This document discusses overlapping generations models with heterogeneous agents. It introduces an overlapping generations model framework where cohorts live for multiple periods and differ in their preferences and decision rules. This allows for analysis of policy instruments aimed at voluntary pension savings. The model includes heterogeneous subcohorts within cohorts that differ in factors like preferences, financial literacy, and pension expectations. The document outlines the standard household problem and possible extensions, as well as the production sector, government policies, and different pension system components that could be analyzed using this type of model framework. The goal is to use this model to analyze potential reforms by first finding an initial stable solution and then examining the transition path after a policy change.

1. Introduction What are these models about? Reforming Future work
Overlapping generations model
and within cohort heterogeneity
Jan Woznica
Beethoven Project meeting, November 2016
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2. Introduction What are these models about? Reforming Future work
Where we are and want to be
Starting idea from the policy angle:
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3. Introduction What are these models about? Reforming Future work
Where we are and want to be
Starting idea from the policy angle:
Plenty of countries have instruments aiming to foster voluntary pension savings
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4. Introduction What are these models about? Reforming Future work
Where we are and want to be
Starting idea from the policy angle:
Plenty of countries have instruments aiming to foster voluntary pension savings
So far in the literature NOBODY analyzed if these instruments make sense
2 / 13

5. Introduction What are these models about? Reforming Future work
Where we are and want to be
Starting idea from the policy angle:
Plenty of countries have instruments aiming to foster voluntary pension savings
So far in the literature NOBODY analyzed if these instruments make sense
Starting idea from the academic angle:
2 / 13

6. Introduction What are these models about? Reforming Future work
Where we are and want to be
Starting idea from the policy angle:
Plenty of countries have instruments aiming to foster voluntary pension savings
So far in the literature NOBODY analyzed if these instruments make sense
Starting idea from the academic angle:
Within cohort heterogeneity is rare, also because it is conceptually difficult
2 / 13

7. Introduction What are these models about? Reforming Future work
Where we are and want to be
Starting idea from the policy angle:
Plenty of countries have instruments aiming to foster voluntary pension savings
So far in the literature NOBODY analyzed if these instruments make sense
Starting idea from the academic angle:
Within cohort heterogeneity is rare, also because it is conceptually difficult
There are no models, where agents within cohort would differ on decision rules
2 / 13

8. Introduction What are these models about? Reforming Future work
Where we are and want to be
Starting idea from the policy angle:
Plenty of countries have instruments aiming to foster voluntary pension savings
So far in the literature NOBODY analyzed if these instruments make sense
Starting idea from the academic angle:
Within cohort heterogeneity is rare, also because it is conceptually difficult
There are no models, where agents within cohort would differ on decision rules
Combining the two together can make a valuable contribution (JPub?)
2 / 13

