Economic Simulations Using Mathematica

1. Economic Simulations Using Mathematica Kota Minegishi

2. Outline • Objectives • Notional Demand Driven Economies • Effective Demand Driven Economies • Conclusions

3. 1. Objectives Q. Why economic simulations? A. Economic simulations allow us to • Understand existing theories better • Change some assumptions in theories • Light existing theories from different angles • Improve our intuitions on economic theories

4. Objectives Our Targets • Setup and compare models for: Notional Demand Driven Economies • The Walrasian Auctioneer Effective Demand Driven Economies • Triangular Trade • To Show Simulations in Mathematica • Iterations • Modified assumptions in theories • Graphical interpretations

5. 2. Notional Demand Driven Economies P1, P2, P3 Auctioneer Excess Demand  P Excess Supply  P S1 Dn2 S2 Dn3 S3 Dn1

6. 2. Notional Demand Driven Economies P1, P2, P3 Auctioneer No Excess Demand or Supply Then, Traders FINALLY trade. S1 Dn2 S2 Dn3 S3 Dn1

7. $ $ $ $ $ $ 2. Notional Demand Driven Economies Final P1, P2, P3 For time = t Auctioneer S1 Dn2 S2 Dn3 S3 Dn1

8. 2. Notional Demand Driven Economies Ideas For Implementation • Define traders’ supply functions • Define traders’ utility functions and budget constraints  derive demand functions • Solve di = si for i = 1, 2, 3 simultaneously for {p1, p2, p3} • With these price equations, define equations for quantities, money holding, and GDP over time.

9. Commodity Higher Utility Level c3(t) Budget Constraint m2(t+1) Money Holding 3D Utility Maximizing Behavior 2D From [1], [2], & [3], obtain local extrema (x, y) and Lagrange multiplier λ

10. Utility Maximizers (Trader 1, 2, & 3) Consider Trader 2;

11. Trader 2;

12. 2. Notional Demand Driven Economies Definitions A1; si[t] = di[t] m1[t] = m1[t - 1] + p1[t] s1[t] - p2[t] d2[t] m2[t] = m2[t - 1] + p2[t] s2[t] - p3[t] d3[t] m3[t] = m3[t - 1] + p3[t] s3[t] - p1[t] d1[t] d1[t] = β2(m3[t] + p3[t] s3[t]) / p1[t] d3[t] = β1 (m2[t] + p2[t] s2[t]) / p3[t] d2[t] = β3 (m1[t]+ p1 [t] s1[t]) / p2[t] s1[t] = γ1 p1[t] s2[t] = γ2 p2[t] s3[t] = γ3 p3[t]

13. 2. Notional Demand Driven Economies • Solving di = si for i = 1, 2, 3, we obtain; • So, the auctioneer can “solve” market equations for the prices for which all excess demands are zero.

14. 2. Notional Demand Driven Economy q3 GDP q2 Real GDP q1 GDP Quantities Traded P1 m1 m2 P2 P3 m3 Prices Money Holdings

15. “Path” of Money Holding Vectors over time

16. 2. Notional Demand Driven Economies • As time [t] elapses, the economy will find the general equilibrium * under well known conditions such as; • the weak axiom of revealed preferences • gross substitutions • a dominant diagonal • At the general equilibrium, all variables stop changing over time [t]. *Roberts and Schultz, Modern Mathematical and Economic Analysis, pp304.

17. 2. Notional Demand Driven Economies Finding The General Equilibrium • set the changes in money holdings = 0 i.e. m1[t] - m1[t - 1] = p1[t] s1[t] - p2[t] d2[t] = 0 • Since si[t] = di[t], we have p1[t] s1[t] = p2[t] s2[t] = p3[t] s3[t] • Solving them gives; where M = m1 + m2 + m3

18. 2. Notional Demand Driven Economies So, for the set of constants where {β1,β2,β3}={.5,.5,.6} {γ1,γ2,γ3}={2,7,10} we have the set of equilibrium values {m1[0], m2[0], m3[0]} = {191.25, 191.25, 127.5}; {p1[0], p2[0], p3[0]} = {9.7788, 5.22699, 4.37321}; {q1[0], q2[0], q3[0]} = {19.5576, 36.5889, 43.7321}; we will use them as initial conditions. Then we will give economies some shocks for different models.

19. Vector field of {m1’[t], m2’[t], m3’[t] } {β1, β2, β3}= {.5, .5, .6} The long run equilibrium

20. Vector field of {m1’[t], m2’[t], m3’[t] } {β1, β2, β3}= {.5, .6, .6} The long run equilibrium

21. Vector field of {m1’[t], m2’[t], m3’[t] } {β1, β2, β3}= {.5, .6, .6} The long run equilibrium

22. 2. Notional Demand Driven Economies Q. Why do prices adjust even when demands are notional? • There is the auctioneer in this economy. Agents trade with the auctioneer.

