Three Essays in the Stochastic Overlapping Generations Models

The dissertation consists of three chapters studying theoretical to applied topics in macroeconomics using stochastic overlapping generations models.Chapter 1 studies the welfare effect of social security across heterogeneous households who differ by education levels, saving propensities and life cycle income in the U.S. I found that the U.S. social security can benefit the rich more than the poor, unlike the policy’s motivation when it was initiated after the Great Depression. The intuition is that thereis a channel, called a general equilibrium capital income effect, generating a significant welfare gain to the rich, which dominates a welfare loss from redistribution. Crowding out effect increases the interest rates and the value of the wealth of the rich rises more than the poor. There are two important contributions to this paper. First, I show the importance of household heterogeneities in the welfare analysis of social security. Without them at all, I found a social security system will reduce the overall welfare. With only income heterogeneity, social security benefits the poor the most. However, after incorporating rich heterogeneities in saving propensities, I found the policy can rather benefit the rich,unlike its purpose. In the calibration, I allow agents to live about 60-periods of lives which requires an efficient and accurate algorithm to compute an equilibrium in this large-scale life cycle model under aggregate shocks. I developed my own algorithm which has advantages in terms of accuracy and speed compared to the existing algorithm. This algorithm can be used in other applied topics such as inequality, asset pricing, and portfolio choices.Chapter 2 examines the consumption risk of households when they face different prices for identical goods. We found that household’s consumption volatility will increase under the price variation following the empirical observation that the poor pay more than the rich for the same goods because of quantity premium. The rationale is that if those receiving high-income shocks face lower effective prices, then the double-lucky house holds can purchase more goods than the economy with homogenous prices. Therefore, there is a larger consumption gap across households and we observe a rise in consumption volatility.This additional consumption risk generates a super-modular welfare loss. The result of this paper implies that the previous studies ignoring the price heterogeneity might under estimate the household consumption risk, and we need to consider the complementary consumption risk from the interaction of incomplete market and imperfect competition frictions.Chapter 3 analyzes the singular invariant Markov distribution in stochastic overlap-ping generations models. Interesting features of singular measures are: 1) they do not permit density functions and 2) the support of the distribution has self-affinity. In this chapter, we provide a sufficient condition for a singular measure to arise in a simple three-period-lived log-monetary model to a general stochastic OLG model where households live arbitrary periods, have general preferences and the Lucas-tree asset. We also both analytically and numerically characterize the set of economies satisfying our sufficient condition.The contribution of our paper is to extend the theoretical knowledge for the stochastic steady-state in stochastic OLG models beyond its existence, uniqueness, and stability. In addition, we provide an economic mechanism/model which generates a fractal pattern in its rational expectation equilibrium as observed in the time-series financial data on stock prices.