Economic Simulations Using Mathematica
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Transcript of Economic Simulations Using Mathematica
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?