23. 3. Effective Demand Driven Economies • Notional Demands • Budget Constraints • Effective Demands • Budget Constraints and Other Constraints • e.g. If a trader could not sell, then he cannot buy as much as he wanted.

24. Trader 1 $ $ $ Trader 2 Trader 3 Triangular Trade

25. 3. Effective Demand Driven Economies Ideas For Implementation • Have Trader 1 be an initiator of trades and Trader 2 and Trader 2 be utility maximizers • Create variables for actual traded quantities ( ai= min[ di, si ] ) so that traders will adjusting budget constrains according to them

26. m1(t) - p2 a2 + p1 a1 = m1(t+1) m1(t) Trader 1 m1(t) - p2 a2 m2(t) + p2 a2 m2(t) Trader 2 m2(t) + p2 a2 – p3 a3 = m2(t+1) m3(t) Trader 3 m3(t) + p3 a3 – p1 a1 = m3(t+1) … Controlled Demand for C2 … Utility Maximizing Between {Commodity, Money Holding} 3. Effective Demand Driven Economies Time = t +1 Time = t m3(t) + p3 a3

27. 3. Effective Demand Driven Economies Definitions B1; ai[t] actual traded q’s m1[t] = m1[t - 1] + p1[t-1] a1[t - 1] - p2[t-1] a2[t - 1] m2[t] = m2[t - 1] + p2[t-1] a2[t - 1] - p3[t-1] a3[t - 1] m3[t] = m3[t - 1] + p3[t-1] a3[t - 1] - p1[t-1] a1[t - 1] d1[t] = β2(m3[t] + p3[t] a3[t]) / p1[t] d3[t] = β1 (m2[t] + p2[t] a2[t]) / p3[t] d2[t] = β3 (m1[t]+ p1[t] s1[t]) /p2[t] s1[t] = γ1 p1[t] s2[t] = γ2 p2[t] s3[t] = γ3 p3[t] a1[t]=min[s1[t], d1[t] a2[t]=min[s2[t], d2[t]] a3[t]=min[s3[t], d3[t]]]

28. 3. Effective Demand Driven Economies Definitions B2; price adjustments z1[t] = d1[t] - s1[t] z2[t] = d2[t] - s2[t] z3[t] = d3[t] - s3[t] p1[t] = p1[t - 1] + k1*z1[t - 1] p2[t] = p2[t - 1] + k2*z2[t - 1] p3[t] = p3[t - 1] + k3*z3[t - 1]

29. GDP a3 Real GDP a2 a1 GDP Actual Quantities Traded P1 m1 m3 P2 P3 m2 Prices Money Holdings Effective Demand Driven Economy

30. Recalling… Notional Demand Driven Economy q3 GDP q2 Real GDP q1 GDP Quantities Traded P1 m1 m2 P2 P3 m3 Prices Money Holdings

31. q1 a1 a2 q2 a3 q1 Quantity Traded Over Time Effective D. Nominal D.

32. P1 Excess Demands = 0 for every commodity for every time = t P1 Excess Demands P2 P2 P3 P3 Prices Over Time Effective D. Nominal D.

33. 3. Effective Demand Driven Economies Excess Demand Traded Amount

34. 3. Effective Demand Driven Economie Price Vector Money Holding

35. Comparison of GDP[t] Paths over time Notional. D Effective. D 2: 0.5 0.6 Trader 2 prefers to buy more and hold less money

36. Comparison of GDP[t] Paths over time Notional. D Half-Notional. D Effective. D 2: 0.5 0.6 Trader 2 prefers to buy more and hold less money

37. Comparison of GDP[t] Paths over time Notional. D Half-Notional. D Effective. D Effective. D. Supplies Fixed 2: 0.5 0.6 Trader 2 prefers to buy more and hold less money

38. Comparison of GDP[t] Paths over time Notional. D Trader 1 expects his sales *P,S-fixed Trader 1 buys a fixed amount 2: 0.5 0.6 Trader 2 prefers to buy more and hold less money

39. Comparison of GDP[t] Paths over time Notional. D Effective. D Initial conditions: For the first two periods, Trader 2 decided to buy less.

40. 4. Conclusions We have shown; The difference b/w Notional and Effective demands • the Walrasian Auctioneer • Triangular Trade Economic simulations • Improve Our Understanding of the Neoclassical theory • Have modified assumptions • Light the theory from different angles • Improve our intuitions on economic theories Economic Simulations using Mathematica • iterations • modified assumptions • graphical interpretations

41. Any Questions?