We begin with an endowment economy like that discussed in the video. • Xavier's utility function is 0.51 ln(C₁) + 0.49 In(C₂2). He begins with an endowment of 1000 pounds for consumption today and 1000 pounds for tomorrow (E1=E2=1000). People consume three to five pounds of food daily, so 1,000 per year is reasonable. Well, at least it is a nice round number. Xavier's demand: C₁ = 0.51*(1000*P₁+1000*P2) P1 C₂ 0.49* (1000*P₁+1000*P₂) P2 • Yuri's utility function is 0.52 In(C₁) + 0.48 In(C2). He starts with the identical endowment of 1000 pounds for consumption today and 1000 pounds for tomorrow (E1=E2=1000). • • 0.52*(1000*P₁+1000*P₂) 0.48*(1000 P₁+1000*P₂) Yuri's demand: C₁ = C2 = P1 P2 The first period market clearing equation is as follows. + P1 (0.52 * (1000 P1+1000*P2)) P1 = 1,000+ 1,000 = 2,000 (0.51* (1000 P1+1000*P₂)) Setting P₁=$1 then the market clearing condition becomes (0.51* (1000+ 1000*P2)) + (0.52 * (1000 + 1000*P2)) = 2000 510+510*P2 + 520 + 520*P2 = 2000 10301030 P2 = 2000 1030 P2 2000-1030-970 P2 = 970 1030 ≈ $0.9417 The real interest rate answers, "By how many percent more is a unit of food valued today than tomorrow?" r = real interest rate = (P1 - P2) P2 = P1 P2 - 1 ≈ 0.06186 or 6.19% 3. Consider the case of everyone being wealthier in the future, such as from a positive productivity shock (computers, internet, robotics, AI). A. Begin from the baseline preferences and endowments. Give both people an endowment of 1000 pounds for the first period and 1100 pounds for the second. AI increases the supply of second period goods by 10%. Note that there is now a total of 2000 pounds in the first period and 2200 pounds in the second. Determine the equilibrium interest rate. r = % B. Begin from the baseline preferences and endowments. Give both people an endowment of 1100 pounds for the first and 1100 pounds for the second periods. AI increases the supply in all periods by 10%. Note that there are now 2200 pounds in the first period and 2200 pounds in the second. Determine the equilibrium interest rate. r = % C. Explain how productivity and the real rate are connected. Write at least five sentences.