Economic growth and economic development 257

Introduction to Modern Economic Growth for some β h ∈ (0, 1) These preferences imply that there are no externalities and preferences are time consistent We also assume that all markets are open and competitive Let an Arrow-Debreu equilibrium be given by (p∗ , x∗ ), where x∗ is the complete list of consumption vectors of each household h ∈ H, that is, x∗ = (x1,0 , xN,0 , , x1,T , xN,T ) , â ê with xi,t = xhi,t h∈H for each i and t) and p∗ is the vector of complete prices ¡ ¢ p∗ = p∗1,0 , , p∗N,0 , , p1,T , , pN,T , with one of the prices, say p∗1,0 , chosen as the numeraire, i.e., p∗1,0 = In the Arrow-Debreu equilibrium, each individual purchases and sells claims on each of the commodities, thus engages in trading only at t = and chooses an allocation that satisfies the budget constraint N T X X p∗i,t xhi,t t=0 i=1 Market clearing then requires N XX h∈H i=1 xhi,t ≤ N XX ≤ N T X X t=0 i=1 p∗i,t ω hi,t for each h ∈ H ω hi,t for each i = 1, , N and t = 0, 1, , T h∈H i=1 In the equilibrium with sequential trading, markets for goods dated t open at time t Instead, there are T bonds–Arrow securities–that are in zero net supply and can be traded among the households at time t = The bond indexed by t pays one unit of one of the goods, say good i = at time t Let the prices of bonds be denoted by (q1 , , qT ), again expressed in units of good i = (at time t = 0) This implies that a household can purchase a unit of bond t at time by paying qt units of good and then will receive one unit of good at time t (or conversely can sell short one unit of such a bond) The purchase of bond t by household h is denoted by bht ∈ R, and since each bond is in zero net supply, market clearing requires that X bht = for each t = 0, 1, , T h∈H Notice that in this specification we have assumed the presence of only T bonds (Arrow securities) More generally, we could have allowed additional bonds, for example bonds traded at time t > for delivery of good at time t0 > t This restriction to only T bonds is without loss of any generality (see Exercise 5.10) 243