Solving Equations And Matrices In Economics

1. Using equation (1) only, write the expressions for x1 and x2 and simplify them to express x2 in terms of x0 and P.
2. Using the transition matrix P and the initial endowment of workers x0, find x1.
3. Find the number of workers in each region after migration in period 1 if
P =0.75 0.25 0.01
0.1 0.65 0.54
0.15 0.1 0.45
? , x0 =81220, where values in x0, are in millions.

4. Find the number of workers in each region after migration in period 2.

5. What would be the number of workers in each region after an influx of international migrants to each region in Loonyland by 5, 000, 000 after period 1 

Equilibrium Price and Quantity

Q_d= K – 4P and Q_s = 1 + 3P

Equilibrium occurs when Q_d = Q_s (Mankiw, 2014).

 K – 4P = 1 + 3P

7P = K – 1

P = 1/7 (K – 1). As the equilibrium price in terms of K……………………...1

Then;

Qty = 1 + 3 (1/7 (K – 1))

 Qty = 1 + 3/7 (K – 1). As the equilibrium quantity in terms of K ………………… 2

But when K = 3 (Mankiw, 2014).

From P = 1/7 (K – 1).

         P = 1/7 (3 – 1) = 2/7.

And Quantity, from Qty = 1 + 3/7 (K – 1).

                                 Qty = 1 + 2/7 (3 – 1)

                                 Qty = 13/7.   

1. K = 4,

By substituting in equations 1 and 2;

P = 3/7 and Qty = 16/7.   

1. K = 2,

Also, by substituting in equations 1 and 2 we get;

P = 1/7 and Qty = 10/7.

When K reduces exponentially to K =1/2, the price will be negative (-1/14) at the intercept of 1/2. This implies the firm is selling the commodities at a price lower than the purchase price which eventually leads the break down of the business firm (Burke  &Abayasekara, 2018).  

Migration and Matrices.

• From X_t= PX_t-1 where t = 1 in this case,

Then, X₁ = PX_1-1 = PX₀

Also, X₂ = PX_t-1 = PX_2-1 = PX₁

But from X₁ = PX₀, X₂ becomes;

X₂ = P*PX₀

                 X₂ = P²X₀

• X₁= PX₀

X₁= P₁₁       P₁₂        P₁₃X₀₁

    P₂₁          P₂₂     P₂₃X₀₂

         P₃₁       P₃₂     P₃₃X₀₃

       = P₁₁ (X₀₁ + X₀₂ + X₀₃)    P₁₂ (X₀₁ + X₀₂ + X₀₃)   P₁₃ (X₀₁ + X₀₂ + X₀₃)

          P₂₁(X₀₁ + X₀₂ + X₀₃)      P₂₂ (X₀₁ + X₀₂ + X₀₃)    P₂₃ (X₀₁ + X₀₂ + X₀₃)      

          P₃₁ (X₀₁ + X₀₂ + X₀₃)     P₃₂ (X₀₁ + X₀₂ + X₀₃)     P₃₃ (X₀₁ + X₀₂ + X₀₃) 

• X₁₁= PX₀

X_{11 =}  0.75  0.25  0.01               8          =        9.2

         0.1    0.65   0.54              12                   19.4      as the value of X₁₁

• 1    0.45              20                   11.4 

• Workers in period 2;

            8

= P² *  12

            20                                    

= 0.589    0.351     0.147            8

   0.221     0.5015    0.595         12

   0.19        0.1475    0.258         20

= 11.864

    19.68

    8.45 

= 9.2        * 5000000

    19.4

     11.4

= 46000000

    97000000    

    57000000

There will be 46000000 workers in region 1, 97000000 in region 2 and 57000000 in region 3 after international migrants (Mankiw, 2014). 

• Endogenous variables are Government G, Taxes T and Investment I while exogeneous variable is Income Y (Mankiw, 2014).
• C = 20 + 0.85Y – 0.85T

T = 25 + 0.25Y…………………….1, I = 155 and G = 100

But from;

Y = C + I + G

Y = 20 + 0.85Y +- 0.85T + 155 + 100

0.15Y = 275 – 0.85T………………………………2

From equations 1 and 2,

Using determinant formula to get the variables (Mankiw, 2014).,

= 0.15     0.85       the determinant becomes;

   -0.25      1

= 0.15     0.85          = (1*0.15) – (-0.25*0.85) =   1.2125.      

   -0.25      1

Therefore, Y = 275     0.85

                          25        1                      =    700

                           0.15     0.85

                           -0.25      1

Hence the value of Y is 700.

T =                     275     0.15

                          25        -0.25                     =    200

                           0.15     0.85

                           -0.25      1

Hence the value of T is 200

By using inverse matrix.

Determining the matrix in form of AX = C

Where, A =     0.15     0.85  ,     X = Y     and  C =      275

                        -0.25      1                   T                         25 

Hence, =     0.15     0.85             Y        = 275

                    -0.25      1               T              25      here we are to determine the values of Y and T. by multiplying both sides by the inverse of A, (A^-1) (Burke  &Abayasekara, 2018).

From AX = C, we get,   A^-1AX = A^-1C, but A^-1A = I, and also, IX = X, this gives us; X = A^-1C. where X = Y

                   T

Y       =    0.15     0.85          275

 T           -0.25      1                25              

Y     =    700

T           200

Therefore, the value of Y and T are 700 and 200 respectively (Burke  &Abayasekara, 2018).

• From Y = 20 + 275 + 0.85Y – 0.85T (Mankiw, 2014).

Y – 0.85Y = 275 – 0.85 (25 + 0.25Y)

0.15Y = 253.75 – 0.2125Y

0.3625Y = 253.75, by dividing both sides by 0.3625, we obtain

Y = 700.

Also substituting this value into, T = 25 + 0.25Y we obtain

T = 200.

Therefore, using inverse matrix method we obtain the same results as 700 and 200 for Y and T respectively (Burke  &Abayasekara, 2018).  

References

Burke, P. J., &Abayasekara, A. (2018). The price elasticity of electricity demand in the United States: A three-dimensional analysis. The Energy Journal, 39(2), 123-145.

Mankiw, N. (2014). Principles of Microeconomics. Cengage Learning. p. 32. ISBN 978-1-305-15605-0.