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# 5solvethefollowingsystemofequationsusingthematrixmetho

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Unformatted text preview: ally, if both constraints bind, we have: 1 8 0 and 0, we have 2 1 and 1. Which one do we want? When and 1 , When 1 and , When 1 and 1, When and This is the one we want. ,1 2 3 4 5 1, 2 3 4 1, , 2 3 4 5 1, 2 8 0 implies 0) the following conditions must be met: 0. If both 0. Solving, we have: 1. But the second condition, 1 3 4 .55093 6 0 and . Here, the second order conditions must be satisfied. We have found all the points in the interval to 0 1 such that the gradient = 0, and we found the points that gave us the greatest value for 0 , 1 and . 5. Solve the following system of equations using the matrix methods we developed in class: 2 1 6 4 3 5 1 2 1 2 ANSWER: First, find the determinant of the matrix 1 6 4 3 5 15 5 28 1 2 1 2 3 5 2 1 4 1 6 2 1 1 6 1 3 5 = 2 (‐3 – 10) – 4 (1‐12) – 1(5 + 18) = ‐26 + 44 ‐23 = ‐5. 2 , Then, using Cramer’s rule: 3 , 1 6. Consider the following consumer optimization problem: Maximize ≡ , ,… ⋯ and 0 . Here, 0 is a preference parameter. to: … 6A. Form the Lagrangian for this problem. 6B. Verify the agent’s demands are given by: 2 ⋯ … ⋯ , subject . The first order conditions for this person’s problem are: … … for i = 1, …k ⋯ We can compute the marginal rate of substitution for all goods, relative to the first good, using the first condition for good i and good 1: … … … ⋯ It then follows from the budget constraint, ∑ ⋯ We can clean this up a bit: 1 ⇒ … ⋯ 1 . So, ∑ ∑ , or ∑ . Likewise, These are the solutions for this problem. Note that the condition ∑ condition applies to our other goods: says we spend a fraction of our income (∑ ∑ , or ∑ on Good 1. A similar percent of our income is spent on good i. … We...
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