Download Kota Triangular Trade#2

Economic Simulations
Using Mathematica
Kota Minegishi
Outline
1.
2.
3.
4.
Objectives
Notional Demand Driven Economies
Effective Demand Driven Economies
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
Auctioneer
P1, P2, P3
Excess Demand  P
Excess Supply  P
S1 Dn2 S2 Dn3 S3 Dn1
2. Notional Demand Driven Economies
Auctioneer
P1, P2, P3
No Excess Demand or Supply
Then, Traders FINALLY trade.
S1 Dn2 S2 Dn3 S3 Dn1
2. Notional Demand Driven Economies
Auctioneer
Final P1, P2, P3
For time = t
S1 Dn2 S2 Dn3 S3 Dn1
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.
Utility Maximizing Behavior
3D
2D
From [1], [2], & [3],
obtain local extrema (x, y)
and Lagrange multiplier λ
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 Economy
GDP
q3
q2
Real GDP
q1
GDP
P1
P2
P3
Prices
Quantities Traded
m1
m2
m3
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
{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.
Vector field of {m1’[t], m2’[t], m3’[t] }
{β1, β2, β3}=
{.5, .5, .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] }
{β1, β2, β3}=
{.5, .6, .6}
The long run
equilibrium
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])a1[t]=min[s1[t],
/p2[t]
s1[t] = γ1 p1[t]
s2[t] = γ2 p2[t]
s3[t] = γ3 p3[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 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]
Effective Demand Driven Economy
GDP
Real GDP
a3
a2
a1
GDP
Actual Quantities Traded
P1
m1
P2
P3
m3
Prices
m2
Money Holdings
Recalling…
Notional Demand Driven Economy
GDP
q3
q2
Real GDP
q1
GDP
P1
P2
P3
Prices
Quantities Traded
m1
m2
m3
Money Holdings
Quantity Traded Over Time
Effective D.
Nominal D.
q1
a1
a2
q2
a3
q1
Prices Over Time
Effective D.
Nominal D.
Excess Demands
P1
ExcessP1Demands
=0
for every
commodity
P2 for
every time = t
P2
P3
P3
3. Effective Demand Driven Economies
Excess Demand
Traded 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
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
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
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
Comparison of GDP[t] Paths over time
Notional. D
Effective. D
Initial conditions: For the first two
periods, Trader 2 decided to buy less.
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
Any Questions?