Welfare Measurements with Heterogenous Agents

Welfare Measurements with Heterogenous Agents
Marek Weretka Marcin Dec
University of Wisconsin Madison and FAME/GRAPE Warsaw
marek.weretka@gmail.com
Milano, August 22, 2022
Marek Weretka, Marcin Dec (FAME/GRAPE) Milano, August 22, 2022 1 / 25

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?

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.

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
PT
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

Preview of the main result
Consider stochastic (stationary) infinite horizon (small open) economy
E
∞
X
t=1
βt
ui
(xt) for i = 1, ..., I
with temporary policies (effects vanish sufficiently fast)

Preview of the main result
Consider stochastic (stationary) infinite horizon (small open) economy
E
∞
X
t=1
βt
ui
(xt) for i = 1, ..., I
with temporary policies (effects vanish sufficiently fast)
As β → 1, the economy converges to a quasilinear limit
λi
xi
0 + E
T
X
t=1
βt
ui
(xt) for some {λi
}I
i=1
in ordinal terms: preferences, choices, prices, ordinal welfare
Significance

Outline of the talk
Simple example (single agent)
Small Open Economy
Application: Polish economy (stimulus packages)
Generalizations

Example: recursive problem of a consumer
Problem
max
∞
X
t=0
βt
ln(ci
t) s.t. ci
t + qt+1wi
t+1 = wi
t + ei
t.

Problem
max
∞
X
t=0
βt
ln(ci
t) s.t. ci
t + qt+1wi
t+1 = wi
t + ei
t.
Two policies
Factual policy p: price q1 = 1, endowment ei
0 = 1
Counterfactual policy p0
: price q1 = 2, endowment ei
0 = 2
in t ≥ 1 price qt+1 = β, endowment ei
t = 2

Problem
max
∞
X
t=0
βt
ln(ci
t) s.t. ci
t + qt+1wi
t+1 = wi
t + ei
t.
Two policies
0 = 1
0 = 2
t = 2
Preference-based welfare (Hicks 1939)
Fix consumption flow d = {dt}∞
t=1 (welfare unit)
Equivalent variation EV i
p,p0 , compensating variation CV i
p,p0

Problem
max
∞
X
t=0
βt
ln(ci
t) s.t. ci
t + qt+1wi
t+1 = wi
t + ei
t.
Two policies
0 = 1
0 = 2
t = 2
Preference-based welfare (Hicks 1939)
Fix consumption flow d = {dt}∞
t=1 (welfare unit)
Equivalent variation EV i
p,p0 , compensating variation CV i
p,p0
Relation
CV i
p,p0 = −EV i
p0,p

Reduced form preferences
Let
vi
(m) ≡ max
{ci
t }t≥1
X
t≥1
βt
ln(ci
t ) :
X
t≥1
βt−1
(ci
t − ei
t) ≤ m
Reduced-form preferences over pairs (m, ci
0) , represented by
Ui
(m, ci
0) ≡ ln(ci
0) + vi
(m)
are sufficient for choice and equivalent variation
Preferences have Cobb-Douglass form
Ui
(m, ci
0) ≡ ln(ci
0) + α ln(m − m) + γ
where α, m and γ are functions of β

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

Choice and ordinal welfare in a reduced form
Choice and income effect
β = 0.5 β = 0.7 β = 0.9 β = 0.99
p : (m, ci
0) (−1.5, 2.5) (−1.3, 2.3) (−1.1, 2.1) (−1.01, 2.01)
p0
: (m, ci
0) (−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 (in units of ci
1)
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

Increasing patience
How are reduced form preferences affected by higher β?
Function
Ui
(m, ci
0) ≡ ln(ci
0) + α ln(m − m) + γ
Effect on γ irrelevant
Effect on m → −∞

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

-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

-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

-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

-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

-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
= m + 2 ln ci
0

Choice and equivalent variation
Choices for the two policies
β = 0.5 β = 0.7 β = 0.9 β = 0.99 Q
p : (m, ci
0) (−1.5, 2.5) (−1.3, 2.3) (−1.1, 2.1) (−1.01, 2.01) (−1, 2)
p0
: (m, ci
0) (−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
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

Example (summary)
In the infinite horizon problem with patient consumer β ≈ 1
Cardinal utility becomes unbounded, limβ→1 Ui
= ∞
Preferences continuously transform into quasilinear ones
Ui
= m + 2 ln ci
0
Observables and welfare close to the quasilinear framework.
The framework acquires all desirable properties
Economic intuition
The mechanism is extremely robust and potentially useful

(Polish) Small Open Economy
A stochastic infinite horizon economy with I heterogenous traders
Preferences represented by
E
∞
X
t=0
βt
ui
(ci
t ).
where ui
is C2
, strictly increasing, strictly convex, Inada.
Complete markets, budget constraint after history ht,
ci
t + E(qt+1wi
t+1|ht) ≤ wi
t + ei
t.
Markov Chains:
Pricing kernel implied by q
Endowments ei
for i = 1, ..., I

Economic policies
Policy: perturbation of q and ei , i = 1, ..., I
Formally: p = {∆q, {∆ei
}i } arbitrary measurable processes

Economic policies
Assumption 1: (Vanishing perturbations) Consider policy p. There exist
constants C > 0 and ∆ ∈ (0, 1) such that |∆q
ht
| ≤ C × (∆)t and
|∆ei
ht
| ≤ C × (∆)t for all periods t, histories ht, and i.

Economic policies
Assumption 1: (Vanishing perturbations) Consider policy p. There exist
constants C > 0 and ∆ ∈ (0, 1) such that |∆q
ht
| ≤ C × (∆)t and
|∆ei
ht
| ≤ C × (∆)t for all periods t, histories ht, and i.
Consider policies p, p0 that satisfy Assumption 1.
Aggregate equivalent variation is
EVp,p0 ≡
1
I
X
i
EV i
p,p0 .
In general, the aggregate index is ill-behaved
is non-additive
depends on individual shocks except for Gorman preferences

Main result
Aggregate income shock ∆GDP =
P
i ∆ei
Theorem
There exists a surplus function S(∆q0
, ∆GDP) such that
lim
β→1
EVp,p0 = S(∆q0
, ∆GDP0
) − S(∆q
, ∆GDP
).
Additive limit, measurable with respect to aggregate shocks.

Applicability?
Experiment β → 1 is the same as period length becoming zero
For what duration of policies our approximation is accurate?

Applicability?
Experiment β → 1 is the same as period length becoming zero
For what duration of policies our approximation is accurate?
“Realistic” toy model of the Polish economy during COVID
Demographic structure and income/unemployment shocks: BAEL
Agents heterogenous in terms of risk aversion
Annual discount factor β = 0.97
Counterfactual policies: stimulus packages of different duration

Accuracy of approximation
Table: EV convergence
number of quarters
1 2 4 8 12
EV 0.0060 0.0163 0.0458 0.1250 0.2162
L 0.0060 0.0164 0.0465 0.1291 0.2272
[EV/L -1] ×100 0.00 -0.48 -1.41 -3.17 -4.82
Note: The first two rows of the table report the actual equivalent variation and the
predicted limit value. The third row gives the difference in percentage terms.

Extensions
Generalizations (some resutls)
Large closed economy
Incomplete markets (bonds with different maturity)
Cournot markets with production (Nash Equilibrium)
Markets with complements/substitutes

The End

Welfare Measurements with Heterogenous Agents

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