ECON864 Mathematical Economics Early Semester Quiz 2014

Instructions: Answer the questions first on paper. Then log on to ilearn to answer the questions in order to be accessed. There are 30 marks in total.

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ECON864 Early Semester Quiz Final Total Demand Output 40 $200 70 $300 120 $250 160 $400

Agriculture Agriculture 10 Manufacturing 40 Services 60 Other 50 Value added 40 Total Input 200

Manufacturing Services 80 40 40 60 25 20 30 60 125 70 300 250

Other 30 90 25 100 155 400

Table 1: Interindustry Transaction Matrix (Values)   1.39 0.51 0.46 0.33 0.67 1.49 0.65 0.57  Leontief Inverse:  0.56 0.33 1.34 0.27 0.73 0.48 0.67 1.61 Question 1. See Lecture Notes Week 4, the textbook Section 5.7, and Chapter 12 Introduction to Mathematical Economics, (Dowling 2001). Use Table 1 and the given Leontief inverse to answer the following. 1.1 [1 Mark] Determine the value of the agriculture goods required to produce $1 dollars worth of manufacturing. 1.2 [1 Mark] Determine the increase in the size of the economy caused by a $1 increase in the final demand for manufacturing.   50 100 1.3 [2 Marks] Determine the total outputs vector x1 for final demands given by y1 =  . The 100 140 amount for each sector should be rounded to the nearest integer. ∂xi 1.4 [2 Marks] See section 7.5 pages 173 – 175 in the textbook. Observe that denote the partial ∂d j derivative of the total output from sector i with respect to final demand from sector j and that ∂x = ∂d j ∂x1 ∂d j ∂x2 ∂d j ∂x3 ∂d j ∂x4 ∂d j
T

denotes the vector of partial derivatives from each sector with respect to final demand from ∂x3 ∂x sector j. Determine and . ∂d2 ∂d4

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ECON864 Early Semester Quiz Question 2. (See Lecture Notes Week 3) Consider the following macroeconomic model. Commodity Market consumption: investment: Money Market C = 0.8Y + 100 I = −24r + 700

∗ money supply: MS = MS = 2400 transaction-precautionary demand for money: L1 = 0.25Y speculative demand for money: L2 = −20r + 2000

2.1 [1 Mark] Determine the equilibrium income Y ∗ . Give your answer to two decimal places. 2.2 [1 Mark] Determine the equilibrium interest rate r∗ . Give your answer to two decimal places.

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ECON864 Early Semester Quiz Question 3. (See Lecture Notes Week 3) Consider the macroeconomic model defined by Commodity Market national income: consumption: investment: Money Market money supply equals money demand : Y =C+I C = b(1 − t)Y I = e− fR (0 < b < 1, 0 < t < 1) (e > 0, f > 0)

M = kY − hR

(M > 0, k > 0, h > 0)

For this problem the endogenous variables are Y, C, I and R, the exogenous variables are e, t, and M, and the parameters are b, f , k, and h. These equations determine the equilibrium values of the endogenous variables in terms of the exogenous variables and the parameters. 3.1 [1 Mark] The system is to be written in the form Ax = b, where A is a 4 × 4 matrix,     Y 0 C 0 x =   and b =   . I  e R M   −1  1 The second column of A is  . Determine the first column.  0 0 3.2 [2 Marks] Which of the following gives the determinant of A? (a) |A| = h[1 − b(1 − t)] + f k (b) |A| = −h[1 + b(1 − t)] − f k (c) |A| = −(1 − (b − t))h − f k (d) |A| = −h[1 + b(1 − t)] − f k (e) |A| = eh + f M 3.3 [2 Marks] Use Cramer’s rule to determine which of the following gives the equilibrium value of Y ? −(1 − b(1 − t))(eh + f M) (a) Y = |A| −(1 − b(1 − t))M − ek (b) Y = |A| −b(1 − t)(eh + f M) (c) Y = |A| b(1 − t)(eh + f M) (d) Y = |A| (e) None of the above.
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ECON864 Early Semester Quiz 3.4 [2 Marks] Use Cramer’s rule to determine which of the following gives the equilibrium value of C? −(1 − b(1 − t))(eh + f M) (a) C = |A| −(1 − b(1 − t))M − ek (b) C = |A| −b(1 − t)(eh + f M) (c) C = |A| −(eh + f M) (d) C = |A| (e) None of the above.

3.5 [2 Marks] Use Cramer’s rule to determine which of the following gives the equilibrium value of R? −(1 − b(1 − t))(eh + f M) (a) R = |A| −(1 − b(1 − t))M − ek (b) R = |A| −b(1 − t)(eh + f M) (c) R = |A| −(eh + f M) (d) R = |A| (e) None of the above.

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ECON864 Early Semester Quiz Question 4. (See Lecture Notes Week 3 and Section 11.3 in the Textbook)   3 −1 −1 0. Let A =  0 −2 −1 5 3 4.1 [1 Mark] Find the characteristic polynomial of A. Your answer will consist of the four coefficients of the polynomial, p(λ) = λ3 + λ2 + λ+ .

4.2 [1 Mark] A has three distinct real eigenvalues. One of the eigenvalues is −2 determine the others.

  x y is an eigenvector corresponding to the least eigen4.3 [2 Marks] Determine x and y so that 1 value of A.

  x y is an eigenvector corresponding to the second least 4.4 [2 Marks] Determine x and y so that 1 eigenvalue of A.

  x 4.5 [2 Marks] Determine x and y so that y is an eigenvector corresponding to the greatest 1 eigenvalue of A.

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ECON864 Early Semester Quiz

REFERENCES

Question 5. For this question see the section on Markov chains in the textbook, pages 78–81 as well as online sources such as Markov Chains and Transition Matrices: Applications to Economic Growth. A society has three states of wealth: rich, middle income and poor. The initial, time t = 0, proportions of the population in each state are rich R0 = 0.1, middle income M0 = 0.4 and poor P0 = 0.5. The transition from one state to another over a single time period is determined by the transition matrix,   0.7 0.3 0 T = 0.3 0.4 0.3 . 0.1 0.5 0.4 Letting vn = (Rn Mn Pn ) denote the vector of proportions after n time periods, one has, vn+1 = (Rn+1 Mn+1 Pn+1 ) = vn T. 5.1 [2 Marks] Determine the portions of the population in each state after one time period. Your answer should have two decimal places so that the format should be 0.12 0.34 0.56.

5.2 [3 Marks] Let vS = (RS MS PS ) denote the vector of steady state portions. Thus, vS T = vS so that T T vS T = vS T and hence vS T is an eigenvector of T T corresponding to eigenvalue 1. Use this fact to determine the steady state portions. Your answer should have two decimal places so that the format should be 0.12 0.34 0.56.
Hint:     0.7 0.3 0.1 0.3 −0.3 −0.1 T T 0.6 −0.5. Row reduction gives T = 0.3 0.4 0.5, and I − T = −0.3 0 0.3 0.4 0 −0.3 0.6     0.3 −0.3 −0.1 0.3 −0.3 −0.1 R2 →R2 +R1 R →10R −0.3 0.6 −0.5 − − − − − →  0 0.3 −0.6 − − 1− − 1 − · · · −−−−− −−−− → − R3 →R3 +R2 0 −0.3 0.6 0 −0.3 0.6

References
Dowling, Edward T. (2001). Introduction to Mathematical Economics. 3rd ed. Schaum’s Outline Series. McGraw-Hill.

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