You are encouraged to work in groups. You can submit the Gradescope (this) part of the assign- ment either as a group or individually. The instructions are available on the LMS.

There are 3 problems in this part of the assignment, with score total equal to 84. Each Grade- scope part of the assignment will be worth 84 points; the total for all five assignments is 420. Your assignment score (35% of your grade) will be calculated as

max ( ai+max(quiz score, 20), 350) ,

where ai is your score for Assignment i (https://canvas.lms.unimelb.edu.au/courses/151236/pages/assignments-philosophy-scoring).

Problem 1 [44pt]

This problem will guide you through one possible model of a consumer who is suffering from ad- diction.

Suppose that the agent consumes two goods, q1 , chocolate, which is an addictive good, and q2 , bread, which is a non-addictive good. What makes chocolate addictive is that you need to eat at least r1 to increase your utility beyond 0. We assume r1 ≥ 0 and treat r1 as a parameter in parts 1 to 6.1 Specifically, the utility function is

u(q1 , q2 ; r1 ) = {q1−r1 )q2

where q1 ≥ 0 and q2 ≥ 0.

if q1 > r1

otherwise,

(1)

1. For given values of q1 > r1 and q2 , how does the agent’s utility depends on r1 ?

2. What is the marginal rate of substitution between q1 and q2 ( )? How does it depend on r1? What is the intuition?

3. Suppose that prices are p1  and p2  and income is Y .  Write down the utility maximisation problem.

4. Solve the problem using Lagrange method.

5. Write down the Marshallian demand.   (Recall that here you need to specify the optimal consumption bundle for all permitted values of p1 , p2 , Y ; that is, for any combination of p1 > 0, p2 > 0 and Y ≥ 0.)

6. How does the consumption of goods 1 and 2 vary with r1? What is the intuition?

Suppose now that the agent optimizes over two periods. That is, she consumes q 1(1) , q2(1) in period 1 and q1(2) , q2(2) in period 2. (Note: superscript denotes period; the quantities are not squared.) In period 1, r1 = 0. In period 2, r1 = αq1 , where 0 < α ≤ 1. The agent understand the mechanism of her addiction – that is, she knows that r1 = αq1(1) – hence take that mechanism into account in her two- period utility function. In other words, her utility function is u(q1(1) , q2(1) , q1(2) , q2(2)) = q 1(1)q2(1) +(q1(2) − αq1(1))q2(2) . The income level is the same in both periods, Y and prices are p1 = p2 = 1 in both periods. Assume that the agent can borrow and save money (hence her total expenditures across two periods must be equal to her total income, 2Y, but she does not need to ensure that she balances income and expenditures in every period).

7. Write down her utility maximisation problem.  Make sure that your maximisation problem matches the description above.

8. Solve the maximisation problem and write down her Marshallian demand.

The formulation above assumes that the agent understands the mechanism of her addiction; that is, she understands that r1 = αq1(1)  and takes that into account.  This is a very sophisticated agent who has completed ECON30010 (or, rather, an advanced course in psychology).  Let us look at a less sophisticated agent.  He understands that he will be addicted to chocolate, but he does not understand the mechanism; that is, he does not understand that r1 = αq1(1) and takes r1 as given. In other words, his utility function is u(q1(1) , q2(1) , q1(2) , q2(2); r1 ) = q 1(1)q2(1) +(q1(2) −r1 )q2(2), where r1 is treated as a parameter.

9. Solve this agent’s utility maximisation problem and write down his Marshallian demand.

10. Once you obtain the solution, plug r1 = αq1(1) . By doing so, you will find the optimal consump- tion level of the agent who correctly anticipates how addicted he will be, but still does not understand the mechanism of the addiction.

11. Compare this result to the result you obtained in part 8.

Problems 2 and 3 are exam problems. When you attempt them, keep in mind that students in 2021 and 2022 solved related problems in their homework assignments.

Problem 2 [11pt]

Suppose that agent’s preferences are represented by a linear utility function

u(m, t) = m +kt ,

where t ∈ R+ is the amount of toilet paper that the agent consumes, m ∈ R+ is the amount of other goods she consumes and k > 0 is a parameter (a given value and not a good consumed by the agent). Goods t and m are perfectly divisible throughout the problem.

Let the price of m be 1 and the price of toilet paper be pt  > 0.

(a) [3pt] Write a Marshallian demand function for this agent for any value of k > 0. (b) [3pt] Write a Hicksian demand function for this agent for any value of k > 0.

Suppose in the remainder of the problem that the price of a roll of toilet paper has the following

schedule:

pt  =

if t ≤ 2;

if t > 2.

(c) [5pt] Let k = 1. What is the consumer’s optimal bundle for any income Y > 0?