1 Two Period Exchange Economy

Transcription

1 University of British Columbia Department of Economics, Macroeconomics (Econ 502) Prof. Amartya Lahiri Handout # 2 1 Two Period Exchange Economy We shall start our exploration of dynamic economies with its simplest form the two period model. In particular, we will specialize the two period model even more by analyzing the two period exchange economy model. Consider a closed economy inhabited by a single agent who lives two periods. The single agent is an abstraction (as are all models and most features of this one for that matter!). You could think of this economy instead as being inhabited by a continuum of identical agents of measure one. As we shall show later, it is easy to aggregate the decisions and allocations of a representative agent from this continuum. Let the agent s preferences be given by V = u (c 1 ) + βu (c 2 ) where the utility function u is continuous and strictly concave with u > 0, u < 0. c 1 denotes consumption now and c 2 is consumption tomorrow. β < 1 denotes the discount factor which captures how much the agent values today relative to tomorrow. Suppose the agent receives a non-storable good as an endowment in each period y 1 now and y 2 tomorrow. We shall assume that agents operate in competitive markets for goods. Let us use and to denote the (utility based) market prices of these goods. Hence, the agent s budget constraint is c 1 + c 2 y 1 + y 2 i.e., the total value of the agent s consumption bundle cannot exceed the value of her total resources (endowments in this case). 1

2 Before proceeding we will first describe our notion of equilibrium. Since agents are operating in competitive markets we define the competitive equilibrium for this economy as a set of allocations and prices {c 1, c 2 ;, } such that: (a) the allocations are consistent with agents maximizing their utility given their budget constraints taking these prices as given; and (b) at these prices all markets clear. In this economy the total endowment of goods available for consumption in periods 1 and 2 are y 1 and y 2. Market clearing implies that total demand for a good must equal its supply. Hence, the second equilibrium condition implies that at the equilibrium prices we must have c 1 = y 1 and c 2 = y 2. The first condition for a competitive equilibrium requires that agents maximize their utility subject to their budget constraint taking prices as given. This maximization problem can be set up as a simple Lagrangean: L = u (c 1 ) + βu (c 2 ) + λ ( y 1 + y 2 c 1 c 2 ) where λ is the Lagrange mulitplier on the agent s budget constraint. Maximizing this with respect to c 1 and c 2 taking and as given gives the standard necessary conditions for utility maximization u (c 1 ) = λ βu (c 2 ) = λ Combining the first two conditions gives c 1 + c 2 = y 1 + y 2 βu (c 2 ) u (c 1 ) =. This should be a familiar condition from undergraduate microeconomics. The left hand side gives the marginal rate of substitution between c 2 and c 1 while the right hand side is the relative price of c 2 in terms of c 1. At an optimum the agent should be indifferent between trading off an additional unit of c 1 for c 2 This will happen if the additional relative utility gain from the trade-off exactly equals the additional relative cost of making that trade. Graphically, this is the point of tangency of the indifference curve with the budget line of 2

3 the agent. The third optimality condition ensures that the relevant tangency point is the one with the highest possible indifference curve. (You should try drawing this to make sure that you remember your undergraduate micro). Now recall the second condition for competitive equilibrium: markets must clear. Imposing c 1 = y 1 and c 2 = y 2 we then have βu (y 2 ) u (y 1 ) =. (1) This expression provides a solution for the equilibrium price ratio in terms of the primitives of the economy the endowments and the discount factor. At this price ratio, the allocation c 1 = y 1 and c 2 = y 2 is consistent with both optimality and market clearing. It is important to note that we can only solve for the price ratio not the price levels themselves. Any multiple of the equilibrium prices will also be an equilibrium price ratio at this allocation. With no loss of generality therefor, we can choose the normalization = 1. Hence, / = is now the price of the good tomorrow in terms of the good today. Combining all these, the competitive equilibrium for this economy then is the set of { } allocations and prices y 1, y 2 ; 1, βu (y 2 ) u (y 1. ) A couple of points about this equilibrium are worth noting. First, suppose y 1 goes up. Clearly, concavity of u and equation (1) jointly imply that / must rise. Why? Starting from an equilibrium, a rise in y 1 implies an excess supply of the good now relative to the good tomorrow. For markets to clear therefore the price of the good today must fall relative to the good tomorrow, i.e., / rises. Second, following a similar logic, notice that the more impatient the agent (the lower the β) the lower is the equilibrium price ratio /. Naturally, the less the agent values the future good the lower must be the market price for it in order to induce the agent to consume the available supply of the future good. While this may seem obvious, it will prove useful to keep in mind when we start talking explicitly about interest rates. As you would recall from your undergraduate days, the two fundamental welfare theorems, which apply to economies such as ours (with no distortions, no externalities, perfect information, competitive markets etc), say that (a) competitive allocations are Pareto opti- 3

4 mal (no one can be made better off without making someone worse off); and (b) all Pareto optimal allocations can be decentralized as competitive equilibria. To see the first welfare theorem here, note that a planning optimum maximizes utility subject to the resource constraints c 1 y 1 and c 2 y 2. Clearly, the best a planner could do given the resource constraints is to give the entire endowment in each period to the agent to consume, i.e., set c 1 = y 1 and c 2 = y 2. Hence, our competitive allocation is Pareto optimal. Moreover the prices implicit in this allocation can be derived by computing the marginal rate of substitution at the planner s chosen allocation. This is just βu (y 2 ) u (y 1 ) = as before. Notice that we have thus far written this two period model in a very similar way to a standard static model economy except for calling good 1 and good 2 goods today and goods tomorrow. We could make the dynamic nature of the economy more explicit by writing the agent s problem as one involving consuming and saving at each date. Thus, letting s denote saving and r denote the rate of interest, we could write the agent s budget constraint in each period of life as c 1 + s = y 1 c 2 = y 2 + (1 + r) s One can combine these two equations by solving out for s to get c 1 + c r = y 1 + y r. Note that this budget constraint looks very similar to the one we had above. In fact if we set 1 + r = then they would be exactly the same. In fact they are. This way of writing it makes clear that the interest rate factor 1 + r is just the relative price of the good today in terms of goods tomorrow, i.e., it gives the number of units of goods tomorrow that have to be offered in order to get one unit of the good today. That is precisely the interest rate: if you borrow an apple from me today how many apples would I have to be promised in exchange tomorrow? It is also useful to use this structure to flesh out the relationship between real and nominal interest rates. Notice that we have written everything in terms of real things so far, i.e., in 4

5 terms of goods. This includes prices as well. Noting that is just the utility cost of goods today while is the utility cost of goods tomorrow, it is easy to see that = 1 + r = Goods tomorrow Goods today This is the real interest interest rate in words. A related notion is the nominal interest rate which describes how many dollars you will get tomorrow in exchange for giving up a dollar today. Instead of describing the payoff in terms of goods, the nominal interest rate describes the payoff from saving in terms of money. Clearly, 1 + R = Dollars tomorrow Dollars today If we use P to denote goods prices then P 2 P 1 = Dollars per good tomorrow Dollars per good today = 1 + π where π denotes the rate of inflation. It is easy then to check that P 2 P 1 = Dollars tomorrow Dollars today Hence, 1 + R = (1 + π) (1 + r) Under moderate inflation and interest rates, the above relation can often be reasonably approximated by R r + π. 5