Extend the Baily model :
incorporating savings
decision
Sang Hyun Park
• How does this affect the Baily formula?
• How do you expect this to affect the consumption smoothing
benefits?
Literature Review
• Baily (1978) and Chetty (2006) suggested a stylized model of unemployment insurance and
optimal solution.
• Gruber (1997) examined the consumption smoothing benefits of unemployment insurance.
• Feldstein and Altman (1998) examined a system of Unemployment Insurance Saving Accounts
(UISAs).
• Card, Chetty and Weber (2007) presented new tests of the permanent income hypothesis and
other widely used models of household behavior using data from the labor market.
• Shimer and Werning (2008) explained that the optimal unemployment policy can be
implemented with a simple policy that keeps unemployment benefits and taxes constant and
gives the employed access to savings.
• Spinnewijn (2014) used a stylized model to discuss the biased belief of individuals on
employment prospect.
• And many others
• Recap: Baily-Chetty Result
• Now, incorporate saving decision?
• Incorporating saving decision
• Max 𝜋𝜋 𝑒𝑒 𝑏𝑏
𝑢𝑢 𝑤𝑤 − 𝜏𝜏(𝑏𝑏) − 𝑠𝑠 + [1-𝜋𝜋 𝑒𝑒 𝑏𝑏 ] 𝑢𝑢 𝑏𝑏 + 𝑠𝑠 1 + 𝑟𝑟
• Government budget constraint remains the same
• IC: 𝜋𝜋’ 𝑒𝑒 𝑢𝑢 𝑤𝑤 − 𝜏𝜏(𝑏𝑏) − 𝑠𝑠 − 𝜋𝜋’ 𝑒𝑒 𝑢𝑢 𝑏𝑏 + 𝑠𝑠 1 + 𝑟𝑟 − 1 = 0
− 𝑒𝑒
• Is S independent of b?
• We can pose that it depends on b, that is, 𝑠𝑠(𝑏𝑏) however it
seems like it does not change the implication of Baily-Chetty
formula. (See this in later slide)
• First order condition -RTC
•
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
= 0 = 1−𝜋𝜋 𝑒𝑒 𝑏𝑏
Note that
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
− 𝜋𝜋 𝑒𝑒 𝑏𝑏
+𝜋𝜋 ′ 𝑒𝑒 𝑏𝑏
𝑢𝑢′ 𝑏𝑏 + 𝑠𝑠 1 + 𝑟𝑟
𝑢𝑢′ 𝑤𝑤 − 𝜏𝜏(𝑏𝑏) − 𝑠𝑠
−𝑢𝑢
𝑏𝑏 + 𝑠𝑠 1 + 𝑟𝑟
𝑑𝑑𝜏𝜏/𝑑𝑑𝑑𝑑
+ 𝑢𝑢
𝑤𝑤 − 𝜏𝜏 𝑏𝑏 − 𝑠𝑠 − 1
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
is 0 (already optimized) hence the third term equals 0.
Rearrange and plugging in the IC from previous page gives the
same equation (MB=MC).
The only difference is
Cu = 𝑏𝑏 + 𝑠𝑠 1 + 𝑟𝑟
Ce = 𝑤𝑤 − 𝜏𝜏(𝑏𝑏) − 𝑠𝑠
• We can clearly see that allowing individuals to make saving
decision promotes the consumption smoothing effect.
(Cu-Ce)/Cu ={ 𝑤𝑤 − 𝜏𝜏 𝑏𝑏 − 𝑠𝑠 − 𝑏𝑏 + 𝑠𝑠 1 + 𝑟𝑟 }/ 𝑤𝑤 − 𝜏𝜏 𝑏𝑏 − 𝑠𝑠
This is smaller than the value without s and s(1+r).
Also note that if b goes up, s needs to go down to satisfy the
equation. Therefore, we find that UI crowds out self-insurance.
When using S(b) instead of S,
we get the FOC (using envelope theorem) (tedious)
•
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
= 0 = 1−𝜋𝜋 𝑒𝑒 𝑏𝑏
− 𝜋𝜋 𝑒𝑒 𝑏𝑏
𝑢𝑢′
𝑢𝑢′
𝑏𝑏 + 𝑠𝑠(𝑏𝑏) 1 + 𝑟𝑟 [
𝑤𝑤 − 𝜏𝜏(𝑏𝑏) − 𝑠𝑠(𝑏𝑏)
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
𝑑𝑑𝜏𝜏
𝑑𝑑𝑑𝑑
1 + 𝑟𝑟 + 1]
+
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
Rearrange, we get the Baily-Chetty result (the same).