9. Introduction What are these models about? Reforming Future work
Where we are and want to be
Starting idea from the policy angle:
Plenty of countries have instruments aiming to foster voluntary pension savings
So far in the literature NOBODY analyzed if these instruments make sense
Starting idea from the academic angle:
Within cohort heterogeneity is rare, also because it is conceptually difficult
There are no models, where agents within cohort would differ on decision rules
Combining the two together can make a valuable contribution (JPub?)
This kind of question cannot be answered with tools other than OLG (low data
availability, difficulties to identify, etc.)
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10. Introduction What are these models about? Reforming Future work
OLG Model
Cohorts live for J periods
⇒ OLG is a GE model, so there are partial equilibrium and general equilibrium
effects (may decompose!)
⇒ We can see spillovers between cohorts (between-cohort redistribution)
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11. Introduction What are these models about? Reforming Future work
OLG Model
Cohorts live for J periods
Simultaneously J cohorts alive
⇒ OLG is a GE model, so there are partial equilibrium and general equilibrium
effects (may decompose!)
⇒ We can see spillovers between cohorts (between-cohort redistribution)
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12. Introduction What are these models about? Reforming Future work
OLG Model
Cohorts live for J periods
Simultaneously J cohorts alive
Consumers working and chose labor supply endogenously
⇒ OLG is a GE model, so there are partial equilibrium and general equilibrium
effects (may decompose!)
⇒ We can see spillovers between cohorts (between-cohort redistribution)
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13. Introduction What are these models about? Reforming Future work
OLG Model
Cohorts live for J periods
Simultaneously J cohorts alive
Consumers working and chose labor supply endogenously
During work years they earn wage income (which may be consumed or saved)
⇒ OLG is a GE model, so there are partial equilibrium and general equilibrium
effects (may decompose!)
⇒ We can see spillovers between cohorts (between-cohort redistribution)
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14. Introduction What are these models about? Reforming Future work
OLG Model
Cohorts live for J periods
Simultaneously J cohorts alive
Consumers working and chose labor supply endogenously
During work years they earn wage income (which may be consumed or saved)
During retirement they obtain pension benefits
⇒ OLG is a GE model, so there are partial equilibrium and general equilibrium
effects (may decompose!)
⇒ We can see spillovers between cohorts (between-cohort redistribution)
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15. Introduction What are these models about? Reforming Future work
OLG Model
Cohorts live for J periods
Simultaneously J cohorts alive
Consumers working and chose labor supply endogenously
During work years they earn wage income (which may be consumed or saved)
During retirement they obtain pension benefits
Voluntary savings used to smoothen consumption over lifetime, but preferences
matter (e.g. really impatient and lazy agents will borrow all working life to
repay from pension benefits)
⇒ OLG is a GE model, so there are partial equilibrium and general equilibrium
effects (may decompose!)
⇒ We can see spillovers between cohorts (between-cohort redistribution)
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16. Introduction What are these models about? Reforming Future work
Standard model: households
“Born” at age 20 (j = 1) and live up to 100 years (J = 80)
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17. Introduction What are these models about? Reforming Future work
Standard model: households
“Born” at age 20 (j = 1) and live up to 100 years (J = 80)
Subject to time and cohort dependent survival probability πj,t
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18. Introduction What are these models about? Reforming Future work
Standard model: households
“Born” at age 20 (j = 1) and live up to 100 years (J = 80)
Subject to time and cohort dependent survival probability πj,t
Choose labor supply lj endogenously
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19. Introduction What are these models about? Reforming Future work
Standard model: households
“Born” at age 20 (j = 1) and live up to 100 years (J = 80)
Subject to time and cohort dependent survival probability πj,t
Choose labor supply lj endogenously
Maximize remaining lifetime utility derived from consumption c and leisure 1 − l:
Uj,t =
J−j
s=0
δs πj+s,t+s
πj,t
cφ
j+s,t+s (1 − lj+s,t+s)1−φ
− 1
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20. Introduction What are these models about? Reforming Future work
Standard model: households
“Born” at age 20 (j = 1) and live up to 100 years (J = 80)
Subject to time and cohort dependent survival probability πj,t
Choose labor supply lj endogenously
Maximize remaining lifetime utility derived from consumption c and leisure 1 − l:
Uj,t =
J−j
s=0
δs πj+s,t+s
πj,t
cφ
j+s,t+s (1 − lj+s,t+s)1−φ
− 1
Subject to the budget constraint
(1 + τc
t )cj,t + sj,t = (1 − τl
t )(1 − τ)wtlj,t ← labor income
+ (1 + (1 − τk
t )rt)sj−1,t−1 ← capital income
+ (1 − τl
t )bj,t ← pension income
+ beqj,t ← bequests
− Υt ← lump-sum tax
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21. Introduction What are these models about? Reforming Future work
Standard model: households
“Born” at age 20 (j = 1) and live up to 100 years (J = 80)
Subject to time and cohort dependent survival probability πj,t
Choose labor supply lj endogenously
Maximize remaining lifetime utility derived from consumption c and leisure 1 − l:
Uj,t =
J−j
s=0
δs πj+s,t+s
πj,t
cφ
j+s,t+s (1 − lj+s,t+s)1−φ
− 1
Subject to the budget constraint
(1 + τc
t )cj,t + sj,t = (1 − τl
t )(1 − τ)wtlj,t ← labor income
+ (1 + (1 − τk
t )rt)sj−1,t−1 ← capital income
+ (1 − τl
t )bj,t ← pension income
+ beqj,t ← bequests
− Υt ← lump-sum tax
There exists a closed-form solution to this problem
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22. Introduction What are these models about? Reforming Future work
Extension: within cohort heterogeneity I
1 Each cohort consists of M subcohorts that can differ in preferences:
Uj,m,t =
J−j
s=0
βm δs
m
πj+s,t+s
πj,t
cφm
j+s,m,t+s (1 − lj+s,m,t+s)1−φm

23. Introduction What are these models about? Reforming Future work
Extension: within cohort heterogeneity II
3 Hands-to-mouth consumers (e.g. financially illiterate population which
consumes all available income): same utility function but different budget
constraint
(1 + τc
t )cj,m,t = (1 − τl
t )(1 − τ)wtlj,m,t ← labor income
+ (1 − τl
t )bj,m,t ← pension income
+ beqj,m,t ← bequests
− Υt ← lump-sum tax
4 Differences in financial literacy to form expectations about net present value of
the future stream of income
5 Differences in “pension system” literacy to form expectations about future
pension benefits (δb/δl)
6 Other possible ideas (related to Konrad’s work)

