Math for Economists Worksheet
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ECON 3154
Homework #3
é10 2 ù
é 6 2 5ù
ê
ú
1. Let A = ê
ú, B = ê11 1 ú
3
1
3
ë
û
êë 1 0 úû
a. Compute BT + A , A * B and B * A
b. Can you compute AT * B? Explain.
2. Suppose that a firm produces two types of output using three types of input. Its output
quantities are given by the column vector
é15, 000 ù
Q=ê
ú
ë 27, 000 û
and the prices of these are given in the row vector P = [10 12]. The amounts of inputs it
uses are given in the column vector
é11, 000 ù
Z = êê15, 000 úú
êë15, 000 úû
and the input prices are given by W = [10 10 8]. Find the profit of this firm using
matrices calculation.
3. Compute the determinant and state whether the matrix is singular or nonsingular.
é 2 3ù
a. A = ê
ú
ë6 9û
é2
B=ê
ë0
é1
c. C = ê 0
ê
êë -5
b.
-4 ù
-5úû
2 -1ù
1 0 úú
2 3 úû
4. Find the inverse of the following matrices.
é3 0 1ù
a. A = ê0 2 1 ú
ê
ú
êë1 2 0 úû
b.
é 2 -4 ù
B=ê
ú
ë 0 -5û
5. Use the matrix inversion or Cramers rule to solve the system of linear equations.
x1 + x3 = 5
x1 + x2 = 5
3×1 + 2 x2 + 6 x3 = 10
6. Test for functional dependence, using the Jacobian:
y1 = 3×1 – 2 x2
y2 = 9 x12 + 12 x1 x2 + 4 x22
7. Given the following profit function for a firm producing three different goods
technically related in production:
p = -5Q12 – 3Q1Q2 + 370Q1 – 3Q22 – Q2Q3 + 170Q2 – 4Q32 – 2Q1Q3 + 230Q3
a. Find the profit-maximizing level of output for each good. Use the matrix
inversion or Cramers rule to solve the first-order condition.
b. Test the second-order conditions using the proper Hessian.
8. Given ?? = 40?? & + 25???? + 23?? & + 746 subject to the production quota, 5?? + 4?? = 220
a. Minimize costs subject to the constraint. Use matrix inversion or Cramers Rule
to solve the first-order conditions.
b. Test the second-order conditions using the proper Hessian.
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Explanation & Answer:
8 Questions
Tags:
matrix inversion
Math for Economists
Cramers rule
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