre, 2006, and Aguirre, Cowan and Vickers, 2010) output effect is positive. If both the share of the strong market under uniform pricing and the elasticity difference between markets are big enough then the output effect dominates to the misallocation effect (see, Aguirre, 2011).

iv. Third-degree price discrimination is a topic covered by any microeconomics text book. However, there is a gap in the literature with respect to an appropriate graphical analysis of the effects on social welfare.11 The above analysis fills this gap and provides a graphic treatment that is accessible for most readers and highlights the welfare effects of third-degree price discrimination.

## **4. Advantages over the Schmalensee's and Varian's analyses**

We next compare our analysis with Schmalensee's and Varian's.

Schmalensee (1981) in his graphical analysis decomposed social welfare into two effects:12

<sup>10</sup> Pass-through is extensively analyzed by Weyl and Fabinger (2011) and shown to be a unifying concept which generalizes and simplifies the analysis of many industrial organization models.

<sup>11</sup> Graphical presentations of third-degree price discrimination typically focus on the comparison of the corresponding profit maximization problems. See for example Round and McIver (2006) and Weber and Pasche (2008) for recent analysis.

<sup>12</sup> In order to facilitate comparison we have rewritten his graphical analysis in terms of our notation.

Welfare Effects of Third-Degree Price Discrimination: Ippolito Meets Schmalensee and Varian 173

loss due to the transfer of 1 *q* units of production from consumers in market 1 to consumers in market 2 and, secondly, it identifies the output effect by stating the social valuation of an increase in output as the increase in total surplus of consumer in market 2,

Varian (1985) obtained upper and lower bounds on the change in welfare when moving from uniform pricing to third-degree price discrimination. By using the property of

' 0 ' '

The upper bound provides a necessary condition for price discrimination to increase social welfare (that is, an increase in output) and the lower bound a sufficient condition. Our approach presents some advantages over Varian's. One crucial advantage relates to the graphical analysis: his graphic treatment goes not very further from the one market case as it appears in Varian (1992)'s text book. On the other hand, the bounds are not very informative. Consider for example the two cases used by Varian (1992) to illustrate the bounds: (i) linear demands and (ii) market opening. In both cases, our approach allows to

22 2 11 2 22 2 [() ] [() ] [() ] .

' ' 11 22 1 0 [ ( ) ( )] .

' '0 ' '0 11 11 1 22 22 2 [ ( ) ( )] [ ( ) ( )] 0

(23)

*W u q u q dq*

*d d*

(21)

*u q c dq W u q c dq u q c dq*

1 12 2 () ()() *d d <sup>p</sup> cq W p cq p cq* (20)

*d d*

(22)

*d*

(24)

1 2 *d d p p p* and

concavity of the utility function the bounds on welfare change are:

0 0 0 2 1 1 1 2 1

0 0 0 2 1 1 2

compute exactly the welfare change. (i) Under linear demands (21) becomes:

and however our analysis states, from (15), (16) and (17), that:

0 1 1

*q q*

0 1

*q*

0 0 1 1 2 1

*q q q q*

0 0 1 2

*W ME u q u q dq u q u q dq* 

' 0 '

22 2 22 2 [() ] [() ] ,

'

22 2 [() ] .

(25)

*u q c dq W u q c dq*

*q q*

On the other hand, when uniform pricing serves to open markets ( <sup>0</sup>

therefore 1 *<sup>q</sup>* 0 and 2 2 <sup>0</sup> *<sup>d</sup> q q* ) the bounds on social welfare are:

while the change in social welfare is given by:

0 0 2 1 2 1

0 0 2 1 2

> 0 2 1

> > 0 2 1

*q q W OE u q c dq* 

*qqq*

*q q q*

*qqq qqq*

*q q q q*

*qqq q q qqq*

0

or in terms of marginal willingness to pay:

the most elastic demand market.

