University of Sydney ECON6003 – MATHEMATICAL METHODS OF ECONOMIC ANALYSIS Mid-semester Exam: Semester 2, 2021 INSTRUCTOR: VLADIMIR SMIRNOV 1) For function f(x)=2x+2 find its inverse. Correct answer: (y-2)/2. 2) Let f ( x ) = − (x2 + 1)/ x where x ∈ R ∖ { 0 }. Find if f(x) concave or convex? Correct answer: Convex for only negative values of x. 3) Find the value at which the following function f ( x ) = x ( x − 1 )3 is minimized. Correct answer: x=0.25. 4) Find the present value of $10 000 which is paid after 5 years and $10 000 which is paid after 10 years if the interest rate is compounded annually at 2% per annum (rounded to the nearest whole number). Correct answer: $17 261. 5) Find the following definite integral (integral from 0 to 2 of x*exp(x2)). Correct answer: (e2-1)/2. 6) What values of x and y satisfy the following system of two equations? 3 x − y = 1 , 2 x + 7 y = 16. Correct answer: x=1, y=2. 7) If f(x,y)=ln(2x+y) and g(x,y) is the second derivative with respect to x and y, what is g(1,1)? Correct answer: -2/9. 8) Let f ( x , y , z ) = x 2 y z . Find the partial derivative with respect to y and evaluate it at ( x , y , z ) = ( 1 , 2 , 3 ). Correct answer: 3. 9) What is the 5th derivative of the following function y = 2 x 5 + 3 x 4 + 2 x 3 + x 2. Correct answer: 240. 10) Find the derivative of the following function y(x)=(x-1)(x+1)/(x2+1) using the quotient rule and evaluate it at x=1. Correct answer: 1. Question 1 : Let p 1 ( x 1 , x 2 ) = x 1 3 + x 2 3 . What is the degree of homogeneity of p 1 ( x 1 , x 2 ) = x 1 3 + x 2 3 ? Correct answer: 3. Let p 2 ( x 1 , x 2 ) = 3 p 1 ( x 1 , x 2 ) . What is the degree of homogeneity of p 2 ( x 1 , x 2 ) ? Correct answer: 3. Find the slope, d x2/d x1, of p 2 ( x 1 , x 2 ) at ( x 1 , x 2) = ( 1 , 1 ). Correct answer: -1. Find the slope, d x2/d x1, of p 2 ( x 1 , x 2 ) at ( x 1 , x 2) = ( 3 , 1). Correct answer: -9. Find the curvature, d2 x2/(d x1) 2 of p 2 ( x 1 , x 2 ) at ( x 1 , x 2) = ( 1 , 1 ). Correct answer: -4. Question 2: Dr. Jones is a billionaire and an amateur mathematician. Today, she wants to invest 40 million dollars at a fixed annual interest rate of 10% and use this fund to set up a “Nobel prize” for mathematicians: in each year, she awards a sum of money to the most outstanding mathematician. To combat inflation, the size of the prize is x for the first year, 1.045x in the second year, and $1.045t-1x in year t. Suppose that the first prize is scheduled to be given out immediately and Dr. Jones wants this to become a legacy that lasts forever, what is the highest possible value of x in millions of dollars? (For example, if your answer is 5 million, please enter 5 without adding "million".) Correct answer: x=40(1-1.045/1.1)=2. Question 3: At a given point in time, the marginal product of labor is 7 and the marginal product of capital is 10, the amount of labor is increasing by 15 per each unit of time and the rate of change of capital is 5. What is the rate of change of output? Correct answer: Q/dt=MPL*dL/dt+MPK*dK/dt=7*15+10*5=155. Question 4: The number of patients, n, that seek treatment from Doctor Michael is a function of the quality of his service, q, and is given by n(q), where n′(q) > 0, n′′(q) = 0, and q ≥ 0. The government pays Doctor Michael p for every patient he treats. The cost to Doctor Michael of treating n patients is c(n), where c′(n) > 0 and c′′(n) > 0, and the cost of delivering quality q is αq. Thus, the total cost is c(n)+ αq. (i) Express Doctor Michael's profit as a function of q. (1 mark) (ii) Doctor Michael chooses q to maximise profit. What is the first order condition for an interior maximum? (2 marks) (iii) Check that the solution to (ii) yields a maximum. (2 marks) (iv) Write down the condition for the solution to Doctor Michael's profit maximisation problem to be interior. Show on a diagram. (3 marks) (v) Take the differential of the first order condition in (ii) or use implicit differentiation to find an expression for dq/dp and sign it. (2 marks)
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