ECON 462 Group Homework
Maple Trade Model #2

Note:  I have asked for Maple to be installed on at least some of the COBA computers on the third floor.  It is still available in the Getchell Dataworks lab, in the Math lab, and online at  is https://apollo.library.unr.edu/ (again, you will need your UNR NetID account information, and you will probably need to download the Citrix MetaFrame Presentation Server Client (Plugin) for 32-bit Windows, which is free).

1)  Using Maple, solve for the following free trade equilibrium.  It is virtually identical to the equilibrium in #3 of the previous Maple homework, though I also assume that there is more labor in foreign (as in #4.3 in the previous assignment) as well.  I add an exogenous variable F to represent international transfers (i.e., a savings flow from Country 1 to Country 2 that goes through the balance of payments.)  Remember that if you get more than one solution set, you need to pick the right one.

Free Trade Equilibrium:

restart;
A11:=100; A12:=100; T11:=150; T12:=50; L1:=200; D1:=1; MV1:=1000000;
A21:=100; A22:=100; T21:=50; T22:=150; L2:=400; D2:=1; MV2:=2000000;
F:=0;
freetrade:= {
Q11=A11*(T11^0.5)*(L11^0.5),
Q12=A12*(T12^0.5)*(L12^0.5),
L1=L11+L12,
W1=P11*0.5*Q11/L11,
W1=P12*0.5*Q12/L12,
C11/C12=D1/(P11/P12),
MV1=P11*C11+P12*C12,
U1=(C11^(D1/(D1+1)))*(C12^(1/(D1+1)))/L1,
Q21=A21*(T21^0.5)*(L21^0.5),
Q22=A22*(T22^0.5)*(L22^0.5),
L2=L21+L22,
W2=P21*0.5*Q21/L21,
W2=P22*0.5*Q22/L22,
C21/C22=D2/(P21/P22),
MV2=P21*C21+P22*C22,
U2=(C21^(D2/(D2+1)))*(C22^(1/(D2+1)))/L2,
C11+C21=Q11+Q21,
C12+C22=Q12+Q22,
P11*(Q11-C11)+P12*(Q12-C12)+F=0,
P11=E*P21,
P12=E*P22,
P1=(P11*P12)^0.5,
Q1=(P11*Q11 + P12*Q12)/P1,
RW1=W1/P1,
NX1=P11*(Q11-C11)+P12*(Q12-C12),
TR1=(P11*(Q11-C11)+P12*(C12-Q12))/(P1*Q1),
P2=(P21*P22)^0.5,
Q2=(P21*Q21 + P22*Q22)/P2,
RW2=W1/P2,
NX2=P21*(Q21-C21)+P22*(Q22-C22),
TR2=(P21*(C21-Q21)+P22*(Q22-C22))/(P2*Q2),
EW2=E*W2 };
freetraderesult:=solve(freetrade,
{L11, L12, Q11, Q12, C11, C12, P11, P12, W1, U1,
 L21, L22, Q21, Q22, C21, C22, P21, P22, W2, U2, E,
 P1, RW1, Q1, NX1, TR1,
 P2, RW2, Q2, NX2, TR2, EW2 });

Assignment:  Summarize your results.
Note that I have dropped the specific factor income equations, and added a number of second-stage calculations that will make the computer take more time to solve the system of equations, but save you the bother of doing it yourself.  These include:
 
Pi=(Pi1*Pi2)^0.5 Price Index for Country i (prices are given equal weights)
RWi=Wi/Pi Real Wage Rate
Qi=(Pi1*Qi1 + Pi2*Qi2)/Pi Real GDP (Nominal GDP divided by the price level)
NXi=Pi1*(Qi1-Ci1)+Pi2*(Qi2-Ci2) Net Exports (the Balance of Trade)
TRi=(EXPi+IMPi)/PQi Trade Ratio
EW2=E*W2 Country 2's nominal wage rate in Country 1's currency

2)  Assume that 100 workers decide to migrate from Country 2 to Country 1, by changing the following variables:

L1:=300;  L2:=300;
Assignment:  How does this migration affect production and consumption patterns in each country, relative prices, wages (both nominal and real), and average consumer utility?  How does it affect real GDP, and real GDP per worker in each country?

Compare the wage rate between these two countries, before and after the migration above (to use comparable currencies, compare W1 to EW2).  Then calculate total combined GDP (in real Country 1 currency):

Total GDP = (P1*Q1 + E*P2*Q2)/P1
and show that migration that equates real wages improves efficiency.
3)  Now consider how the amount of migration necessary for equilibrium would be affected by differences in productivity.  Do this by assuming that productivity in Country 1 is twice that of Country 2 (so that A11=A12=200).  You might use the following code, in which I drop a number of secondary variables in order to speed up processing (you may need to run this a number of times to find the solution):

restart;
A11:=200; A12:=200; T11:=150; T12:=50; L1:=200; D1:=1; MV1:=1000000;
A21:=100; A22:=100; T21:=50; T22:=150; L2:=400; D2:=1; MV2:=2000000;
freetrade:= {
Q11=A11*(T11^0.5)*(L11^0.5),
Q12=A12*(T12^0.5)*(L12^0.5),
L1=L11+L12,
W1=P11*0.5*Q11/L11,
W1=P12*0.5*Q12/L12,
C11/C12=D1/(P11/P12),
MV1=P11*C11+P12*C12,
Q21=A21*(T21^0.5)*(L21^0.5),
Q22=A22*(T22^0.5)*(L22^0.5),
L2=L21+L22,
W2=P21*0.5*Q21/L21,
W2=P22*0.5*Q22/L22,
C21/C22=D2/(P21/P22),
MV2=P21*C21+P22*C22,
C11+C21=Q11+Q21,
C12+C22=Q12+Q22,
P11*(Q11-C11)+P12*(Q12-C12)=0,
P11=E*P21,
P12=E*P22,
P1=(P11*P12)^0.5,
P2=(P21*P22)^0.5,
EW2=E*W2 };
freetraderesult:=solve(freetrade,
{L11, L12, Q11, Q12, C11, C12, P11, P12, W1,
 L21, L22, Q21, Q22, C21, C22, P21, P22, W2, E,
 P1, P2, EW2 });

Note that increases in Country 1's productivity and factor migration affect the equilibrium in both countries, and changes the price level so that nominal values cannot be easily compared.  Still, you don't need to worry about all that.  Instead, just use trial and error to find the amount of migration that equates comparable wages (i.e., W1=EW2).  Remember that the total amount of labor (L1+L2=600) does not change with migration.  To get within 10 units is close enough.

Assignment:  How much labor must migrate to achieve a migration equilibrium, and from where to where, when Country 1 is twice as productive as Country 2?  Why is this amount different than in Problem #2?  Does this migration improve efficiency?  (Hint:  Calculate real combined GDP again.  You may need to rerun the original equations with your labor migration result to get Q1 and Q2.)  

4)  Going back to the original equilibrium in Problem #1, consider instead the effects of a financial transfer (e.g., a loan, repayment, foreign investment, or other foreign savings flow) from Country 2 to Country 1, by changing the following variable:
F:=100000;
Assignment:  How does this transfer affect production and consumption patterns in each country, relative prices, the price level, wages (both nominal and real), real GDP, consumer utility, and the exchange rate?  How does the transfer affect the balance of trade and the trade ratio in both Country 1 and Country 2?