Fig. 4. Schmalensee (1981)'s Output and Misallocation Effects.

$$
\Delta V = \int\_{q\_1^0}^{q\_1^0 + \Delta q\_1} [\dot{u}\_1(q\_1) - \dot{u}\_1(q\_1^0)] dq\_1 + \int\_{q\_2^0}^{q\_2^0 + \Delta q\_2} [\dot{u}\_2(q\_2) - \dot{u}\_2(q\_2^0)] dq\_2
$$

$$
+ \int\_{q\_2^0 - \Delta q\_1}^{q\_2^0 - \Delta q\_1 + \Delta q} [\dot{u}\_2(q\_2^0) - c] dq\_2. \tag{19}
$$

where the third term on the right hand in (19) may be equivalently written as <sup>0</sup> ( ) *p c q* . Schmalensee (1981) named the first two terms the (negative) distribution effect and the last term the output effect. In contrast, in our paper the output effect is the social valuation of an increase in output (that is the valuation of the consumers in the most elastic market). Figure 4 shows how in the Schmalensee's approach the distribution effect and the output effect are overstated and overlapped. His output effect (the green area plus the blue area), <sup>0</sup> ( ) *p c q* , exaggerates the social valuation of the increase in total output. It is more reasonable to define the output effect as the valuation of the additional output by the elastic market consumers (that is, those consumers enjoying the increase in output); i.e. the green area. On the other hand, the Schmalensee's distribution effect overestimates the negative effect of distributional inefficiency: the red areas plus the (negative) blue area. Our approach presents two advantages: firstly, it allows to interpret the misallocation effect as the welfare loss due to the transfer of 1 *q* units of production from consumers in market 1 to consumers in market 2 and, secondly, it identifies the output effect by stating the social valuation of an increase in output as the increase in total surplus of consumer in market 2, the most elastic demand market.

Varian (1985) obtained upper and lower bounds on the change in welfare when moving from uniform pricing to third-degree price discrimination. By using the property of concavity of the utility function the bounds on welfare change are:

$$(p^0 - c)\Delta q \ge \Delta W \ge (p\_1^d - c)\Delta q\_1 + (p\_2^d - c)\Delta q\_2 \tag{20}$$

or in terms of marginal willingness to pay:

172 Social Welfare

*p* 

2 *d p*

Fig. 4. Schmalensee (1981)'s Output and Misallocation Effects.

*q*

1 *<sup>d</sup> q* <sup>0</sup>

<sup>2</sup> *D p*( ) <sup>1</sup>

*p* 

*d p*

0 *p*

<sup>1</sup> *D p*( )

0 0 1 1 2 2

*q q q q*

 

0 0 1 2

*q q*

*qqq*

0 2 1

> 0 2 1

*q q*

' '0 ' '0 11 11 1 22 22 2 [ ( ) ( )] [ ( ) ( )]

*u q c dq*

(19)

<sup>2</sup> *q* <sup>2</sup> *d q* *q*

*c*

*W u q u q dq u q u q dq*

<sup>1</sup> *q* <sup>1</sup> *q q*

<sup>1</sup> *<sup>q</sup>* <sup>0</sup>

' 0 22 2 [() ] ,

where the third term on the right hand in (19) may be equivalently written as <sup>0</sup> ( ) *p c q* . Schmalensee (1981) named the first two terms the (negative) distribution effect and the last term the output effect. In contrast, in our paper the output effect is the social valuation of an increase in output (that is the valuation of the consumers in the most elastic market). Figure 4 shows how in the Schmalensee's approach the distribution effect and the output effect are overstated and overlapped. His output effect (the green area plus the blue area), <sup>0</sup> ( ) *p c q* , exaggerates the social valuation of the increase in total output. It is more reasonable to define the output effect as the valuation of the additional output by the elastic market consumers (that is, those consumers enjoying the increase in output); i.e. the green area. On the other hand, the Schmalensee's distribution effect overestimates the negative effect of distributional inefficiency: the red areas plus the (negative) blue area. Our approach presents two advantages: firstly, it allows to interpret the misallocation effect as the welfare

$$\int\_{q\_2^0-\Delta q\_1}^{q\_2^0+\Delta q\_1+\Delta q} [u\_2^+(q\_2^0) - c] dq\_2 \ge \Delta \mathcal{W} \ge \int\_{q\_1^0}^{q\_1^0+\Delta q\_1} [u\_1^+(q\_1^d) - c] dq\_2 + \int\_{q\_2^0}^{q\_2^0-\Delta q\_1+\Delta q} [u\_2^+(q\_2^d) - c] dq\_2. \tag{21}$$

