Need questions 1 all parts and questions 2,3
Department of Economics Trent University ECON3250H – Mathematical Economics Fall Semester, 2022 Peterborough Campus Assignment #2 Due in Class October 31, 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. (18 marks) Consider general models and implicit functions: a. State the three conditions of the Implicit Function Theorem for ? functions of the form ??(?1, … , ??; ?1, … , ??) where ?? is endogenous for all ? ∈ {1,2,… , ?} and ?? is exogenous for all ? ∈ {1,2,… ,?}. That is, for a point (?10, … , ??0; ?10, … , ??0), state the three conditions by which there exists a neighbourhood around this point for which ?? = ? ?(?1, … , ??) is an implicitly defined and continuously differentiable function, and where ??(?1, … , ??; ?1, … , ??) = 0 is an identity, for all ? ∈ {1,2,… , ?}. b. Assuming the conditions referred to in Part (a) hold, take the total differential of ??(?1, … , ??; ?1, … , ??) = 0 for each ? ∈ {1,2,… , ?} to derive the Implicit Function Derivative Rule ??? ??? = |??| |?| for all ? ∈ {1,2,… , ?} where ? is the Jacobian matrix ? = [ ??1 ??1 ⋯ ??1 ??? ⋮ ⋱ ⋮ ??? ??1 ⋯ ??? ??? ] and ?? is the matrix ? wherein the ? ?ℎ column is replaced with the negative of the gradient vector [ ??1 ??? , … , ??? ??? ] ? . c. Apply the concepts involved in Parts (a) and (b) to the basic national income model ? = ? + ?0 + ?0 ? = ? + ?(? − ?) ? = ? + ?? to verify the comparative static ?? ??0 = ?(1 − ?) 1 − ?(1 − ?) > 0 where ?, ? ∈ (0,1), ?, ? > 0 and ?0 and ?0 are exogenous. You may assume without verification that the conditions of the Implicit Function Theorem are satisfied for the equilibrium point of the model. 2. (12 marks) Find the critical and inflection points of ?(?) = 2 sin3 ? + 3 sin? for ? ∈ [0, ?], making sure to classify the critical points. 3. (4 marks) Apply Taylor’s Theorem to prove that (i) sin ? = ? − ?3 3! + ?5 5! − ?7 7! + ?9 9! − ?11 11! + ⋯ and (ii) ln(? + 1) = ? − ?2 2 + ?3 3 − ?4 4 + ?5 5 − ?6 6 + ⋯. 4. (4 marks) Let the exponential ?(?) be the inverse of the logarithm ?(?). Use the properties of logarithms to prove that (i) ?(? + ?) = ?(?)?(?) and (ii) ?′(?) = ?(?)/?′(1). 5. (12 marks) Suppose a developer owns a parcel of land worth ?(?) = ?0? 2√? (measured in dollars) when developed at its highest and best use at time ? ≥ 0 (measured in years from today), where ?0 > 0 is the value of the parcel today and ? > 1 is a parameter. Assume the only cost to the developer of owning the parcel is foregone interest, at an annual rate of ? > 0, on the capital tied up in the parcel. Letting ?(?) be the present value of the parcel, the developer seeks to maximize ?(?) by choosing the time at which the parcel is developed at its highest and best use. a. Derive the optimal time ?∗ at which to develop the parcel at its highest and best use, and then check the second-order condition to confirm that ?∗ indeed provides for a relative maximum. b. Use the result of Part (a) to evaluate ?∗ for the case of ? = √? 5 and ? = 5%, where ? is Euler’s number.