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In applied economics, we are often interested in migration of workers from one geographical region to another. Suppose we divide a hypothetical country Loonyland into three regions: R1, R2, R3. We can find the proportion of the workers of those three regions who stay put or migrate to another region. For instance, p12 is the proportion of the workers in region 2 that move to region 1 while p33 is the proportion of workers in region 3 who do not migrate to any of the two other regions. We can denote these transition proportions by pij , i, j = 1, 2, 3 and can state these proportions of workers that stay put or migrate to another region in terms of the following transition matrix P = p11 p12 p13 p21 p22 p23 p31 p32 p33 . 3 Suppose the number of workers in these regions (in millions) at a point in time t is denoted by (xti) and presented in the vector xt = xt1 xt2 xt3 , where at some initial time t = 0, we have x0 = x01 x02 x03 . The regional migration over n periods in time can be determined using the following equation: xt = Pxt−1, t = 1, 2, …n. (1) 1. (1 Mark) Using equation (1) only, write the expressions for x1 and x2 and simplify them to express x2 in terms of x0 and P. 2. (1 Mark) Using the transition matrix P and the initial endowment of workers x0, find x1. 3. (2 Mark) 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 = 8 12 20 , where values in x0, are in millions. 4. (2 Marks) Find the number of workers in each region after migration in period 2. 5. (1 Mark) 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.
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