24. Introduction What are these models about? Reforming Future work
Producers
Perfectly competitive representative firm, with Cobb-Douglas (or other
production) function
Yt = Kα
t (γtLt)1−α
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25. Introduction What are these models about? Reforming Future work
Producers
Perfectly competitive representative firm, with Cobb-Douglas (or other
production) function
Yt = Kα
t (γtLt)1−α
Profit maximization
max
(Yt,Kt,Lt)
Yt − wtLt − (rt + d)Kt
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26. Introduction What are these models about? Reforming Future work
Producers
Perfectly competitive representative firm, with Cobb-Douglas (or other
production) function
Yt = Kα
t (γtLt)1−α
Profit maximization
max
(Yt,Kt,Lt)
Yt − wtLt − (rt + d)Kt
Results in
wt = γt(1 − α)ˆkα
t
rt = αˆkα−1
t − d
where d is the capital depreciation rate
and ˆk is capital per effective unit of labor
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27. Introduction What are these models about? Reforming Future work
Government
Spends a fixed share of GDP g on government consumption
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28. Introduction What are these models about? Reforming Future work
Government
Spends a fixed share of GDP g on government consumption
Collects taxes T
Tt =
J
j=1
M
m=1
Nj,m,t · τc
t · cj,m,t + τk
t · rt · sj−1,m,t−1
+τl
t ((1 − τ)wtlj,m,t + bj,m,t)
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29. Introduction What are these models about? Reforming Future work
Government
Spends a fixed share of GDP g on government consumption
Collects taxes T
Tt =
J
j=1
M
m=1
Nj,m,t · τc
t · cj,m,t + τk
t · rt · sj−1,m,t−1
+τl
t ((1 − τ)wtlj,m,t + bj,m,t)
Closes the gap between pension system contributions and benefits (subsidyt)
8 / 13

30. Introduction What are these models about? Reforming Future work
Government
Spends a fixed share of GDP g on government consumption
Collects taxes T
Tt =
J
j=1
M
m=1
Nj,m,t · τc
t · cj,m,t + τk
t · rt · sj−1,m,t−1
+τl
t ((1 − τ)wtlj,m,t + bj,m,t)
Closes the gap between pension system contributions and benefits (subsidyt)
Can take on debt D
Tt + Dt = (1 + rt)Dt−1 + gYt + subsidyt
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31. Introduction What are these models about? Reforming Future work
Pension system – we can have it any way we like, with multiple components
Pay As You Go Defined Benefit (PAYG DB)
b ¯J,k,t = ρ · gross wage ¯J−1,k,t−1
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32. Introduction What are these models about? Reforming Future work
Pension system – we can have it any way we like, with multiple components
Pay As You Go Defined Benefit (PAYG DB)
b ¯J,k,t = ρ · gross wage ¯J−1,k,t−1
Pay As You Go Defined Contribution (PAYG DC) pension system
b ¯J,m,t =
accumulated sum of contributions ¯J,m,t
expected remaining lifetime ¯J,t
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33. Introduction What are these models about? Reforming Future work
Pension system – we can have it any way we like, with multiple components
Pay As You Go Defined Benefit (PAYG DB)
b ¯J,k,t = ρ · gross wage ¯J−1,k,t−1
Pay As You Go Defined Contribution (PAYG DC) pension system
b ¯J,m,t =
accumulated sum of contributions ¯J,m,t
expected remaining lifetime ¯J,t
Voluntary pension saving schemes
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34. Introduction What are these models about? Reforming Future work
Reforming something
Initial solution
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35. Introduction What are these models about? Reforming Future work
Reforming something
Initial solution
Stable solution of model
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36. Introduction What are these models about? Reforming Future work
Reforming something
Initial solution
Stable solution of model
Calibrated to replicate what we know about the economy (e.g. depreciation and
discount rate to match investment rate in the economy)
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37. Introduction What are these models about? Reforming Future work
Reforming something
Initial solution
Stable solution of model
Calibrated to replicate what we know about the economy (e.g. depreciation and
discount rate to match investment rate in the economy)
Solution satisfies kt+1 = kt which implies stability of other variables – steady
state
10 / 13

38. Introduction What are these models about? Reforming Future work
Reforming something
Initial solution
Stable solution of model
Calibrated to replicate what we know about the economy (e.g. depreciation and
discount rate to match investment rate in the economy)
Solution satisfies kt+1 = kt which implies stability of other variables – steady
state
10 / 13

39. Introduction What are these models about? Reforming Future work
Reforming something
Initial solution
Stable solution of model
Calibrated to replicate what we know about the economy (e.g. depreciation and
discount rate to match investment rate in the economy)
Solution satisfies kt+1 = kt which implies stability of other variables – steady
state
Transition path
10 / 13

40. Introduction What are these models about? Reforming Future work
Reforming something
Initial solution
Stable solution of model
Calibrated to replicate what we know about the economy (e.g. depreciation and
discount rate to match investment rate in the economy)
Solution satisfies kt+1 = kt which implies stability of other variables – steady
state
Transition path
Path between two steady states: with and without reform
10 / 13

