Identification in multidimensional hedonic models
Lars Nesheim
CeMMAP, UCL and IFS
March 2015
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Multidimensional hedonic models
March 2015
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What is a hedonic model?
Hedonic models are models of markets for differentiated products.
They are used to analyse a wide range of markets including those for
a) workers, b) houses, c) automobiles and many others.
In a hedonic market, each product is described by z, a vector of
observable (and sometimes unobservable) characteristics that affect
either costs or utility.
Hedonic models are used to forecast prices for new products, to
adjust consumer price indexes for changes in product quality, and to
estimate how consumer demand and consumer utility depend on z.
Nesheim (CeMMAP)
Multidimensional hedonic models
March 2015
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Multidimensional hedonic models
Nearly all empirical applications of hedonic models involve products
with multidimensional characteristics.
Labor markets: wages depend on a vector of skills, occupational
characteristics, risk of injury/death.
Housing: housing quality is measured by its location vector, size of
house, size of lot, quality of materials, age of dwelling, etc.
Computers: prices depends on processing cores, speed, memory, brand,
display, weight, ...
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Multidimensional hedonic models
March 2015
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2010 London housing prices
2010 London house prices
6
x 10
2.5
2
1.5
1
0.5
0
2
5.45
1.9
5.4
1.8
5.35
5.3
1.7
5.25
5.2
north
Nesheim (CeMMAP)
1.6
5.15
east
Multidimensional hedonic models
March 2015
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2010 London housing prices
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Multidimensional hedonic models
March 2015
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Nonseparable hedonic models: the identification and
estimation problem
Nonseparable models:
Hedonic demand z = d (x, ε) for vector z implicitly defined by system
of first-order conditions
Dz p (z, m ) = Dz u (z, x, ε) .
Prices and demand generated from matching equilibrium.
Identification and estimation problem?
Given data on (z, x, p, m ) , identify and estimate (p, u, Fε ) .
Nonparametric identification results for scalar nonseparable hedonic
models worked out in Heckman, Matzkin, Nesheim (2010).
This paper extends this work to multidimensional hedonic models.
Related work in Chernozhukov, Galichon, and Henry (2014).
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Multidimensional hedonic models
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Results: multi-market data
Fully nonseparable hedonic model is point identified:
Requires policy invariant normalizations.
Requires data from multiple markets with “rich” price variation.
Requires observable multidimensional aggregate supply or demand
shifters.
Estimation based on nonparametric estimation of the joint
distribution F Z |XM and p (z, m ) .
Identified structural functions can be used to estimate the impact of
counterfactual policy changes or changes in the economic
environment.
Nesheim (CeMMAP)
Multidimensional hedonic models
March 2015
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Results: single market data
1
Additivity restrictions
Additive specification is over-identified (up to normalizations)
Dz u (z, x, ε) = u0 (z ) + v (x ) + ε.
Resulting econometric model is a multidimensional transformation
model
Dz T (z ) = v (x ) + ε.
2
Work in progress
Simulations of two dimensional model and estimation.
Applications to 1) wage data and injury/death risk, 2) housing market
data and location quality are in progress.
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Multidimensional hedonic models
March 2015
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Outline
1
2
Model.
Equilibrium.
Existence, uniqueness and purity, differentiability of price function.
3
Identification: single market data.
4
Identification: multiple market data.
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Multidimensional hedonic models
March 2015
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Model: products and prices
Products (jobs, houses) have payoff relevant characteristics
z ∈ Z ⊆ R2 .
Traded in many markets indexed by m ∈ M.
Price in market m is p (z, m ) .
Buyers and sellers cannot move between markets.
If they do, we are in the single market case.
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Multidimensional hedonic models
March 2015
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Assumption 1: Distribution of buyers
Set of buyers with characteristics (x, ε) ∈ X × E ⊆ Rnx × R2 .
Random variables (X , M, ε) continuously distributed on their
supports.
With ε ⊥
⊥ (X , M ) .
Density functions (fXM , fε ) ∈ C 2 (X × M) × C 2 (E) with
0 < bXL ≤ (fXM , fε ) ≤ bXH < ∞.
Can allow for discrete X ; potential loss of point identification.
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Assumption 2: Buyer utility
Buyers have reservation utility u0 (x, ε, m ) = 0.
