Marek Weretka, the principal investigator in the INFERENCE project was a seminar speaker at the Peking University in Shenzhen (China) during The Second PHBS Workshop in Macroeconomics and Finance.

1. Quasilinear Approximations
Marek Weretka
University of Wisconsin Madison
marek.weretka@gmail.com
Shenzen, December 15, 2019
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 1 / 29

2. Motivation
Economics: Axiom of rationality
General Equilibrium (Consumption Based) approach
(Macroeconomics, Finance, Labor, (some) Trade)
Preferences specified over primitive consumption
Complex, income effects, multiple equilibria, no additive ordinal welfare
Partial Equilibrium Approach
(IO, Auction Theory, Public Finance, (some) Trade)
Simple, no income effect, unique equilibrium, additive welfare (surplus)
Consumption of money?
Can the two approaches be reconciled?
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 2 / 29

3. Literature
Marshall “[. . . ] expenditure on any one thing, as, for instance, tea, is
only a small part of his whole expenditure” (Principles of Economics
p.17). If so, policy that affects one market should have negligible
impact on the marginal utility of money.
Vives (Restud 1987)
Considers a consumer T
t=1 ui
(xi
t ) with wealth wi
Q: What happens to λi
and Slutsky matrix as T → ∞?
Shadow price λ insensitive to wealth ∂λi
∂wi → const
Income effect vanishes in each market
Limitations
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 3 / 29

4. Preview of the main result
Consider stochastic infinite horizon economy
E
∞
t=1
βt
ui
(xt) for i = 1, ..., I
with temporary policies (affecting fundamentals in t ≤ T)
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 4 / 29

5. Preview of the main result
Consider stochastic (stationary) infinite horizon economy
E
∞
t=1
βt
ui
(xt) for i = 1, ..., I
with temporary policies (affecting fundamentals in t ≤ T)
As β → 1, the economy converges to a quasilinear limit
λi
xi
0 + E
T
t=1
βt
ui
(xt) for some {λi
}I
i=1
in ordinal terms: preferences, choices, prices, ordinal welfare
Significance
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 5 / 29

6. Outline of the talk
Simple example (single agent)
Technical result: Ordinal Minimum Theorem (OMT)
Complete Markets
Small Open Economy (prices determined outside of the model)
Large Closed Economy (General Equilibrium with endogenous prices)
Generalizations
Persitent but vanishing policy effects
Incomplete Financial Markets
Cournot economy
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 6 / 29

7. Example: A problem of a consumer
Preferences { i
β}β∈[0,1) represented by ∞
t=1 βt
ln(xi
t )
Two (temporary) policies
Factual policy p: price ζ1 = 1, endowment ei
1 = 1
Counterfactual policy p : price ζ1 = 0.5, endowment ei
1 = 2
in t ≥ 2 price ζt = βt−2
, endowment ei
t = 2
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 7 / 29

8. Example: A problem of a consumer
Preferences { i
β}β∈[0,1) represented by ∞
t=1 βt
ln(xi
t )
Two (temporary) policies
Factual policy p: price ζ1 = 1, endowment ei
1 = 1
Counterfactual policy p : price ζ1 = 0.5, endowment ei
1 = 2
in t ≥ 2 price ζt = βt−2
, endowment ei
t = 2
Preference-based (ordinal) welfare (Hicks 1939)
Fix welfare unit d = {dt}∞
t=1
Equivalent variation EV i
p,p ,d , compensating variation CV i
p,p ,d
Relation
CV i
p,p ,d = −EV i
p ,p,d
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 8 / 29

9. Example: A problem of a consumer
Preferences { i
β}β∈[0,1) represented by ∞
t=1 βt
ln(xi
t )
Two (temporary) policies
Factual policy p: price ζ1 = 1, endowment ei
1 = 1
Counterfactual policy p : price ζ1 = 0.5, endowment ei
1 = 2
in t ≥ 2 price ζ1 = βt−2
, endowment ei
t = 2
Ordinal welfare (Hicks 1939)
Fix welfare unit d = {dt}∞
t=1
Equivalent variation EV i
p,p ,d , compensating variation CV i
p,p ,d
Relation
CV i
p,p ,d = −EV i
p ,p,d = EV i
p,p ,d
Additivity of equivalent variation → equality of the indices
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 9 / 29

