First semester 2022 Midsemester examination
FOUNDATIONS OF ECONOMIC AND FINANCIAL MODELS EMET1001
Information and instructions:
• The exam begins at 3:30 pm and the writing time ends at 5:00 pm Canberra time (90 minutes). You must upload your answers by 5:20 pm (20 minutes). There is a grace period of 10 minutes where no penalty is applied.
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Students with special exam arrangements and longer writing times should follow the in- structions they received by email.
• Permitted materials include all course materials (textbook, exercises, notes) and non- programmable calculators, but no Internet browsing and no collaboration of any kind.
• Attempt to answer all questions. Show your workings for all questions. The maximum for this exam is 54 points. Partial credit is given for partial answers. Illegible or ambiguous answers will get 0 points.
• You must submit handwritten answers and workings as a single pdf file through the Tur- nitin link on the Wattle course page. Please see the ‘Week 7’ block on the Wattle course page for instructions on how to create and upload your answer file. We must receive your submission no later than 5:30 pm. If you experience technical problems with Turnitin, use the non-Turnitin link on Wattle or email your answer file to before 5:30 pm, and upload the identical file to Turnitin when it is up and running again.
• Use your time wisely! Do not get stuck on a particular question. Move on, and return to it again at the end if you have time.
Submissions received with a time stamp later than 5:30 pm will get a mark of 0.
Page 1 of 3 — FOUNDATIONS OF ECONOMIC AND FINANCIAL MODELS — EMET1001
Question 1 [9 points]
(a) Find the domain of j(x) = ln(x − 11) .
x−15 (b) Solve the equation 3x23x = 17 for x.
(c) Divide the polynomial n(x) = x2 − x + 5 by the polynomial d(x) = x + 2. What is the remainder?
Question 2 [6 points]
(a) Supposet(x)=exp(6+22x).Findaformulafortheinversefunctiont−1.
(b) Let f (x) = x2 − 6x + 5. Examine where f is increasing/decreasing and concave/convex.
Question 3 [9 points] The function f is defined by f (x) = x for all x. x2 −2x+4
(a) Show that x2 −2x+4 is never 0.
(b) Compute f ′(x) and find where f ′(x) is 0.
(c) Find the equation for the tangent to the graph of f at the point where x = 1.
Question 4 [6 points] Consider the equation system x1 + 2×2 = 0 and px1 − 4×2 = q.
(a) Set up the augmented coefficient matrix for the system and perform elementary row op-
erations to solve for x1 and x2.
(b) Find the values of p and q for which the equation system has (i) one solution; (ii) several
solutions; (iii) no solution.
Question 5 [6 points] Suppose A = XY , where
1/3 0 0 3 0 0 X=2 10,Y=−2−10.
−2/3 1 1/2 6 4 −2
(a) Compute A.
(b) Compute the inverse of A using elementary operations.
Question 6 [6 points] Consider the market for tax law experts. The demand and supply for experts is given by P = 200 − Q/10 and P = 20 + Q/20, where P is hourly wage (price) an expert receives and Q is the number of experts employed (quantity).
(a) Find the wage and the number of experts employed in equilibrium.
(b) Suppose the Tax Expert Association (TEA) persuades the government that experts must be licensed. It is agreed that the number of licenses available will be limited to 600. What is the wage that experts receive and how many experts are employed?
Page 2 of 3 — FOUNDATIONS OF ECONOMIC AND FINANCIAL MODELS — EMET1001
Question 7 [3 points] Consider the production function which, for fixed inputs N and K, depends on the parameter α as follows:
F(α) = NK where b, N, and K are positive constants. (Nα +bKα)1/α
Use logarithmic differentiation to find an expression for F′(α).
Question 8 [3 points] Let n be a given integer. Let I be the n × n identity matrix; that is, the diagonal elements of I are all 1 and the off-diagonal elements are all 0. Let i and z be n × 1 column vector of 1s and 0s, respectively. Define the matrix D by D = I − 1nii′. Show that Di=z.
Question 9 [6 points]
(a) The function g is defined on the interval (1, ∞) by g(x) = 14 (ln(x) − 8/ ln(x)). Find the
inverse function g−1.
(b) Show that √x−1−1 > 12 ln(x−1) for x > 2.
END OF QUESTIONS
Page 3 of 3 — FOUNDATIONS OF ECONOMIC AND FINANCIAL MODELS — EMET1001
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