Buyer (x, ε) in m who chooses z obtains u (z, x, ε) − p (z, m ) .
Utility function u ∈ C 5 (Z × X × E) .
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Assumption 3: Distribution of sellers
Set of sellers with characteristics (y , η ) ∈ Y × H ⊆ Rny × R2 .
Random variables (Y , M, η ) continuously distributed on their
supports.
With η ⊥
⊥ (Y , M ) .
Density functions fYM , fη ∈ C 2 (Y × M) × C 2 (H) and with
0 < bYL ≤ fYM , fη ≤ bYH < ∞.
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Assumption 4: Seller profits
Sellers have reservation profit π0 (y , η, m ) = 0.
Seller (y , η ) in m who chooses z obtains p (z, m ) − c (z, y , η ) .
Cost function c ∈ C 5 (Z × Y × H) .
Alternate (easier) case: endowment economy
Distribution of sellers is FZM and profits are p (z, m ) .
Alternate (harder) case:
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Monopoly case
Multidimensional hedonic models
March 2015
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Buyer’s problem
In market m, problem of buyer (x, ε) is to find z to solve
v (x, ε, m ) = max {u (z, x, ε) − p (z, m )} .
z ∈Z
First-order condition is
Dz u (z, x, ε) − Dz p (z, m ) = 0.
Solution defines demand function
z = d (x, ε, m ) .
For the moment, assume second-order condition is satisfied.
Analogous problem for seller.
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Equilibrium
Definition (Equilibrium)
Let α be a measure on the space of buyers, sellers and products.
A pair (α, p ) is an equilibrium if
1
Markets clear: The marginals of α equal the marginals of buyers and
sellers respectively.
2
Agents optimise: If (z, x, ε, y , η ) is in the support of α, then z is
optimal for (x, ε) and for (y , η ).
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Assumption 5: Generalized single-crossing
5a. For almost all (x, ε, y , η ) , the problem
s (x, ε, y , η ) = max {u (z, x, ε) − c (z, y , η )}
z ∈Z
has a unique interior optimizer, z = h (x, ε, y , η ) .
5b. The matrix M + M T is positive definite where
2
Dxy s Dx2η s
M=
.
2 s D2 s
Dεy
εη
2 u > 0 and D 2 c < 0.
5c. Moreover, Dzε
zη
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Optimal transportation formulation
Find equilibrium by matching buyers and sellers together to maximise
social welfare.
Social welfare maximisation problem is
Z
max
s (x, ε, y , η )d γ
(1)
subject to γ ∈ Γ(FX ,ε , FY ,η ).
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Multidimensional hedonic models
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Equilibrium
Theorem 1 (Existence, uniqueness and purity of equilibrium)
An equilibrium exists and is unique. In this equilibrium, matching is pure.
Each buyer matches with a unique optimal seller.
Second-order conditions are satisfied for everyone who enters the
market.
If s (x, ε, y , η ) > 0 almost surely, everyone enters the market almost
surely.
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Equilibrium
Proof.
Under the conditions stated:
Social welfare maximisation problem is an optimal transport problem
with a unique solution.
Matching is pure.
Multipliers on constraints can be used to construct prices that
decentralise the optimal allocation.
Prices are unique for trades that occur in equilibrium.
See Chiappori, McCann and Nesheim (2010).
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Multidimensional hedonic models
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Assumption 6: Curvature of surplus
To ensure smoothness of equilibrium payoffs, fourth derivatives
cannot be too big.
Specifically, there exists a constant c0 > 0 such that for any
(x, ε) ∈ X × E, (y , η ) ∈ Y × H, and ξ ∈ X × E, ζ ∈ Y × H, ξ ⊥ ζ
∑
(s p,q sij,p sq,rs − sij,rs ) s r ,k s s,l ξ i ξ j ζ k ζ l ≤ −c0 |ξ |2 |ζ |2 .
i,j,k,l,p,q,r ,s
See Ma, Trudinger and Wang (2005).
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Multidimensional hedonic models
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Assumption 7: s-convexity of spaces of buyers and sellers
If optimal map from buyers to sellers hits boundary of space, can lead
to failure of smoothness.
To avoid this, require spaces of buyers and sellers to be s-convex and
s ∗ -convex (these are generalisations of convexity).
For all (y , η ) ∈ Y × H, the set
Dy η s
−1
(X × E, y , η )
is convex.