10. Reduced form preferences
Let
vi
(xi
0, β) ≡ max
{xi
t }t≥2
t≥2
βt
ln(xi
t ) :
t≥2
βt−2
(xi
t − ei
t) ≤ xi
0
Reduced from preferences { i∗
β }β∈(0,1) over (xi
0, xi
1), represented by
Ui∗
(xi
, β) ≡ vi
(xi
0, β) + β ln(xi
1)
are sufficient for choice and equivalent variation
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 10 / 29

11. Reduced form preferences
Let
vi
(xi
0, β) ≡ max
{xi
t }t≥2
t≥2
βt
ln(xi
t ) :
t≥2
βt−2
(xi
t − ei
t) ≤ xi
0
Reduced from preferences { i∗
β }β∈(0,1) over (xi
0, xi
1) represented by
Ui∗
(xi
, β) ≡ vi
(xi
0, β) + β ln(xi
1)
are sufficient for choice and equivalent variation
They admit closed form (Cobb-Douglass)
Ui∗
(xi
, β) ≡ α ln(xi
0 − xi
0) + β ln(xi
1) + γ
where α, xi
0 and γ are functions of β
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 11 / 29

12. Choices and ordinal welfare
-5 -4 -3 -2 -1 0 1 2
0
1
2
3
4
5
6
Figure: Choice and welfare, β = 0.7
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 12 / 29

13. Choice and ordinal welfare in a reduced form
Choice and income effect
β = 0.5 β = 0.7 β = 0.9 β = 0.99
xi
p (−1.5, 2.5) (−1.3, 2.3) (−1.1, 2.1) (−1.01, 2.01)
xi
p (−1.5, 5) (−1.3, 4.6) (−1.1, 4.2) (−1.01, 4.02)
IE 1.1 0.7 0.3 0.06
Equivalent and compensating variation
β = 0.5 β = 0.7 β = 0.9 β = 0.99
EV i
2.07 1.77 1.5 1.4
CV i
1.46 1.44 1.4 1.39
%Gap 29% 18% 6% 0.7%
Positive income effects and non additive welfare
Marshallian conjecture does not hold
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 13 / 29

14. Increasing patience
How are reduced form preferences affected by higher β?
Function
Ui∗
(xi
, β) ≡ α ln(xi
0 − xi
0) + β ln(xi
1) + γ
Effect on γ irrelevant
Effect on xi
0
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 14 / 29

15. Convergence of preferences β → 1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
3.5
4
0.5
0.5
0.5
Figure: Reduced form preferences, β = 0.5
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 15 / 29

16. Convergence of preferences β → 1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
3.5
4
0.5
0.5
0.5
0.7
0.7
0.7
Figure: Reduced form preferences, β = 0.7
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 16 / 29

17. Convergence of preferences β → 1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
3.5
4
0.9
0.7
0.5
0.5
0.7
0.5
0.9
0.7
0.9
Figure: Reduced form preferences, β = 0.9
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 17 / 29

18. Convergence of preferences β → 1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
3.5
4
0.7
0.5
0.5
0.7
0.9
0.5
0.9
0.7
0.9
0.99
0.99
Figure: Reduced form preferences, β = 0.99
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 18 / 29

19. Convergence of preferences β → 1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
3.5
4
0.7
0.5
0.9
0.99
0.5
0.7
0.99
0.5
0.7
0.9
Figure: Reduced form preferences, β = 0.99
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 19 / 29

20. Convergence of preferences β → 1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
3.5
4
0.7
0.5
0.9
0.99
0.5
0.7
0.99
0.5
0.7
0.9
Figure: Reduced form preferences, Ui
= 1
2 xi
0 + ln xi
1
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 20 / 29