41. Introduction What are these models about? Reforming Future work
Reforming something
Initial solution
Stable solution of model
Calibrated to replicate what we know about the economy (e.g. depreciation and
discount rate to match investment rate in the economy)
Solution satisfies kt+1 = kt which implies stability of other variables – steady
state
Transition path
Path between two steady states: with and without reform
Reform introduced unexpectedly in the first period
10 / 13

42. Introduction What are these models about? Reforming Future work
Reforming something
Initial solution
Stable solution of model
Calibrated to replicate what we know about the economy (e.g. depreciation and
discount rate to match investment rate in the economy)
Solution satisfies kt+1 = kt which implies stability of other variables – steady
state
Transition path
Path between two steady states: with and without reform
Reform introduced unexpectedly in the first period
Let economy adjust ...
10 / 13

43. Introduction What are these models about? Reforming Future work
Reforming something
Initial solution
Stable solution of model
Calibrated to replicate what we know about the economy (e.g. depreciation and
discount rate to match investment rate in the economy)
Solution satisfies kt+1 = kt which implies stability of other variables – steady
state
Transition path
Path between two steady states: with and without reform
Reform introduced unexpectedly in the first period
Let economy adjust ...
... compare final steady states ...
10 / 13

44. Introduction What are these models about? Reforming Future work
Reforming something
Initial solution
Stable solution of model
Calibrated to replicate what we know about the economy (e.g. depreciation and
discount rate to match investment rate in the economy)
Solution satisfies kt+1 = kt which implies stability of other variables – steady
state
Transition path
Path between two steady states: with and without reform
Reform introduced unexpectedly in the first period
Let economy adjust ...
... compare final steady states ...
but also observe the process of adjustment
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45. Introduction What are these models about? Reforming Future work
Consumption equivalence
Compute transition paths: status quo and reform
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46. Introduction What are these models about? Reforming Future work
Consumption equivalence
Compute transition paths: status quo and reform
Calculate welfare for each cohort in each period for both paths
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47. Introduction What are these models about? Reforming Future work
Consumption equivalence
Compute transition paths: status quo and reform
Calculate welfare for each cohort in each period for both paths
Find for each cohort λj that satisfies
Wbase
j (cbase
j ) = Wreform
j ((1 − λj)creform
j )
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48. Introduction What are these models about? Reforming Future work
Consumption equivalence
Compute transition paths: status quo and reform
Calculate welfare for each cohort in each period for both paths
Find for each cohort λj that satisfies
Wbase
j (cbase
j ) = Wreform
j ((1 − λj)creform
j )
λj is a fraction of lifetime consumption that a given cohort is willing to give up
in order to introduce the reform
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49. Introduction What are these models about? Reforming Future work
What should we discuss
Which dimensions of heterogeneity we want to introduce (relationship to
bounded rationality promised in project)
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50. Introduction What are these models about? Reforming Future work
What should we discuss
Which dimensions of heterogeneity we want to introduce (relationship to
bounded rationality promised in project)
Which policy instruments to analyze (voluntary pension saving schemes
promised in project)
12 / 13

51. Introduction What are these models about? Reforming Future work
What should we discuss
Which dimensions of heterogeneity we want to introduce (relationship to
bounded rationality promised in project)
Which policy instruments to analyze (voluntary pension saving schemes
promised in project)
Possible instruments
12 / 13

52. Introduction What are these models about? Reforming Future work
What should we discuss
Which dimensions of heterogeneity we want to introduce (relationship to
bounded rationality promised in project)
Which policy instruments to analyze (voluntary pension saving schemes
promised in project)
Possible instruments
Tax exemption on private retirement savings (with and without limits)
12 / 13

53. Introduction What are these models about? Reforming Future work
What should we discuss
Which dimensions of heterogeneity we want to introduce (relationship to
bounded rationality promised in project)
Which policy instruments to analyze (voluntary pension saving schemes
promised in project)
Possible instruments
Tax exemption on private retirement savings (with and without limits)
Does not affect HtM consumers
12 / 13

54. Introduction What are these models about? Reforming Future work
What should we discuss
Which dimensions of heterogeneity we want to introduce (relationship to
bounded rationality promised in project)
Which policy instruments to analyze (voluntary pension saving schemes
promised in project)
Possible instruments
Tax exemption on private retirement savings (with and without limits)
Does not affect HtM consumers
Mandatory private savings
12 / 13

55. Introduction What are these models about? Reforming Future work
What should we discuss
Which dimensions of heterogeneity we want to introduce (relationship to
bounded rationality promised in project)
Which policy instruments to analyze (voluntary pension saving schemes
promised in project)
Possible instruments
Tax exemption on private retirement savings (with and without limits)
Does not affect HtM consumers
Mandatory private savings
???
12 / 13

56. Introduction What are these models about? Reforming Future work
Thank you for your attention
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