For all (x, ε) ∈ X × E, the set
(Dx ε s )−1 (x, ε, Y × H)
is convex.
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Multidimensional hedonic models
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Differentiability of price function
Theorem 2 (Differentiability of price function)
p (z ) is twice continuously differentiable in z.
1
Envelope theorem implies s (x, ε, y , η ) ∈ C 4 .
2
Thus, under assumptions 1 - 7, Ma, Trudinger and Wang (2005)
Theorem 2.1 implies v (x, ε) ∈ C 3 and π (y , η ) ∈ C 3 .
3
Price function satisfies
p (z ) = u (x, z, ε) − v (x, ε).
4
So, using envelope theorem again
Dz p (z ) = Dz u + (Dx u − Dx v ) Dz x
Dz p (z ) = Dz u.
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Multidimensional hedonic models
March 2015
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Single market estimation problem
Data on (xi , zi , p
ei )N
i =1 .
Price function
p
e = p (z ) + ξ
where ξ is measurement error.
Demand function z = d (x, ε) satisfies
Dz p (z ) = Dz u (z, x, ε) .
Want to estimate (p, u, Fε ) .
Data provide direct estimates of p and F Z |X .
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Multidimensional hedonic models
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Failure of point-identification in a single market
Even if Fε is known, multiple models for d are consistent with data.
Suppose ε is bivariate independent uniform.
A demand model consistent with data is
−1
d1A (x, ε 1 ) = F Z1 |X
(x, ε 1 )
h
i
−1
−
1
x, F Z1 |X
d2A (x, ε 1 , ε 2 ) = F Z2 |Z1 X
(x, ε 1 ) , ε 2 .
An observationally equivalent demand model is
i
−1 h
−1
x, F Z2 |X
d1B (x, ε 1 , ε 2 ) = F Z1 |Z2 X
(x, ε 2 ) , ε 1
−1
d2B (x, ε 2 ) = F Z2 |X
(x, ε 2 ) .
Neither model is structural. Can be used to predict distribution of
demand if price held fixed but not otherwise.
Not a normalisation in a single market.
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Multidimensional hedonic models
March 2015
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Failure of point-identification (2)
Even if d (x, ε) were identified, still u is not.
The first-order condition is
Dz p (d (x, ε)) = Dz u (d (x, ε) , x, ε) .
u is a function of nx + 4 variables, but, in a single market, we only
observe its values on a nx + 2 dimensional manifold.
Without further structure, it is impossible to estimate
Dz u (d (x, ε) + ∆z, x, ε) .
Need data from multiple markets or further restrictions on u.
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Multidimensional hedonic models
March 2015
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Point-identification using additive marginal utility
restriction
Suppose utility takes an additive form
Dz u (z, x, ε) = u0 (z ) + v (x ) + ε.
First-order conditions become
Dz p (z ) = u0 (z ) + v (x ) + ε.
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Point-identification – Bivariate transformation model
Rewrite this as
Dz T ( z ) = v ( x ) + ε
where
D z T ( z ) = D z p ( z ) − u0 ( z ) .
This is a bivariate ”transformation” model with the restriction that
Dz T (z ) is the gradient of a convex function and T (z ) is one-to-one
and onto.
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Multidimensional hedonic models
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Implications of transformation model
Density of data is
fZ |X (z, x ) = fε (Dz T (z ) − v (x )) |det (Dzz 0 T (z ))| .
CDF of data is
FZ |X (z1 , z2 , x ) =
Zz1 Zz2
fε (Dz T (s ) − v (x )) |det (Dzz 0 T (s ))| ds
−∞ −∞
=
Z
fε (ε) d ε.
Dz T (A)−v (x )
where
A = {Z ∈ Z : Z1 ≤ z1 and Z2 ≤ z2 }
Dz T (A) − v (x ) = {ε : ε = T (z ) − v (x ) for some z ∈ Z } .
Multivariate formula for change of variable.
