This thesis consists of three chapters studying dynamic economies in general equilibrium. The first chapter considers an economy in business cycles with potentially imperfect financial markets. The second chapter investigates an economy in its balanced growth path with heterogeneous firms. The third chapter analyzes dynamic competitions that these firms are potentially engaged in. The first chapter, "Asset Price and Real Investment Volatility with Heterogeneous Beliefs," sheds light on the role of imperfect financial markets on the economic and financial crisis 2007-2008. This crisis highlights the role of financial markets in allowing economic agents, including prominent banks, to speculate on the future returns of different financial assets, such as mortgage-backed securities. I introduce a dynamic general equilibrium model with aggregate shocks, potentially incomplete markets and heterogeneous agents to investigate this role of financial markets. In addition to their risk aversion and endowments, agents differ in their beliefs about the future aggregate states of the economy. The difference in beliefs induces them to take large bets under frictionless complete financial markets, which enable agents to leverage their future wealth. Consequently, as hypothesized by Friedman (1953), under complete markets, agents with incorrect beliefs will eventually be driven out of the markets. In this case, they also have no influence on asset prices and real investment in the long run. In contrast, I show that under incomplete markets generated by collateral constraints, agents with heterogeneous (potentially incorrect) beliefs survive in the long run and their speculative activities drive up asset price volatility and real investment volatility permanently. I also show that collateral constraints are always binding even if the supply of collateralizable assets endogenously responds to their price. I use this framework to study the effects of different types of regulations and the distribution of endowments on leverage, asset price volatility and investment. Lastly, the analytical tools developed in this framework enable me to prove the existence of the recursive equilibrium in Krusell and Smith (1998) with a finite number of types. This has been an open question in the literature. The second chapter, "Innovation from Incumbents and Entrants," is a joint work with Daron Acemoglu. We propose a simple modification of the basic Schumpeterian endogenous growth models, by allowing incumbents to undertake innovations to improve their products. This model provides a tractable framework for a simultaneous analysis of entry of new firms and the expansion of existing firms, as well as the decomposition of productivity growth between continuing establishments and new entrants. One lesson we learn from this analysis is that, unlike in the basic Schumpeterian models, taxes or entry barriers on potential entrants might increase economic growth. It is the outcome of the greater productivity improvements by incumbents in response to reduced entry, which outweighs the negative effect of the reduction in creative destruction. As the model features entry of new firms and expansion and exit of existing firms, it also generates an equilibrium firm size distribution. We show that the stationary firm size distribution is Pareto with an exponent approximately equal to one (the so-called "Zipf distribution"). The third chapter, "Racing: when should we handicap the advantaged competitor?" studies dynamic competitions, for example R&D competitions used in the second chapters. Two competitors with different abilities engage in a winner-take-all race; should we handicap the advantaged competitor in order to reduce the expected completion time of the race? I show that if the discouragement effect is strong, i.e., both competitors are discouraged from exerting effort when it becomes more certain who will win the race, we should handicap the advantaged. We can handicap him either by reducing his ability or by offering him a lower reward if he wins. Doing so induces higher effort not only from the disadvantaged competitor because of his higher incentive from a higher chance of winning the race but also from the advantaged competitor because of their strategic interactions. Therefore, the expected completion time is strictly shortened. To prove the existence and uniqueness of the equilibria (including symmetric and asymmetric equilibria) that leads to the conclusion, I use a boundary value problem formulation which is novel to the dynamic competition literature. In some cases, I obtain closed-form solutions of the equilibria.
