Lagrangian Function - System of First-Order Condition - Economics Assignment Help

Assignment Task

This is the fifth out of five quizzes. The four best quizzes will count towards your grade at 10% each.

1. You can form groups up to 3 people. However, each group member has to submit an individual solution.

2. Please submit your solution via email to Benjamin.Balzer@uts.edu.au . Note that your email must be sent from your official UTS email account.

3. In your email, please use the Subject line: Submission Q5, 23565.

4. Attach a single file including your solutions to the questions and a cover page on which you write your name and your student ID and those of your group members.

5. The quiz consists of 5 exercises. There is a total of 40 points. 30 points, however, is enough to achieve 100 percents. Please find a detailed explanation of how your mark is calculated below.

Your solutions can either be typeset or handwritten. If you submit handwritten solutions, you might scan your solutions. If you decide to take photos with your mobile phone, please ensure that the quality is sufficiently good and attach all images to a single file. (For example, copy all images to a single word-document and then attach this single document to your email).

You may solve this quiz together with one or two other students who are enrolled in the subject. If you decide to work in a group, each group member still has to submit an individual solution (even though the individual solutions of the group members might be the same). When submitting your individual solution, please write the names of your group members (together with the student IDs) on the cover page.

Exercise 1 asks you to solve a constrained-optimisation problem and Exercise 3 asks you to solve an unconstrained-optimisation problem. These two exercises are the most relevant ones in the light of the final exam. I highly recommend everyone to try at least Exercise 1 and Exercise 3, which can be solved even without solving Exercise 2. Exercise 2, Exercise 4 and Exercise 5 put these optimisation problems into an economic perspective. At the very beginning of the course we focused on a micro model. This model was pretty stylized because we took the demand and the supply function as given. These functions, however, don’t necessarily fall from the sky. Rather, they can be derived from primitive maximisation problems. By now, after a semester of (hopefully not too painful) math, we have access to the tools that are necessary to solve (constrained- )maximization problems. Thus, we can solve each individual consumer’s (firm’s) utility (profit) maximization problem for given prices. This is Exercise 1 and Exercise 3. Then, we can use the solutions to these problems to derive demand (supply) functions, that is, functions that tell us how many units of a good all consumers (all firms) demand (supply) in total for given prices (this is what we do in Exercise 2 and Exercise 4). In the end, we can calculate for equilibrium (Exercise 5).

There is a total of 40 points. However, you only need 30 points to receive 100 percent. The function that maps your points, say x, into a mark is piece-wise linear and looks like this:

For example, suppose a student receives 40 points. Then x = 40 and this student receives 100 percent on this quiz. Suppose a student receives x = 31 points. Then, again, this student receives 100 percent on this quiz. Now, suppose a student receives x = 20 points. Then, the student receives 20/30 · 100, that is, approximately 67 percent. Therefore, it is possible to receive full points even without answering all the exercises. However, if you try all exercises you learn more. Moreover, trying all exercises provides you with an insurance against making small mistakes.

If you submit your solution before the deadline but for some reason want to change your solutions, feel free to resubmit your solution. I will correct the latest version.

This is an open book quiz, calculators, class notes, the textbook, and other materials are allowed.

Please show your work clearly and be concise. No credit is given for results without derivations. Wherever possible, use the notation from the lecture and show your ability to apply the methods that were introduced in the lecture.

Exercises 1 and 2: Demand

Consider an economy that consists of 1000 consumers and two goods. Let p1 be the price of good 1 and p2 that of good 2, where p1 ≥ 0 and p2 ≥ 0.

1. Exercise 1

Each individual consumer takes the prices as given and chooses her consumption bundle, (x1, x2) ∈ R 2 +, by maximizing the utility function

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