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Derivative w.r.t. z
Z
Dzi FZ |X (z1 , z2 , x ) =
fε ( ε )
∂Dz T (A)−v (x )
Z
=
Dε 1 fε ( ε )
Dz T (A)−v (x )
Z
+
∂2 T (z )
∂2 T (z )
d ε2 −
d ε1
∂z1 ∂z i
∂z2 ∂z i
∂2 T
∂z1 ∂zi
Dε 2 fε ( ε ) d ε
Dz T (A)−v (x )
Z
Dz T (A)−v (x )
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∂2 T
∂z2 ∂zi
T


= 
 ∂ ( Dz T )
Dε fε ( ε ) d ε 
.
∂zi
Multidimensional hedonic models
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Derivative w.r.t. x

∂FZ |X (z1 , z2 , x )
=−
∂xi
=−
∂ 

∂xi

Z

fε ( ε ) d ε 
Dz T (A)−v (x )
Z
∂Dz T (A)−v (x )
=−
Z
∂v1
∂v2
fε ( ε )
d ε2 −
d ε1
∂xi
∂xi
(Dε fε )T Dxi v .
Dz T (A(z ))−v (x )
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Derivatives of CDF
Restrictions of separability are
Dz FZ |X = (Dzz 0 T )
Dx FZ |X = − (Dx v )
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Z
Dε fε d ε
(2)
Dε fε d ε.
(3)
Z
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Normalization
For some x0 , normalize Dx v (x0 ) = I and define
FZ0 |X = FZ |X (z, x0 )
Fε0 = Fε (T (A) − v (x0 )) .
Then, using (2) and (3)
Dz FZ0 |X = − (Dzz 0 T ) Dx FZ0 |X .
Linear system of PDE’s in unknown Dzz 0 T .
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Final result
Restriction on T is
−Fz01
Fx02 Fz02
+
2
Fx01
Fx01
!
+
Fx02
Fx01
2
2 T22 = T11 .
Linear second order elliptic differential equation in the unknown T .
Identified up to boundary conditions. For example, one could impose
that for all z ∈ Z,
Dz1 T ( z ) = Φ − 1 ( z 1 )
Dz2 T ( z ) = Φ − 1 ( z 2 )
.
Once T is known, equation (3) is a system of first-order linear PDE’s
in Dx v . Looking at two different values of z, this can be solved for
v (x ) with an initial condition such as v (0) = 0.
With T and v , known, recover fε from change of variables formula.
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Multidimensional hedonic models
March 2015
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Single market estimation
max
{fε ,T ,v1 ,v2 }
{∑i (ln fε (ε 1i , ε 2i ) + ln det(Dzz T (zi )}
subject to
ε 1i
ε 2i
Dz1 T ( z )
Dz2 T ( z )
v1 (0)
v2 (0)
=
=
=
=
=
=
Dz1 T (zi ) − v1 (xi )
Dz2 T (zi ) − v2 (xi )
Φ−1 (z1 )∀z ∈ ∂Z
Φ−1 (z2 )∀z ∈ ∂Z
0
0
Simulation
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Multimarket identification (1)
Recall the FOC
Dz p (z, m ) = Dz u (z, x, ε) .
∗ = d (x, ε, m ) . In single market, it is impossible to learn the
Let zm
precise value of
∗
Dz u (zm
+ ∆z, x, ε)
∗ + ∆z is never observed.
because outcome z 0 = zm
With multiple markets, can learn marginal utility at outcome z 0 by
finding a market m0 with price p (z, m0 ) such that
z 0 = d x, ε, m0
= zm∗ + ∆z.
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Multidimensional hedonic models
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Multimarket identification (2)
Equilibrium implies that marginal price varies across markets due to
observable cost or demand shifters.
That is, variation in distributions of observables F Y |M and/or F X |M .
Use this variation to estimate p (z, m ) and F Z |XM .
Need support and price variation conditions.
Also need normalisation on distribution Fε .
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Assumptions 8 and 9: Full support and price variation
Assumption 5 implies that for each (z, x ), Dz u (z, x, ε) : E → V is
one-to-one where Vzx = Range (Dz u ) .
Assumption 8: Range of Dz u
Assume that Range (Dz u ) is independent of (z, x ) .
Define the random variable v = Dz p (z, m ) and let
V = Range (Dz u ).
Assumption 9: Distribution of V | ZX
Suppose that (V , Z , X ) has support on V × Z × X.
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Normalization (1): Distribution of unobservables
Vector ε are not observed and have no natural units.
Enter model in non-additive fashion.
Natural normalization: each element of ε measures percentile rank of
consumer in some dimension.