The upper bound provides a necessary condition for price discrimination to increase social welfare (that is, an increase in output) and the lower bound a sufficient condition. Our approach presents some advantages over Varian's. One crucial advantage relates to the graphical analysis: his graphic treatment goes not very further from the one market case as it appears in Varian (1992)'s text book. On the other hand, the bounds are not very informative. Consider for example the two cases used by Varian (1992) to illustrate the bounds: (i) linear demands and (ii) market opening. In both cases, our approach allows to compute exactly the welfare change. (i) Under linear demands (21) becomes:

$$0 > \Delta V > \int\_{q\_1^0}^{q\_1^0 + \Delta q\_1} [\mu\_1^{\cdot}(q\_1^d) - \mu\_2^{\cdot}(q\_2^d)] d\eta\_1. \tag{22}$$

and however our analysis states, from (15), (16) and (17), that:

$$
\Delta \mathcal{W} = \text{ME} = \int\_{q\_1^0}^{q\_1^0 + \Delta q\_1} [\stackrel{\circ}{u\_1}(q\_1) - \stackrel{\circ}{u\_1}(q\_1^0)] dq\_1 + \int\_{q\_2^0}^{q\_2^0 - \Delta q\_1} [\stackrel{\circ}{u\_2}(q\_2) - \stackrel{\circ}{u\_2}(q\_2^0)] dq\_2 < 0 \tag{23}
$$

On the other hand, when uniform pricing serves to open markets ( <sup>0</sup> 1 2 *d d p p p* and therefore 1 *<sup>q</sup>* 0 and 2 2 <sup>0</sup> *<sup>d</sup> q q* ) the bounds on social welfare are:

$$\int\_{q\_2^0-\Lambda q\_1}^{q\_2^0-\Lambda q\_1+\Lambda q} \left[ u\_2^{\cdot}(q\_2^0) - c \right] dq\_2 \ge \Delta \mathcal{W} \ge \int\_{q\_2^0}^{q\_2^0-\Lambda q\_1+\Lambda q} \left[ u\_2^{\cdot}(q\_2^0) - c \right] dq\_2,\tag{24}$$

while the change in social welfare is given by:

$$
\Delta \mathcal{W} = OE = \int\_{q\_2^0 - \Delta q\_1}^{q\_2^0 - \Delta q\_1 + \Delta q} [u\_2^+(q\_2) - c] dq\_2. \tag{25}
$$

Spain.

**8. References** 

6.

54.

99, 206-208.

1254-1262.

Welfare Effects of Third-Degree Price Discrimination: Ippolito Meets Schmalensee and Varian 175

Vasco (IT-223-07) is gratefully acknowledged. I would like to thank Ilaski Barañano, Simon Cowan and Ignacio Palacios-Huerta for helpful comments. *BRiDGE* Group, Departamento de Fundamentos del Análisis Económico I, Avda. Lehendakari Aguirre 83, 48015-Bilbao,

Aguirre, Iñaki. 2006. "Monopolistic Price Discrimination and Output Effect Under

Aguirre, Iñaki. 2008. "Output and Misallocation Effects in Monopolistic Third-Degree Price

Aguirre, Iñaki. 2011. "Monopolistic Price Discrimination Under Constant Elasticity Demand:

Aguirre, Iñaki, Simon Cowan and John Vickers. 2010. "Monopoly Price Discrimination and Demand Curvature." *American Economic Review* 100 (4), pp. 1601-1615. Armstrong, Mark. 2008. "Price Discrimination," in P. Buccirossi (ed) *Handbook of Antitrust* 

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Cheung, Francis K., and Xinghe Wang. 1994. "Adjusted Concavity and the Output Effect

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Formby, John P., Stephen Layson and W. James Smith. 1983. "Price Discrimination,

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*Economics*, MIT Press, Cambridge Mass..

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*Bulletin*, vol. 29, 2951-2956.

Output and Misallocation Effects," unpublished manuscript.

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#### **5. Concluding remarks**

Based on a pioneering paper by Ippolito (1980) we construct a simple model which allows the welfare effects of third-degree price discrimination to be well understood and explained. The decomposition of the change in welfare into a misallocation effect and an output effect has advantages over the well-established analyses by Schmalensee (1981) and Varian (1985). In particular, our approach provides an earlier and easier proof of the theorem that an increase in output is a necessary condition for welfare to improve and a graphic analysis which clarifies the welfare analysis of third-degree price discrimination.

