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Department of Economics Trent University ECON3250H – Mathematical Economics Fall Semester, 2022 Peterborough Campus Assignment #3 Due in Class November 21, 2022 General Information and Instructions: Worth 10% of the course grade, this assignment has 5 questions for a total of 50 marks. Marks for each question are as indicated and are evenly divided among the parts of the question. Answers to the questions must be word-processed or hand-written and they must be legible, orderly and concise; written explanations, where requested, should have no more than a very brief paragraph of content. While students may work together on the assignment, they must develop, write up and submit answers independently. On the due date noted above, answers to the assignment must be submitted in a single hard copy package at the beginning of class. Questions: 1. (2 marks) Define definiteness of a quadratic form ?. 2. (24 marks) Consider unconstrained optimization of multi-variable functions: a. State Young’s Theorem as it relates to ?(?1, … , ??). b. Determine the definiteness of the quadratic form ? = 3?2 + 4?? − 2?? − ?2. c. Define concavity and convexity as they relate to ?(?1, … , ??). d. State the Absoluteness and Uniqueness of Extrema Theorem as it relates to ?(?1, … , ??). e. Define convexity as it relates to a set ?. Illustrate the set {(?, ?): ?2 ≤ ? ≤ √?} in the ?? plane and prove that it is convex. f. Find and classify the extrema of ?(?, ?, ?) = ?2 + ?(? − 2) + 3(? − 1)2 + ?2. 3. (8 marks) Consider the optimization of ? = ?(?) subject to the constraints ??(?) = ?? for ? ∈ {1,… ,?}, where ? = (?1, … , ??) are choice variables, the ?? are parameters and ? > ?. Let the Lagrangian be ℒ(?), the Lagrange multiplier for constraint ? be ??, the bordered Hessian be �̅�, the optimal choice variables be ?∗ = (?1 ∗, … , ?? ∗) and the optimized objective function be ?∗ = ?(?∗). a. State the Lagrangain form of the optimization problem and its first-order and second-order conditions for a maximum as well as for a minimum. b. Consider the case of ? = 2 and ? = 1, whereby the objective function is ? = ?(?, ?), the sole constraint is ?(?, ?) = ?, the Lagrangian is ℒ(?, ?), the sole Lagrangian multiplier is ?, the optimal choice variables are ?∗ = ?∗(?) and ?∗ = ?∗(?), the optimal Lagrangian multiplier is ?∗ = ?∗(?) and the optimized Lagrangian is ℒ∗ = ℒ∗(?∗, ?∗). Prove that ?ℒ∗ ?? = ?∗ and very briefly interpret this result. 4. (8 marks) A competitive firm uses labour ? and capital ? to produce output according to the production function ?(?, ?) = ???? where ? > 0, ? > 0 and 0 < +="">< 1.="" the="" prices="" for="" output,="" labour="" and="" capital="" are="" ,="" and="" ,="" respectively.="" a.="" derive="" the="" factor="" demand="" functions="" ∗(?,="" ,="" )="" and="" ∗(?,="" ,="" ),="" where="" dependence="" on="" the="" parameters="" and="" is="" suppressed="" for="" simplicity="" of="" notation.="" b.="" use="" the="" hessian="" of="" the="" optimization="" problem="" to="" verify="" that="" the="" above-noted="" factor="" demand="" functions="" indeed="" maximize="" profit.="" 5.="" (8="" marks)="" a="" consumer="" lives="" for="" two="" periods,="" 1="" and="" 2,="" for="" which="" consumption="" is="" denoted="" as="" and="" ,="" respectively="" in="" units="" of="" expenditure="" (i.e.="" consumption="" is="" measured="" in="" the="" same="" units="" as="" expenditure="" and="" income).="" the="" consumer="" seeks="" to="" maximize="" his="" or="" her="" utility,="" which="" is="" given="" by="" (?,="" )="?" ln="" +="" ln?="" where=""> 0 is a parameter. The consumer has exogenous income of ? > 0 in period 1, has no exogenous income in period 2 and is able to borrow and lend income across the two periods at the interest (i.e. discount) rate ? > 0. a. Derive the demand functions ?∗(?, ?, ?) and ?∗(?, ?, ?). b. Use the bordered Hessian of the optimization problem to verify that the above- noted demand functions indeed maximize utility.