Reference no: EM132844115

Question 1. Consider the following model of housing using a simple exchange econ¬omy. Suppose there are two tradeable goods: money m, and housing h. Agents may also derive utility from a non-tradable good, their superannuation s. All agents i in the economy have the utility function

u_i(m, h) = ln(m) + h + ln(s_i)

Note that utility is written as a function of money m and housing used h. As superannu¬ation s is not tradeable, it has not been included as a choice variable. It will never have a 'price' attached to it. It simply adds to utility if the agent has it, and doesn't if they don't .

Suppose further that there are 100 agents in the economy. Agents 1 to 50 (Type-1 agents) looking to buy a home, and are endowed with w_i = (10, 0). Agents 51 to 100 (Type-2 Agents) already own homes, and are endowed with w_i = (5, 2). Throughout your work, assume the price of money is p_m, = 1.

a) Why can we assume the price of money is 1 in our modelling?

b) What is the Marshallian Demand function x_i(p, pw_i) for the Type-1 agents?

c) What is the Marshallian Demand function x_j (p, pw_j) for the Type-2 agents?

d) What is the Walrassian Equilibrium in this simple exchange economy? (You should provide both the equilibrium price and consumed bundles for all agent-types.)

e) Suppose s_i = 10 for the Type-1 agents, and s_i = 10 for the Type-2 agents. What is the utility of the two agent-types in the Walrassian Equilibrium?

Suppose now that the government introduces a new policy, allowing Type-1 agents to "access part of their superannuation". This reduces si to si = 5 for Type-1 agents, and increases the endowment for the Type-1 agents to (A = (15, 0).

f) What is the new Walrassian Equilibrium in this simple exchange economy? What is the utility of the two types of agents in equilibrium? How does this compare to the utility prior to the policy change?

The purpose of this model was to investigate an actual government proposed policy to allow renters to access their retirement funds when buying their first property.

g) Maximum word Limit: 150 words. Describe at most three problems with this specific model as a model to investigate such a policy. For each problem, briefly describe, in words, a potential change to the model to address it. [Note general comments about weaknesses of every simple exchange economy model which make no reference to the housing market will not attract marks.]

Question 2. Suppose a consumer has a utility function such that the income-elasticity of demand for good i is zero.

Show that

∂x_i/∂p_i, ∂h_i/∂p_i,

where x_i is the Marshallian demand for good i, and h_i is the Hicksian demand for good i.

Question 3. Consider a demand problem with three goods x = (a, b, c) and price vector p = (pa, pb, pc). Let the agent have utility function
u(a, b, c) = -γ ln(α) + (1 - γ) ln(b) + c

and facing the usual budget constraint px = m

For what values of income m does the agent demand a strictly positive quantity of all goods?

Question 4. Suppose Ann has preferences given by the Constant Elasticity of Substitution (CES) utility function over two goods u(a, b) = (aP + bP)11 P with p E (0, 1).

a) For each of the following, determine whether Ann's preferences have this property. If they do, prove this. If not, provide a counter-example.
i. Rational
ii. Weakly Monotone
iii. Strongly Monotone
iv. Locally Non-satiated
b) Suppose Ann faces the usual budget constraint px = m. By forming the maximisa-
tion (or minimisation) problem, and solving through the Lagrangian, show that:
i. Ann's Marshallian Demand for good a is

a*(p, m) = m.(P_a1(p-1))/p_ap/(p-1) + P_b^p(p-1)

ii. Ann's Hicksian Demand function for good a is

a_h*(p, u¯) = u.[1 + (P_b/p_a)^ρ/(p-1)]^-1/ρ

Question 5. Maximum Word limit: 100 words. A choice function C satisfies Property Alpha when: for all menus A and B, if A ⊆ B, x ∈ A, and x  ∈ C(B), then x ∈ C(A).

Informally: A choice function satisfies Property Alpha if, whenever option x is chosen from a larger menu and available in a more limited menu, then x must also be chosen from the more limited menu.

Suppose C is a choice function which satisfies WARP. Show that C also satisfies Property Alpha.

Question 6. Let u(a, b, c) = min[(a^ρ + b^ρ)^1/ρ, c]. For each of the following, determine whether u satisfies this property. If it does, prove this. If not, prove it does not (either by providing a counterexample or otherwise).
a) Homogeneous of degree 1
b) Homothetic
c) Quasi-linear
d) Additively separable
e) Weakly separable [Hint: u is weakly separable. You need to show this.]
Suppose now that an agent with this utility function faces the usual budget constraint P_aa ± p_bb + p_cc = m.
f) Using the 2-step maximisation process for weakly separable utility (or otherwise), find the Marshallian demand.

Attachment:- Demand function.rar