Solving Equations And Matrices In Economics

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.