#### **6. Appendix**

Here we decompose the change in welfare into two effects for cases where third-degree price discrimination does not increase total output. When total output does not increase *q* 0 it is more illustrative to express the change in welfare as:

$$
\Delta \mathcal{W} = \int\_{q\_1^0}^{q\_1^0 - \Delta q\_2} p\_1(q\_1) dq\_1 + \int\_{q\_2^0}^{q\_2^0 + \Delta q\_2} p\_2(q\_2) dq\_2 + \int\_{q\_1^0 - \Delta q\_2}^{q\_1^0} [p\_1(q\_1) - c] dq\_1. \tag{A1}
$$

which under quasilinear utility ' ( ) ( ), 1,2, *ii ii pq uq i* becomes:

$$
\Delta \mathcal{W} = \int\_{q\_1^0}^{q\_1^0 - \Delta q\_2} u\_1^\dagger(q\_1) dq\_1 + \int\_{q\_2^0}^{q\_2^0 + \Delta q\_2} u\_2^\dagger(q\_2) dq\_2 + \int\_{q\_1^0 - \Delta q\_2}^{q\_1^0 - \Delta q\_2 + \Delta q} [u\_1^\dagger(q\_1) - c] dq\_1. \tag{A2}
$$

By adding and subtracting <sup>0</sup> <sup>2</sup> ( ) *p c q* , the misallocation effect and the output effect can be expressed as follows:

$$ME = \int\_{q\_1^0}^{q\_1^0 - \Delta q\_2} [\dot{u\_1^\cdot}(q\_1) - \dot{u\_1^\cdot}(q\_1^0)] dq\_1 + \int\_{q\_2^0}^{q\_2^0 + \Delta q\_2} [\dot{u\_2^\cdot}(q\_2) - \dot{u\_2^\cdot}(q\_2^0)] dq\_2. \tag{A3}$$

$$OE = \int\_{q\_1^0 - \Delta q\_2}^{q\_1^0 - \Delta q\_2 + \Delta q} [\mu\_1^\cdot(q\_1) - c] d\eta\_1. \tag{A4}$$

The misallocation effect can be written as:

$$ME = -[\boldsymbol{\mu}\_1(\boldsymbol{q}\_1^0) - \boldsymbol{\mu}\_1(\boldsymbol{q}\_1^0 - \Delta \boldsymbol{q}\_2)] + [\boldsymbol{\mu}\_2(\boldsymbol{q}\_2^0 + \Delta \boldsymbol{q}\_2) - \boldsymbol{\mu}\_2(\boldsymbol{q}\_2^0)]\,\tag{A5}$$

and may therefore be interpreted as the welfare loss due to the transfer of 2 *q* units of production from market 1 to market 2. The output effect, *OE*, can be interpreted as the effect of the reduction in output on social welfare. It is negative because the social valuation of the increase in output exceeds the marginal social cost.

#### **7. Acknowledgments**

Financial support from Ministerio de Ciencia e Innovación and FEDER (ECO2009-07939), and from the Departamento de Educación, Universidades e Investigación del Gobierno Vasco (IT-223-07) is gratefully acknowledged. I would like to thank Ilaski Barañano, Simon Cowan and Ignacio Palacios-Huerta for helpful comments. *BRiDGE* Group, Departamento de Fundamentos del Análisis Económico I, Avda. Lehendakari Aguirre 83, 48015-Bilbao, Spain.

## **8. References**

174 Social Welfare

Based on a pioneering paper by Ippolito (1980) we construct a simple model which allows the welfare effects of third-degree price discrimination to be well understood and explained. The decomposition of the change in welfare into a misallocation effect and an output effect has advantages over the well-established analyses by Schmalensee (1981) and Varian (1985). In particular, our approach provides an earlier and easier proof of the theorem that an increase in output is a necessary condition for welfare to improve and a graphic analysis

Here we decompose the change in welfare into two effects for cases where third-degree price discrimination does not increase total output. When total output does not increase

'' '

*W u q dq u q dq u q c dq* 

*ME u q u q dq u q u q dq*

*OE u q c dq* 

and may therefore be interpreted as the welfare loss due to the transfer of 2 *q* units of production from market 1 to market 2. The output effect, *OE*, can be interpreted as the effect of the reduction in output on social welfare. It is negative because the social valuation of the

Financial support from Ministerio de Ciencia e Innovación and FEDER (ECO2009-07939), and from the Departamento de Educación, Universidades e Investigación del Gobierno

11 1 22 2 11 1 () () [() ] ,

11 1 22 2 11 1 () () [() ] .

