5. Suppose that, at an old allocation, (35) is violated, in particular: u11(x1)/u21(x1)>f12(z2)/f11(z1). From the viewpoint of the social planner, design a reallocation that Pareto improves the old allocation. 3.5 Efficiency Definition 5 (An Allocation) An allocation a=({xi}iI,{yk,zk}kK) is a vector describing how all consumers and producers behave. In particular, {xi}iI and {yk,zk}kK are called consumption allocation and production allocation, respectively. Definition 6 (A Feasible Allocation) An allocation a is feasible if demand equals supply for all goods and inputs at the allocation. Definition 7 (A Pareto Improvement) An allocation a Pareto improves another allocation a if: ui(xi)ui(xi)i,ui(xi)>ui(xi)i. Definition 8 (A Pareto Efficient Allocation) A feasible allocation a is Pareto efficient if there exists no other feasible allocation that Pareto improves the allocation. characterizing a Pareto efficient allocation (from the viewpoint of the central planner): amaxs.t.given::u1=u1(x1),ui(xi)uii=1,:fk(zk)=iIxki,k=1,,n,:iIzhi=k=1nzhk,h=1,,m,{ui}i=1. Lagrangian method (see section 7.2, Math Apps): L(x,,)u1(x1)+i=1i(ui(xi)ui)+k=1nk(fk(zk)iIxki)+h=1mh(iIzhik=1nzhk).uki(xi)/uli(xi)=k/l=fhl(zl)/fhk(zk)iIh=1,,mk,l=1,,n.fhk(zk)/fgk(zk)=h/gk=1,,nh,g=1,,m. 5. Suppose that, at an old allocation, (35) is violated, in particular: u11(x1)/u21(x1)>f12(z2)/f11(z1). From the viewpoint of the social planner, design a reallocation that Pareto improves the old allocation. 3.5 Efficiency Definition 5 (An Allocation) An allocation a=({xi}iI,{yk,zk}kK) is a vector describing how all consumers and producers behave. In particular, {xi}iI and {yk,zk}kK are called consumption allocation and production allocation, respectively. Definition 6 (A Feasible Allocation) An allocation a is feasible if demand equals supply for all goods and inputs at the allocation. Definition 7 (A Pareto Improvement) An allocation a Pareto improves another allocation a if: ui(xi)ui(xi)i,ui(xi)>ui(xi)i. Definition 8 (A Pareto Efficient Allocation) A feasible allocation a is Pareto efficient if there exists no other feasible allocation that Pareto improves the allocation. characterizing a Pareto efficient allocation (from the viewpoint of the central planner): amaxs.t.given::u1=u1(x1),ui(xi)uii=1,:fk(zk)=iIxki,k=1,,n,:iIzhi=k=1nzhk,h=1,,m,{ui}i=1. Lagrangian method (see section 7.2, Math Apps): L(x,,)u1(x1)+i=1i(ui(xi)ui)+k=1nk(fk(zk)iIxki)+h=1mh(iIzhik=1nzhk).uki(xi)/uli(xi)=k/l=fhl(zl)/fhk(zk)iIh=1,,mk,l=1,,n.fhk(zk)/fgk(zk)=h/gk=1,,nh,g=1,,m
