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Economics Dept programmes

LI Mathematical Methods for Economics

33189

Assignment:

Answer Q1-Q5 below. 

Grading Criteria:

These are listed and explained in the nearby documents ‘MME Assessment Criteria’ and ‘Mathematical Writing, Explanation, And Proof’.

Module Learning Outcomes:

· demonstrate knowledge and critical understanding of the most fruitful mathematical methods of economics;

· identify and apply these methods logically and accurately;

· solve mathematical problems arising in intermediate-level economics and econometrics;

· write and organise solutions to mathematics problems in a clear and structured manner.

as applied to the discussion of methods of mathematical economics that occur in unconstrained and constrained optimization.

Feedback to Students:

Both Summative and Formative feedback is given to encourage students to reflect on their learning that feed forward into following assessment tasks. The preparation for all assessment tasks will be supported by formative feedback within the tutorials/seminars. Written feedback is provided as appropriate.  Please be aware to use the browser and not the Canvas App as you may not be able to view all comments.

Plagiarism:

It is your responsibility to ensure that you understand correct referencing practices. You are expected to use appropriate references and keep carefully detailed notes of all your information sources, including any material downloaded from the Internet. It is your responsibility to ensure that you are not vulnerable to any alleged breaches of the assessment regulations. More information is available at https://intranet.birmingh am.ac.uk/as/studentservices/conduct/misconduct/plagiarism/index.aspx.

QUESTIONS

Q1.       (a) [6 marks] Show that the function defined by is a vector quadratic function. HINT: Start by saying how ‘vector quadratic function’ is defined.

(b) [6 marks] Show that the function defined by is a vector quadratic function.

(c) [8 marks] Let be a vector quadratic function. Define the function as , where is a variable vector, is a parameter matrix, and is a parameter vector, and and have suitable numbers of rows and columns. Show that is a vector quadratic function and hence find any critical points of . HINT. If you can’t initially do the problem as it stands, attempt it for the case where all quantities are scalar.

Q2.       (a) [12 marks] Let be defined by . Show that can be written as a vector quadratic function in the standard form, and hence solve the unconstrained problem . HINT: Pay attention to second-order conditions.

(b) [8 marks] Recall the following result and its proof in Problems and Answers:

Let , where are parameters, be a (scalar quadratic function). Then without using calculus methods (differentiation) we can show that if , then has a unique global minimum at .

Show how to generalize the earlier proof – again, without using differentiation - so that it applies to vector quadratic functions.

HINT. To simplify the algebra, you might find it useful to write the expression in the form , using the fact that is symmetric in a vector quadratic, if that is expressed in standard form.

Q3.        (a) [10 marks] The inverse supply function for makeover consultations is . Supply is measured in hours per week; the price is measured in dollars per hour.

The inverse demand function is . Demand is measured in hours per week, .

Use Newton’s numerical equation-solving method to find the equilibrium price and quantity in the market for makeover consultations.

HINT: The first step in the method is to define a function , where we want to solve the equation in order to obtain the equilibrium price . Initiate the algorithm using the starting-value .

(b) [6 marks] Use numerical methods, implemented on a spreadsheet, to find a critical point of the function

, where .

Explain the algebraic aspects of your answer systematically, as well as presenting the numerical calculations neatly, applying suitable editing.

HINT: A similar example is discussed in the Lecture Slides. Suitable column headings in your spreadsheet are those that appear in one of the examples:

(DET is the determinant of the Hessian.) Note that it is not necessary to work out full algebraic formulae for the last three columns, since they can all be expressed in terms of the previous columns.

HINT: Don’t copy spreadsheet files to each other – it will only end in tears.

(c) [4 marks] Discuss whether the point you have found is a strict local minimum of the function.

Q4       (a) [12 marks] HINT: A similar example is discussed in the Course Notes. My firm produces a single product, nails, but uses 2 plants, each of which requires inputs of labour and capital.

Plant 1 has production function .

Plant 2 has production function .

Total capital and labour are fixed at and respectively, but I can decide how to allocate each of the factors between the two plants, maximizing total production subject to the resource constraints.

(i) Write down a formal statement of the resulting optimization problem. Say what the choice variables and the parameters are.

(ii) Write down the Lagrangean function, and thereby obtain first-order conditions for the problem.

(iii) Obtain an expression for in terms of the parameters. HINT: You will need to use one of the two constraints.

(iv) Obtain expressions for .

(v) Describe and obtain the extreme-value function for this problem.

(vii) Interpret any Lagrange multipliers in the problem. HINT: The question does not ask you to find the values of these multipliers.

(b) [8 marks] A function is strictly concave if for all and , where , and for all scalar ,

.

Using this definition, show that the function is strictly concave.

HINTS: For clarity and brevity, I recommend you use, as well as , the notation . If you cannot manage (b) for general , attempt to work it through for the special (scalar) case , where . HINT: There is a similar problem among the Problems and Answers.

5. A company has a linear cost function and a linear production function .

The company has signed a contract, saying that it will produce an amount . The two factors of production and cannot be negative.

(a) [4 Marks] Formulate the firm’s minimum cost problem as a formal constrained optimization problem.

(b) [4 Marks] Express the problem in standard Kuhn-Tucker format, and hence obtain the Lagrangean function for the problem.

(c) [4 Marks] State all Kuhn-Tucker necessary conditions for to be a solution of the problem.

(d) [3 marks] Express the problem in (a) as a primal-format linear programming problem. HINT: A primal-format linear programming problem is of the form subject to

(e) [2 Marks] Obtain the corresponding dual-format problem.

(f) [3 Marks] Show that if a primal-format linear programming problem subject to and has two (optimal) solutions and , and is a convex combination of and , then is also an optimal solution of the problem.