Economic Simulations Using Mathematica

Kota MinegishiKota Minegishi

Outline

1. Objectives 2. Notional Demand Driven

Economies 3. Effective Demand Driven

Economies 4. Conclusions

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

1. 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

2. Notional Demand Driven Economies

P1, P2, P3

S1 S2 S3Dn2 Dn1Dn3

Auctioneer

Excess Demand PExcess Supply P

2. Notional Demand Driven Economies

P1, P2, P3

S1 S2 S3Dn2 Dn1Dn3

Auctioneer

No Excess Demand or SupplyThen, Traders FINALLY trade.

2. Notional Demand Driven Economies

Final P1, P2, P3For time = t

S1 S2 S3Dn2 Dn1Dn3

Auctioneer

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.

3D 2D

From [1], [2], & [3], obtain local extrema (x, y) and Lagrange multiplier λ

Utility Maximizing Behavior

Utility Maximizers (Trader 1, 2, &

3) Consider Trader 2;

Trader 2;

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]

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.

2. Notional Demand Driven EconomyGDP

Real GDP

q3

q2

q1

P3

P2

P1

m3

m2

m1

GDP Quantities Traded

Prices Money Holdings

“Path” of Money Holding Vectors over time

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.

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

2. Notional Demand Driven Economies

So, for the set of constants where{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};

we will use them as initial conditions. Then we will give economies some shocks for different models.

{p1[0], p2[0], p3[0]} = {9.7788, 5.22699, 4.37321};

{q1[0], q2[0], q3[0]} = {19.5576, 36.5889, 43.7321};

{β1,β2,β3}={.5,.5,.6}

Vector field of {m1’[t], m2’[t], m3’[t] }

{β1, β2, β3}= {.5, .5, .6}

The long run equilibrium

{β1, β2, β3}= {.5, .6, .6}

The long run equilibrium

Vector field of {m1’[t], m2’[t], m3’[t] }

{β1, β2, β3}= {.5, .6, .6}

The long run equilibrium

Vector field of {m1’[t], m2’[t], m3’[t] }

2. Notional Demand Driven Economies

Q. Why do prices adjust even when demands are

notional?

A. There is the auctioneer in this economy.Agents trade with the auctioneer.

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.

Triangular Trade

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

3. Effective Demand Driven Economies

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]]]

3. Effective Demand Driven Economies

Definitions B2; price adjustmentsz1[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]

Effective Demand Driven Economy

GDP

Real GDP

a3

a2

a1

P3

P2

P1

m3

m2

m1

GDP Actual Quantities Traded

Prices Money Holdings

Notional Demand Driven EconomyGDP

Real GDP

q3

q2

q1

P3

P2

P1

m3

m2

m1

GDP Quantities Traded

Prices Money Holdings

Recalling…

a1

a2

a3

q1

q2

q1

Effective D. Nominal D.Quantity Traded Over Time

Effective D. Nominal D.Prices Over Time

P1

P3

P2

P1

P3

P2

Excess Demands Excess Demands = 0

for every commodity for every time = t

3. Effective Demand Driven Economies

Excess DemandTraded Amount

3. Effective Demand Driven Economie

Price Vector

Money Holding

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

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

Comparison of GDP[t] Paths over time

Notional. D

Effective. DHalf-Notional. D

Effective. D. Supplies Fixed

2: 0.5 0.6 Trader 2 prefers to buy more and hold less money

Comparison of GDP[t] Paths over time

Notional. D

Trader 1 expects his sales

Trader 1 buys a fixed amount

*P,S-fixed

2: 0.5 0.6 Trader 2 prefers to buy more and hold less money

Notional. D

Effective. D

Comparison of GDP[t] Paths over time

Initial conditions: For the first two periods, Trader 2 decided to buy less.

4. ConclusionsWe 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

Any Questions?