' '0 ' '0 11 11 1 22 22 2 [ ( ) ( )] [ ( ) ( )] ,

'

11 1 [() ] .

00 0 0

(A2)

 (A1)

(A3)

11 11 2 22 2 22 *ME u q u q q u q q u q* [ ( ) ( )] [ ( ) ( )], (A5)

<sup>2</sup> ( ) *p c q* , the misallocation effect and the output effect can be

(A4)

1 2 2 2 1

*<sup>d</sup> q q q q q*

0 00 1 2 1 2

00 0 1 2 2 2 1 2 000 1 2 1 2

0 0 1 2 2 2

*q q q q*

0 0 1 2

*q q*

0 1 2

> 0 1 2

*q q*

*qqq*

*q q q q W p q dq p q dq p q c dq*

*q q q q qqq*

*q q q q*

which clarifies the welfare analysis of third-degree price discrimination.

 *q* 0 it is more illustrative to express the change in welfare as: 0 0

which under quasilinear utility ' ( ) ( ), 1,2, *ii ii pq uq i* becomes:

**5. Concluding remarks** 

By adding and subtracting <sup>0</sup>

The misallocation effect can be written as:

increase in output exceeds the marginal social cost.

expressed as follows:

**7. Acknowledgments** 

**6. Appendix** 


Pigou, Arthur C. 1920. *The Economics of Welfare*, London: Macmillan, Third Edition.

Pindyck, Robert S., and Daniel L. Rubinfeld. 2008. *Microeconomics*. 7th edition. Prentice Hall.

**1. Introduction**

process.

Automated negotiation provides an important mechanism to reach agreements among distributed decision makers (Beer et al., 1999; Kraus et al., 1998; Lai et al., 2004; Rosenschein & Zlotkin, 1994). It has been extensively studied from the perspective of e-commerce (Guttman et al., 1998; He et al., 2003; Lopez-Carmona et al., 2006; Sierra, 2004), though it can be seen from a more general perspective as a paradigm to solve coordination and cooperation problems in complex systems (Jennings, 2001; Klein et al., 2002), providing a mechanism for autonomous agents to reach agreements on, e.g., task allocation, resource sharing, or surplus division (Fatima et al., 2004; Kersten & Noronha, 1998; Zhang et al., 2005). Most research in multiparty automated negotiation has been focused on building efficient mechanisms and protocols to reach agreements among multiple participants, being an objective to optimize some type of social welfare measurement (Hindriks et al., 2009). Examples of such measurements would be the *sum or product of utilities*, the *min* utility, etc... However, social welfare has not been usually placed itself as an integral part of the negotiation

**Mediated Heuristic Approaches and** 

**Highly Uncorrelated Utility Spaces** 

Ivan Marsa-Maestre1, Miguel A. Lopez-Carmona1,

Enrique de la Hoz1 and Mark Klein2

*2Massachusetts Institute of Technology* 

*1Universidad de Alcala* 

*1Spain 2USA* 

**Alternative Social Welfare Definitions for** 

**Complex Contract Negotiations Involving** 

**9**

There are remarkable works which incorporate a social welfare criterion within the search process (Ehtamo et al., 1999; Heiskanen et al., 2001; Li et al., 2009a). In these works, the authors build mechanisms to obtain fair agreements by using fair direction improvements in the joint exploration of the negotiation space. These proposals present however several limitations. Firstly, they work only when utility functions are derivable and quasi-concave. Secondly, the protocols are usually prone to untruthful revelation to bias the direction generated by the mediator. Finally, we argue that, in some scenarios, the classic notions of social welfare do not adequately represent the social goal of the negotiation, and that the type of consensus by which an agreement meets in some specific manner the concerns of all

Robinson, Joan. 1933. *The Economics of Imperfect Competition*, London: Macmillan.

