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Ryerson University Department of Economics Angelique Bernabe CECN230 Mathematics for Economics Final Spring 2021 Instructions: - Individually or collaboratively (up to 5 students - write the names of all team members on the final copy). - Hand written or typed. - Steps count! - Scan the final copy (if you do not have access to a scanner, you can use the “scannable” app (download it for free)). - Submit the final copy on D2L before 9pm on Wednesday, June 16 2021. - Multiple photos submitted on D2L WILL NOT be accepted. Question 1. (11 points) (a) (3 points) Judge whether the following functions are homogeneous. If yes, show their degrees and verify that Euler’s Theorem holds. i)f(x, y) = (12x2)/(15y2) (b) (4 points) Consider the function F (K,L) = H(y), where H(y) = ln(y) and y = f(K,L) = KaLb. Show that F is homothetic, but not homogeneous. (c) (4 points) Find z′x, z ′ y when x 3 + y3 + z3 − 3z = 0 Question 2. (38 points) (a) Suppose the following: π is function of Q1, Q2 and Q3: π = 4Q21 − 11Q2 +Q2Q3 +Q1Q3 +Q22 +Q23 i) (3 points) Write down the first order conditions. ii) (20 points) Rewrite (i) in its matrix form and find the stationary point (Q1, Q2, Q3). Find the optimum USING ALL of the following methods: Gaussian elimination, Cramer’s rule and inverse matrix. 1 iii) (7 points) Use Hessian Matrix and its leading principle minors to determine if this optimum is at maximum or minimum or neither. (b) (8 points) Find the extreme value and determine if at the point the function is at maximum, minimum or neither. i) f(x, y, z) = x2 + 3y2 − 3xy + 4yz + 6z2 Question 3. (35 points) Consider the following optimization problem: maxU = U(x, y) subject to pxx+ pyy = B where px,py and B are constants. (i) (4 points) Write the Lagrangian and write the first order conditions for the La- grangian. (ii) (12 points) Now assume that U(x, y) = x2 + y2, Px = 2, Py = 4 and B = 100. Use both Gaussian elimination and Cramer’s rule to find λ∗ along with the optimal levels of x∗ and y∗. Calculate U∗. (iii) (9 points) Using the Bordered Hessian matrix, verify that the second-order con- ditions for a maximum or minimum are satisfied. (iv) (10 points) Suppose all parameters are the same as (ii) except B changes from 100 to 101. Based on result in (ii), use λ∗ to obtain the estimated value of the new maximized value of utility? Question 4. (8 points) (a) Using the integration by substitution method to determine the following integrals. i) ∫ x2(5x3 − 5)2dx (b) Using the integration by parts method to determine the following integrals. i) ∫ 5x (x+1)5 dx. Question 5. (8 points) 2 Find all the first and second order partial derivatives for the following functions with two or three variables: i) f(x, y) = 10x+y 2 ∗ (3x+ 5y) ii) g(x, y, z) = 8x2y2z − 3(x+ y + z) Bonus. (1 points) (1 point) What accomplishment are you most proud of this semester? Have a good rest of the summer! 3