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Econ 308: Math for Econ

Fall 2021: Prof. Matthew B. Ross

Problem Set 2

Instructions: This problem set will be due prior to the in-class final exam on October 22nd at

12:20PM. Students are encouraged to work together to solve these problems but each student must

turn in their own answers. Final submissions can be handwritten or typed in LaTex but must be

electronically scanned and submitted via the assignments link in Canvas.

Linear Algebra

1 0?1 2

12

3?2 0 5

0

]

1) Let ?? = [ 3 1 ?1], ?? = [3?1], and ?? = [

240 3

0?2

1 0 ?34

?31?1 2

41

1.a) Does the matrix D=ABC exist? If so, then what is d34? What is the det(D)? What is the D-1? What is DT?

1.b) Does the matrix E=BAC exist? If so, then what is e22? What is the det(E)? What is the E-1? What is ET?

1.c) Does the matrix F=BCA exist? If so, then what is f43? What is the det(F)? What is the F-1? What is FT?

1.d) Does the matrix G=ACB exist? If so, then what is g31? What is the det(G)? What is the G-1? What is GT?

1.e) Does the matrix H=CAB exist? If so, then what is h21? What is the det(H)? What is the H-1? What is HT?

1.f) Does the matrix J=CBA exist? If so, then what is j13? What is the det(J)? What is the J-1? What is JT?

11

2) Let ?? = [2121], ?? = [

10

], C=AB

0?1

22

2.a) What is A37?

2.b) What is B63?

2.c) What is B138?

2.d) What is C42?

132

3) Consider the matrix ?? = [?? 62] where a is a real number.

095

3.a) For what values of a will the matrix A be singular?

3.b) What is A-1? For what values of a will the matrix A-1 be singular?

3.c) What is AT? For what values of a will the matrix AT be singular?

100

1

0

4 10

4) Let A= 1 1 1 0 .

0

332

1

1

1

1

[2 2 2 ]

4.a) What is A-1?

4.b) What is the det(A)?

4.c) What is the det(A-1)

4.d) What is AT?

Logic and Proofs

5) State the contrapositive of the following statement: I cant go sailing if I dont have a boat.

6) Show that one or both of the following two statements is a tautology (hint: use truth tables): ?? ? (?? ? ??)

and/or ?? ? (?? ? ??)

7) Prove the following by induction: For all positive integers x, x is even if and only if x2 is even.

8) Prove the following (by any method): (|??| = ?? ?????? |??| = ??) ? ?? = ?? where |??| = ?? means

|??|~{1,2,3,
, ??} (hint: use the theorem that no finite set is equivalent to a proper subset of itself)

Advanced Optimization

9) Consider a representative firms profit maximization (or cost minimization) problem. Use the properties of

supply functions and unconditional input demand functions to determine the effect on output and input

demands from imposing each of the following taxes (assume the affected sector is small relative to the market

such that input and output prices, gross of the tax, does not change as a result of the tax).

9.a) What is the effect of a proportional tax on economic profit?

9.b) What is the effect of a per unit tax on the firms output?

9.c) What if the effect of a per unit tax on one of the firms inputs?

10) Consider a consumer who gets utility from two consumption goods (x1 and x2) ad air quality (A). The

consumer buys the two consumption goods at prices (p1 and p2) with income w but gets air quality for free.

Suppose the consumers preference can be represented by a utility function that takes a strongly separable

form ?? = ?? (??1 , ??2 ) + ?(??) where ??(??1 , ??2 ) is a quasi-concave function.

10.a) Will the Walrasian demand functions depend on A? Explain why or why not.

10.b) Will the Hicksian demand functions depend of A? Support your answer mathematically?

10.c) What is the marginal utility of income for this consumer?

10.d) Suppose the level of air quality increases from A 0 to A1, what would you expect to happen to the

demand for x1 and x2?

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Math for Econ

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