Exercise 3 - Yioryos Makedonis PDF

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Yioryos Makedonis...

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Queen Mary, University of London BSc. (Econ), ECN-124 MMEB2 Exercises L03 1. Determine the total demand for industries 1, 2, 3, given the matrix of technical coefficients 𝑨 and the final demand vector 𝑩 below 0.2 0.3 0.2 150 𝐴 = |0.4 0.1 0.3|, 𝐵 = | 200| 210 0.3 0.5 0.2

2. Use discriminants to determine whether the following quadratic functions are positive or negative definite. 𝑦1 = −3𝑥12 + 4𝑥1 𝑥2 − 4𝑥22 , 𝑦2 = 5𝑥12 − 2𝑥1 𝑥2 + 7𝑥22 . 3. A firm produces two goods in pure competition and has the following total revenue and total cost functions 𝑇𝑅 = 15𝑄1 + 18𝑄2 , TC= 2𝑄12 + 2𝑄1 𝑄2 + 3𝑄22 . Maximise the profits for the firm by using (a) Cramer’s rule for the 1st order condition and (b) the Hessian for the 2nd order condition. 4. Optimise the function 𝑦 = 3𝑥12 − 5𝑥1 − 𝑥1 𝑥2 + 6𝑥22 − 4𝑥2 + 2𝑥2 𝑥3 + 4𝑥32 + 2𝑥3 − 3𝑥1 𝑥3 , using (a) Cramer’s rule for the 1st order condition and (b) the Hessian for the 2nd order condition. 5. Maximise utility 𝑢 = 2𝑥𝑦 subject to a budget constraint equal to 3𝑥 + 4𝑦 = 90 by (a) finding the critical values of 𝑥, 𝑦, and λ and (b) using the bordered Hessian |𝐵𝐻| to test the 2nd order condition. 6. Calculate the Eigenvalues and the Eigenvectors of the matrix below 𝐴=|

6 6 |. 6 −3...