70207notes_week1 - Jan 23rd 2007 What is a model A model is...

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Jan 23rd, 2007 What is a model? A model is ’Toy World’: a simplified specification of the world, endowed with (i) an environment, (ii) agents and (iii) characteristics of the agents. Once a model is defined, we need to know what happens, i.e., an equilibrium. Generally, by equilibrium we mean that we want to know what happens with allocations: Are they (in some sense) ’optimal? Do they exist? What is an equilibrium? An equilibrium is a statement about what the outcome of an economy is. Tells us what happens in an economy, and by an ecomomy we mean a well defined environment in terms of primitives such as preferences and technology. Then an equilibrium is a particular mapping from the environment (preference, technology, information, market structure) to allocations where, 1. Agents maximize 2. Agents’ actions are compatible with each other. One of the important questions is, given the environment what type of equilibria we should look at. The economist doesn’t have the right to choose what happens, but is free to define the environment. For the theory to be able to predict precisely what is going to happen in a well defined environment, the outcome we define as the equilibrium needs to exist and must be unique . For this reason uniqueness is property that we want the equilibrium to have. We also know with certain assumptions that will be covered we can ensure the existence and uniqueness of an equilibrium outcome. 1 Growth Model The basic model we deal with in 702 is the neoclassical growth model. We will discuss the basic environment and then ask what happens in this ’toy world’: does an equilibrium exist? is it optimal? 1
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1.1 Technology Agents have 1 unit of labor and own capital which can be transformed in output. Production function: f : R 2 + R + such that c t + k t +1 = f ( k t , n t ) (1) We assume (i) Constant Returns to Scale (CRS, or homogeneous of degree one, meaning f ( λk, λn ) = λf ( k, n )), (ii) strictly increasing in both arguments, and ((iii) INADA condition, if necessary) 1.2 Preferences We assume infinitely-lived representative agent (RA). 1 We assume that preference of RA is (i) time-separable (with constant discount factor β < 1), (ii) strictly increasing in consumption (iii) strictly concave Our assumptions let us use the utility function of the following form: t =0 β t u ( c t ) (2) Initial capital stock k 0 is given. With these in hand the problem is, max { c t ,n t ,k t +1 } t =0 t =0 β t u ( c t ) (3) subject to 2 k t +1 + c t = f ( k t , n t ) (4) 1 For now, let’s treat the economy as if there were only one agent in the economy. We might interpret it as the result of normalization (so the number of population is 1) of the economy with FINITE number of identical (sharing the same technology, preference, and allocation) agents. If we proceed to the economy with mass of zero measure agents, things will be not so trivial because changing allocation of one agent does not change the aggregate amount of resources in the economy (since, by assumption, measure of an agent is zero), but let’s forget it for now.
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