Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

ECON4019 ADVANCED MICROECONOMIC THEORY

Specific instructions:

Handwritten Equations: For handwritten equations, they should be inserted using an app such as Adobe Scan, which allows you to capture an image using a smartphone and save it is a compressed PDF file. When digitally capturing a handwritten diagram, it is important that you do not use an excessively large file size, e.g. a high-resolution photograph, as this will make uploading and downloading your submission difficult. In all cases, your submission must be a single .pdf file and it is your responsibility to insure it contains and reliably displays all desired components. Further guidance please see THE submission instructions.

Section A

Answer the following question

1. (a) In this part of the question you are required to build a simple model.

Consider a firm which needs employees to produce its output.

The potential employees are drawn from a population and amount in number to 1. They are distinct in their type θ. Type characterises their utility cost of exerting the amount of e↵ort necessary to produce one unit of output. So a better type can be interpreted as being less lazy or more productive.

Assumption 1 The distribution of θ in the population. θ is distributed in , according to  As standard, we impose monoticity of the hazard rate: 

The firm offers contracts with salary as a function of output, which is equivalent to salary as a function of effort  Effort is effectively observable. So contracts are of (w, e) kind, or equivalently of the (U, e) kind.

The firm maximises profit, obtained from the sale of the output. Output is sold in a competitive market at given price p > 0. Employees have a reservation utility uR > 0, exogenously given.

Answer the questions below (35 Marks overall).

i. Determine the optimal (that is profiit maximising) contract offered to the employed workers.

ii. Determine which workers are employed.

iii. Determine the firm’s profit.

(b) Refer to De Fraja’s paper “The Design of Optimal Education Policies”, Review of Economic Studies, 69, 2002, pp 437-466. On slide 18 of the lecture handout you find the assertion that (the first order condition part of) the mother’s incentive compatibility constraint is

Explain carefully its meaning, and then derive it rigorously. (15 Marks)

Section B Answer the following question

2. Recall the following definitions:

● A preference is homothetic if x y implies that αx αy for all α ≥ 0.

● A preference is quasi-linear in commodity in 1 if x y implies x + εe1 y + εe1 where e1 = (1 , 0,..., 0) and α ∈ R.

● A utility u satisfies Walras’ law of demand if it’s Walrasian demand function, xu (p, w), leaves no slack in the budget constraint, p · x(p, w) = w, for all w ∈ and p ∈ 

● A choice correspondence, C, satisfies the weak axiom of revealed preference if whenever x, y ∈ A ∩ B, x ∈ C(A), and y ∈ C(B) it is also true that x ∈ C(B).

(a) Consider a utility u(x) = u1(x1) + u2(x2), where ui(xi) : is a function for each i. That is, we assume that u(x) is additively separable. What are the necessary and sufficient conditions for u to represent a homothetic preference? Show your work. (10 Marks)

(b) Consider a utility v(x) = v1(x1) ⇥ v2(x2) where vi(xi) : is a function for each i. That is, we assume that v(x) is multiplicatively separable. What are the necessary and sufficient conditions for v to represent a quasi linear preference? Show your work. (10 Marks)

(c) Consider a utility function w(x) = min{u(x), v(x)} where u(x) and v(x) are defined as above. Show that if u(x) and v(x) are both continuous functions such that  and  for all i ∈ {1, 2}, then w(x) represents a preference that satisfies Walras’ law of demand. (10 Marks)

(d) Consider an arbitrary utility function f such that the Walrasian demand of f, xf (p, w) satisfies Walras’ law as described above. Does xf (p, w) necessarily satisfy the weak axiom of reveal preference? If so, prove it. If not, find a counter example. (20 Marks)