For example, ε 1 may be the percentile rank of z1 conditional on (x, m )
and ε 2 may be the percentile rank of z2 conditional on (x, m ) and ε 1 .
Choose units of measurement so that elements of ε are independent
uniform random variables.
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The identified set
There is a family of demand functions, distributions Fε and utility
functions Dz u that are consistent with the data.
Let
d (x, m, u ) =
Q Z1 |XM (u1 , x, m )
Q Z2 |Z1 XM u2 , Q Z1 |XM (u1 , x, m ), x, m
.
The identified set of demand functions includes all functions of the
form
de (x, m, ε) = d (x, m, R [Fε (ε), x ])
where Fε is a distribution function and R is any invertible measure
preserving transformation satisfying det ∂R
= 1.
∂u
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Identification of Dz u
Suppose I choose Fε ∼ Uniform and R (ε, x ) = ε. Then
Dz u (z, x, ε) = Dz p (d (x, m∗ , ε), m∗ )
where m∗ solves z = d (x, m∗ , ε).
e and d,
e then
If instead I choose an alternative Feε , R,
Dz u
e(z, x, e
ε) = Dz p de (x, m∗ , e
ε ), m ∗
where m∗ solves z = de (x, m∗ , e
ε ).
Observationally
equivalent models, different interpretations of ε since
e
e
ε = R Fε ( ε ), x .
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Normalization (2)
Choice between d and de is a normalisation.
Both predict same distribution of demand in each market m.
Demand of consumer (x, ε) predicted by
h d (x, ε)i is same as demand of
e
e
consumer (x, e
ε) under d where e
ε = R Feε (ε), x .
By construction, equilibrium price in market m0 using d is the same as
e R, Fε ).
that obtained using (d,
Conditions under which this is NOT a normalisation:
Change in economic environment changes u.
Change in Fε .
If ε is NOT independent of x.
If support conditions are not met.
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Conclusion
Multimarket data required for point identification of unrestricted
model.
Estimation based on nonparametric estimation of a multidimensional
distribution function.
Computationally simple but highly demanding of data.
Restrictions that reduce dimension required for point identification
using single market data.
Applications in progress
Willingness-to-pay for locational quality and housing quality using
English Housing Survey and American Housing Survey.
Willingness to accept risk of injury and death on the job using US CPS.
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Monopoly case
Equilibrium price p (z, m ) in market m could be determined by
monopolist or by oligopoly competition.
For the monopoly case, given (u, FXM , Fε ) and product cost c (z, m ) ,
monopolist chooses p (z ) .
Monopolist solves
Z
max
{p }
(p (d (x, ε)) − c (d (x, ε))) fxm (x, m) fε (ε) dxd ε
subject to
d (x, ε) ∈ arg max {u (z, x, ε) − p (z )} .
z
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Monopoly: mechanism design formulation
Or, equivalently, choose (v , d ) to maximize profits subject to
participation and incentive constraints (Rochet and Chone, 1998,
Carlier, 2001 or Figalli, McCann and Kim, 2011).
That is,
Z
max
{v ,d }
[u (d, x, ε) − v (x, ε) − c (d (x, ε))] fx (x ) fε (ε) dxd ε
subject to
= Dx u (d, x, ε)
Dε v = Dε u (d, x, ε)
v ≥ 0
v (x, ε) =
sup
u z 0 , x, ε − u z 0 , x 0 , ε0 − v x 0 , ε0
Dx v
{z 0 ,x 0 ,ε0 }
Solution p (z ) depends on c (z, m ), FXM , Fε and u.
back to demand
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Example: Distribution of buyers
Density of x vs. (x1,x2)
2.5
2
1.5
1
0.5
0
1
0.8
1
0.6
0.8
0.6
0.4
0.4
0.2
x2
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0.2
0
0
x1
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Example: Distribution of sellers
Density of y vs. (y1,y2)
70
60
50
40
30
20
10
0
1
0.8
1
0.6
0.8
0.6
0.4
0.4
0.2
y2
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0.2
0
0
y1
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Example: Optimal matching
back to estimation
Y1(x) vs. (x1,x2)
6
5
4
3
2
1
0
1
0.8
1
0.6
0.8
0.6
0.4
0.4
0.2
x2
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0.2
0
0
x1
Multidimensional hedonic models
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