Instructions:

1)  Figure out how to get access to Maple.  It is a mathematical program that can solve complex systems of equations, among other things, and the university has a site license.  Here are four options:

• The easiest method is to access it via the web at:  http://hercules.library.unr.edu/.  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.  
• You can use it on campus computers where it is already installed (i.e., in the DATAWORKS Lab, located in the Getchell Library, Lower Level room 8A, and also, I think, in the Math Lab on the 6th floor of the Ansari Business Building).
• Graduate students can get a copy for installation on a campus computer, assuming they have access to one.
• You can buy it at a substantial discount in the university bookstore.

Note also:  Nonlinear systems of equations may have multiple solutions, and some may involve the imaginary number I (i.e., the square root of -1). Thus, Maple may report multiple solutions sets, in apparently random order.  Choose the one solution set where all numbers are positive and nonimaginary, and are not reported as functions of other variables.  Round off your answers.  The free trade equilibrium may take time to solve.

Country 1 Autarky:

restart;
A11:=100; A12:=100; T11:=100; T12:=100; L1:=200; D1:=1; MV1:=1000000;
autarky1:=     {
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,
R11=P11*0.5*Q11/T11,
R12=P12*0.5*Q12/T12,
U1=(C11^(D1/(D1+1)))*(C12^(1/(D1+1)))/L1,
C11=Q11,
C12=Q12 };
autarky1result:=solve(autarky1,
{L11, L12, Q11, Q12, C11, C12, P11, P12, W1, R11, R12, U1 });

Country 2 Autarky:

restart;
A21:=100; A22:=100; T21:=100; T22:=100; L2:=200; D2:=1; MV2:=2000000;
autarky2:=     {
Q21=A21*(T21^0.5)*(L21^0.5),
Q22=A22*(T22^0.5)*(L22^0.5),
L2=L21+L22,
C21/C22=D2/(P21/P22),
MV2=P21*C21+P22*C22,
W2=P21*0.5*Q21/L21,
W2=P22*0.5*Q22/L22,
R21=P21*0.5*Q21/T21,
R22=P22*0.5*Q22/T22,
U2=(C21^(D2/(D2+1)))*(C22^(1/(D2+1)))/L2,
C21=Q21,
C22=Q22 };
autarky2result:=solve(autarky2,
{L21, L22, Q21, Q22, C21, C22, P21, P22, W2, R21, R22, U2 });

Free Trade Equilibrium:

restart;
A11:=100; A12:=100; T11:=100; T12:=100; L1:=200; D1:=1; MV1:=1000000;
A21:=100; A22:=100; T21:=100; T22:=100; L2:=200; 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,
R11=P11*0.5*Q11/T11,
R12=P12*0.5*Q12/T12,
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,
R21=P21*0.5*Q21/T21,
R22=P22*0.5*Q22/T22,
U2=(C21^(D2/(D2+1)))*(C22^(1/(D2+1)))/L2,
C11+C21=Q11+Q21,
C12+C22=Q12+Q22,
P11*(Q11-C11)+P12*(Q12-C12)=0,
P11=E*P21,
P12=E*P22 };
freetraderesult:=solve(freetrade,
{L11, L12, Q11, Q12, C11, C12, P11, P12, W1, R11, R12, U1,
 L21, L22, Q21, Q22, C21, C22, P21, P22, W2, R21, R22, U2, E });
 

3)  Now redo #2 above, but change the following parameters in both autarky and free trade:

T11:=150; T12:=50;
T21:=50;  T22:=150;

That is, we now assume that the home country 1 is relatively abundant in land used to produce good 1, while the foreign country 2 is relatively abundant in land used to produce good 2.  In this answer, it is not necessary to give actual quantities, but instead just give the direction of change.  Compare the autarky conditions.  How do the countries differ in their production and consumption patterns, relative prices, wages and rental rates (both nominal and real), and consumer utility?  For each country, next compare autarky against the free trade equilibrium, explaining what happens to the patterns of specialization and trade, relative prices, wages and rental rates (both nominal and real), and consumer utility.

Hint:  You might consider cutting the correct solutions and pasting them into Word, changing commas to line breaks, sorting, converting to tables, and then moving them into Excel.  You can put the variables for country 1 autarky, country 2 autarky, and free trade in rows, and each version you run into columns.  Just a suggestion.

4)  Now consider each of the following other possible bases of trade, by starting with the original values in #2 each time, and changing only the following variable values in both autarky and free trade.  In this answer, it is again not necessary to give actual quantities, but instead just give the direction of change.  Compare each autarky against the other, and against the free trade equilibrium, explaining what happens to the patterns of specialization and trade, relative prices, wages and rental rates (both nominal and real), and consumer utility.

1. Higher technological level for Home -- A11:=200 and A12:=200
2. Different technological ratios -- A11:=200 and A22:=200
3. More labor in Foreign -- L2:=400
4. Different preferences -- D1:=2 and D2:=0.5
5. Higher money stock at home -- MV1:=2000000

Exogenous Variables (Parameters):

 

Aij	- Multifactor productivity of country i for good j.
Tij	- Land allocated in country i to produce good j.
Li	- Total labor supply of country i.
Di	- Relative demand parameter (a.k.a. alpha) for country i, equal to the income share for good 1 divided by the income share for good 2.
MVi	- Money stock times velocity for country i.

Endogenous Variables:

 

Qij	- Production output by country i of good j.
Cij	- Consumption by country i of good j.
Lij	- Labor allocated in country i to produce good j.
Pij	- Nominal price in country i of good j.
Wi	- Labor wage rate in country i.
Rij	- Rental rate in country i for good j.
Ui	- Average consumer utility in country i from consumption of goods 1 and 2.
E	- Direct exchange rate, i.e. the price of country 2's currency, denominated in country 1's currency.

Equations:

 

Qij = Aij (Kij^bj)*(Lij^(1-bj))	Cobb-Douglas production function for both countries, both goods.  The technological share (bj, a.k.a. the beta for sector j) for land is assumed to be 0.5 in both sectors.  (In a Heckscher Ohlin model, we might set bj equal to 0.75 in sector 1, and 0.25 in sector 2).
Li = Li1 + Li2	Labor allocation (for each country).
Wi=Pij*(1-bj)*Qij/Lij	Wage equals Value of Marginal Product for labor, for each good j in each country i.  Because labor is mobile, the wage is equal across sectors.
Rij=Pij*bj*Qij/Tij	Rental Rate equals Value of Marginal Product for land, for each good j in each country i.  Land is the specific (fixed) factor, so Ri1 may not equal Ri2.
Ci1/Ci2=Di/(Pi1/Pi2)	Relative demand equation, as an inverse function of the relative price (both countries).  Note that this comes from a Cobb-Douglas utility function of the form Ui = Ci1^(Di/(Di+1))*Ci2^(1/(Di+1)), which has the odd characteristic of having price elasticities always equal to -1 and income elasticities always equal to +1.
MVi=Pi1*Ci1+Pi2*Ci2	Money quantity equation (for both countries).
Ui=(Ci1^(Di/(Di+1)))*(Ci2^(1/(Di+1)))/Li	Utility function in country i, based on consumption of each good weighted exponentially by shares, per worker.
P11*(Q11-C11)+P12*(Q12-C12)=0	Balance of payments for country 1 (this also identifies balance of payments for country 2).
P1j = E*P2j	Purchasing power parity (for both goods).

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