21. Choice and equivalent variation
Choices for the two policies
β = 0.5 β = 0.7 β = 0.9 β = 0.99 Q
xi
p (−1.5, 2.5) (−1.3, 2.3) (−1.1, 2.1) (−1.01, 2.01) (−1, 2)
xi
p (−1.5, 5) (−1.3, 4.6) (−1.1, 4.2) (−1.01, 4.02) (−1, 4)
IE 1.1 0.7 0.3 0.06 0
Ordinal welfare
β = 0.5 β = 0.7 β = 0.9 β = 0.99 Q
EV i
2.07 1.77 1.5 1.4 1.39
CV i
1.46 1.44 1.4 1.39 1.39
%Gap 29% 18% 6% 0.7% 0
Limit: zero income effects and additive welfare
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 21 / 29

22. Infinite horizon example (summary)
In the infinite horizon problem with patient consumer β ≈ 1
Cardinal utility becomes unbounded, limβ→1 Ui∗
(xi
, β) = ∞
Preferences continuously transform into quasilinear ones
Ui∗
(xi
, 1) =
1
2
xi
0 + ln xi
1
Observables and welfare close to the quasilinear framework.
The framework acquires all desirable properties
Economic intuition
This project: the mechanism is very very robust and useful
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 22 / 29

23. Technical Challenges
Challenges:
Notion of continuity of family of preferences in β?
Continuity of equivalent variation
Local character of preference convergence
Representation by a jointly continuous utility Ui
(xi
, β)? No!
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 23 / 29

24. Ordinal Minimum Theorem
Consider a family of preferences { i
β}β∈[0,1] over compact X ⊂ RN.
Def 3: Family of preferences { i
β}β∈[0,1] is jointly continuous on Xi
× [0, 1]
whenever correspondence Ψi
xi
, β ≡ yi
∈ Xi
|yi i
β xi
is upper and
lower hemicontinuous (i.e., it is continuous).
Ordinal Minimum Theorem (OMT)
Extension to non-compact sets X.
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 24 / 29

25. Small Open Economy (exogenous prices)
A stochastic infinite horizon economy with I traders
Preferences over consumption { i
β}β∈[0,1) represented by
Ui
xi
, β = E
∞
t=1
βt
ui
(xi
t )
where ui
is C2
, strictly increasing, strictly convex, Inada.
Fundamentals: Exogenous stationary Markov chains
Pricing kernel ζ = {ζt}∞
t=1
Endowments e = {ei
}i where ei
= {ei
t}∞
t=1
Temporary policies: perturbations of ζ, e in periods t ≤ T
Budget correspondence fully determined by policy p = (˜ζ, ˜e)
E
∞
t=1
βt ˜ζt(xi
t − ˜ei
t) ≤ 0
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 25 / 29

26. Small Open Economy: Result
Consider i∗
1 represented by
Ui∗
(xi
, β) ≡ λi
xi
0 + E
T
t=1
ui
(xi
t ).
where λi
satisfies ¯E(ζtui −1
(ζtλi
)) = ¯E(ζtei
t).
Theorem
(Reduced-form) preferences, choices, and (preference based) welfare all
converge to the ones predicted by quasilinear limits i∗
1 .
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 26 / 29

27. Large Closed Economy (General Equilibrium)
Infinite horizon economy with i = 1, ..., I traders (consumers)
Von Neumann-Morgernstern preferences
satisfying previous assumptions
are in Gorman polar form (HARA).
Temporary policies: perturbations of e in periods t ≤ T
Theorem
Preferences, equilibrium allocation, prices and social welfare all converge
to the ones predicted by quasilinear limits { i∗
1 }i
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 27 / 29

28. Extentions
Generalizations (almost completed)
Infinite-horizon, vanishing effects of policies
Incomplete markets (bonds with different maturity)
Cournot markets with production (Nash Equilibrium)
Work in progress
Quantitative predictions for relevant policies
Aiyagari framework
Conclusions
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 28 / 29

29. The End
Marek Weretka (UW Madison) Quasilinear Approximations Shenzen, December 15, 2019 29 / 29