Equilibrium Price and Quantity
Qd= K – 4P and Qs = 1 + 3P
Equilibrium occurs when Qd = Qs (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.
- K = 4,
By substituting in equations 1 and 2;
P = 3/7 and Qty = 16/7.
- 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 Xt= PXt-1 where t = 1 in this case,
Then, X1 = PX1-1 = PX0
Also, X2 = PXt-1 = PX2-1 = 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
X11 = 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.
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