Myopia and pensions in general equilibrium

Abstract

The US social security tax rate has doubled in the last half century. Does the degree of myopic behavior that we observe in the US justify the size of the social security program? To study this question we build a computable general equilibrium model that is composed of life-cycle permanent-income consumers who save optimally and “hand-to-mouth” consumers who just consume their disposable income. Our model is a continuous-time, general equilibrium extension of the model by Cremer et al. (Int Tax Public Financ 15(5):547–562, 2008), though we abstract from the redistributive function of social security to focus on myopia. Retirement is a choice variable in our model and the social security program is designed to mimic the US program in which the annuity value of benefits increases with the retirement age. Also, we allow for delayed claiming beyond the date of retirement. The model matches a variety of important data targets relating to saving and retirement. We find that small reductions in the social security tax rate provide significant welfare gains to both groups of consumers.

Appendices

Appendix A: More on synchronization

Here we take a more direct approach to illustrating the robustness of our analysis to the synchronization constraint by relaxing it directly. Now the consumer’s problem consists of choosing the optimal control variable c(t) and the optimal control parameters T and T ₀

$$ \underset{\{c(t)\},T,T_{0}}{\max }\int_{0}^{T}\exp \left[ -\rho t\right] [\sigma \ln c(t)+(1-\sigma )\ln l_{-}]dt+\int_{T}^{\bar{T}}\exp \left[ -\rho t\right] \sigma \ln c(t)dt, $$

(28)

subject to

$$ \frac{dk(t)}{dt}=rk(t)+(1-\theta )(1-l_{-})e(t)w-c(t)\text{, for }t\in \lbrack 0,T], $$

(29)

$$ \frac{dk(t)}{dt}=rk(t)-c(t)\text{, for }t\in \lbrack T,T_{0}], $$

(30)

$$ \frac{dk(t)}{dt}=rk(t)+b(T_{0})-c(t)\text{, for }t\in \lbrack T_{0},\bar{T}], $$

(31)

$$ k(0)=k(\bar{T})=0, $$

(32)

$$ b(T_{0})=z_{0}+\frac{z_{1}}{1+\exp \left[ z_{2}-z_{3}\times T_{0}\right] }, \text{ ~}z_{0},z_{1},z_{2},z_{3}\in \mathbb{R} ^{+}, $$

(33)

$$ T_{0}\geq T. $$

(34)

This is a three-stage optimal control problem with two endogenous switch points T and T ₀ (we could have split the objective functional into three integrals, as we might expect from a three-stage problem, but the integrand is the same in the second and third phases so we can combine). We can decompose the control problem into an inner control subproblem of choosing c(t) optimally, conditional on T and T ₀, and then an outer problem of jointly choosing T and T ₀ given the optimized value of c(t).

Thus, for a fixed retirement date T and a fixed initiation date T ₀, the solution to the control subproblem is

$$ c_{T,T_{0}}(t)=X\exp \left[ (r-\rho )t\right] , $$

(35)

where

$$ X\equiv \frac{\displaystyle \int_{0}^{T}(1-\theta )(1-l_{-})we_{\max }\exp \left[ -\left( \mu +r\right) s\right] ds+\displaystyle \int_{T_{0}}^{\bar{T}}b(T_{0})\exp \left[ -rs \right] ds}{\displaystyle \int_{0}^{\bar{T}}\exp \left[ -\rho s\right] ds}. $$

(36)

Then, the optimal retirement and initiation dates are chosen jointly according to (substitute Eq. 35 into the objective functional in Eq. 28 and drop terms unrelated to both T and T ₀)

$$ \underset{T,T_{0}}{\max }\left[ \!\displaystyle \int_{0}^{T}\!\exp \left[ -\rho t\right] (1-\sigma )\ln l_{-}~dt+\sigma \displaystyle \int_{0}^{\bar{T}}\!\exp \left[ -\rho t\right] \ln Xdt\right] \!\text{, subject to }T_{0}\geq T. $$

(37)

Note the maximand in Eq. 37 is the value functional from the control subproblem (with terms unrelated to T and T ₀ omitted).

Appendix B: Computational procedure for the stationary competitive equilibrium

After assigning values to all parameters, the following steps are used to find an equilibrium.

• Step 1 Guess values for the aggregate capital stock K and retirement dates T _P and T _H and call these three guesses the vector v _guess.

• Step 2 Based on v _guess, compute aggregate labor L, aggregate output Y, factor prices r and w, and the social security scale parameter z ₀.

• Step 3 Using r, w, and z ₀ compute the LCPI choice of consumption, saving, and retirement and the hand-to-mouth choice of retirement.

• Step 4 Aggregate the LCPI saving profile to find K and put this value, together with the retirement choices from the previous step in a vector called v _feedback.

• Step 5 Let v ≡ v _guess − v _feedback. Iterate on v _guess until v·v < 0.01.