5. Consider a consumer whose utility u: R2 R is given by u(x1, x2) = x1+tx2,...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
5. Consider a consumer whose utility u: R2 R is given by u(x1, x2) = x1+tx2, where x1, x2 0 denote the consumed amounts of good 1 and 2 (the only two goods available to the consumer), and t > 0 is a free parameter. The wealth of the consumer is equal to 4 and the price of each good is 1. Due to medical reasons, the consumer is not allowed to consume more than 2 units of good 2 for every consumed unit of good 1. The objective of the consumer it to maximise their utility given the budget and the medical constraint. (a) Write down the consumer optimisation problem. Sketch the set of all feasible consumption bundles (x1, x2) in a two-dimensional graph and determine if it is closed, or convex, or compact. Precisely motivate your answer. (b) [4 marks] Determine whether the utility function of the consumer is continuous, or con- cave, or strictly concave. What can you say about the number of optimal bun- dles? Precisely motivate your answer. [3 marks] (c) Write down the Lagrange function corresponding to the optimisation problem of the consumer. Write down the Kuhn-Tucker conditions for this problem. You may neglect the non-negativity constraints. [4 marks] (d) By examining each possible combination of binding constraints, derive all so- lutions to the Kuhn-Tucker conditions for any strictly positive value of the pa- rameter t. In each case, clearly state the corresponding values of x1, x2, and the Lagrange multipliers. Determine which solutions to the Kuhn-Tucker con- ditions are solutions to the consumer optimisation problem. [6 marks] (e) Suppose that a recent medical research concluded that a marginal increase in consumption of good 2 for every unit of good 1 causes no health issues. This is to say that now the consumer may consume 2 units of good 2 for every 1 + units of good 1, where is an infinitesimally small number. How would this affect the utility of the agent in the optimum? [3 marks] 5. Consider a consumer whose utility u: R2 R is given by u(x1, x2) = x1+tx2, where x1, x2 0 denote the consumed amounts of good 1 and 2 (the only two goods available to the consumer), and t > 0 is a free parameter. The wealth of the consumer is equal to 4 and the price of each good is 1. Due to medical reasons, the consumer is not allowed to consume more than 2 units of good 2 for every consumed unit of good 1. The objective of the consumer it to maximise their utility given the budget and the medical constraint. (a) Write down the consumer optimisation problem. Sketch the set of all feasible consumption bundles (x1, x2) in a two-dimensional graph and determine if it is closed, or convex, or compact. Precisely motivate your answer. (b) [4 marks] Determine whether the utility function of the consumer is continuous, or con- cave, or strictly concave. What can you say about the number of optimal bun- dles? Precisely motivate your answer. [3 marks] (c) Write down the Lagrange function corresponding to the optimisation problem of the consumer. Write down the Kuhn-Tucker conditions for this problem. You may neglect the non-negativity constraints. [4 marks] (d) By examining each possible combination of binding constraints, derive all so- lutions to the Kuhn-Tucker conditions for any strictly positive value of the pa- rameter t. In each case, clearly state the corresponding values of x1, x2, and the Lagrange multipliers. Determine which solutions to the Kuhn-Tucker con- ditions are solutions to the consumer optimisation problem. [6 marks] (e) Suppose that a recent medical research concluded that a marginal increase in consumption of good 2 for every unit of good 1 causes no health issues. This is to say that now the consumer may consume 2 units of good 2 for every 1 + units of good 1, where is an infinitesimally small number. How would this affect the utility of the agent in the optimum? [3 marks]
Expert Answer:
Related Book For
Posted Date:
Students also viewed these mathematics questions
-
The Hyster - Yale Company is a leading lift truck designer and manufacturer that focuses on consumers' needs for lift trucks to move products efficiently. Their goal is to provide various material -...
-
PLEASE GIVE CORRECT ANSWERS Prove that the number of comparators in any sorting network is (n log n). [4 marks] (ii) What does Part (d)(i) imply in terms of the depth of any sorting network? [1 mark]...
-
MUST BE CORRECT ANSWERS A small software company has the following simplified cashflow, funded by shareholders' equity of 20,000 and a bank overdraft of 5000: Invoiced money received 2 months after...
-
Explain factors governing recruitment?
-
Given that f(1) = 2, f'(1) = - 1, g(1) = 0 and g'(1) = 1, find F'(1) where F(x) = f(x) cos g(x).
-
Three liquid samples with known masses are heated to their boiling points with the use of a heater rated at 500. W. Once their boiling points are reached, heating continues for 4.0 min and some of...
-
Car insurance companies charge higher rates for male teenagers than female teenagers.
-
Data for Morton Chip Company and its industry averages follow. a. Calculate the indicated ratios for Morton. b. Construct the extended Du Pont equation for both Morton and the industry. c. Outline...
-
^Not that question...THIS ONE In May direct labor was 35% of conversion cost. If the manufacturing overhead for the month was $102,050 and the direct materials cost was $23,800, the direct labor cost...
-
Karen is flying home from Dusseldorf and paid $2,600 for two tickets to Greenville, SC. On Dec. 1 she arranged a 19%, 60-day simple interest note as she is expecting to pay for the tickets with her...
-
Fantastic Three Ltd. has 20,000 shares outstanding that are currently trading at $20. Both Yamato Inc and Zoolander International have shown interest in the acquisition of Fantastic Three. Yamato has...
-
Name three types of Software Development Methodologies Describe in detail each software development methodology Make a case of why you would use one over the other
-
6. (15 pts) In logistic regression, the response variable is sometimes labeled y (-1,+1) rather than y {0, 1). Using a maximum likelihood approach, show that when y (-1, +1}, the logistic regression...
-
Consider the following transactions for UpFront Corporation. April 1 Paid cash for one-year policy, $18,000. Dec 31 Unexpired/unused premiums on the insurance policy amounted to $4,500. Requirements:...
-
What is the magnitude of vector labeled with " M " ? Provide your answer in cm , with 2 significant digits ( not in scientific notation ) . What is the magnitude of vector labeled with " N " ?...
-
In the post-pandemic era, there is a stronger pressure for all firms to be more sustainable in their practices. Discuss the challenges of improving sustainable practices among firms operating in less...
-
When the Department of Homeland Security created a color-coded system to prepare government officials and the public against terrorist attacks, what did it do right and what did it do wrong?
-
In Exercises use implicit differentiation to find dy/dx. y ln x + y = 0
-
In Exercises use a graphing utility to graph the polar equation. Find the area of the given region analytically. Inner loop of r = 1 + 2 cos
-
In Exercises, find the indefinite integral. s (cos 30 - 1) de
-
1. ETHICS Suppose you were on a state supreme court and faced with a restaurant- choking case. Should you require restaurant employees to know and employ the Heimlich maneuver to assist a choking...
-
Question: When Hiller Systems, Inc., was performing a safety inspection on board the M/V Cape Diamond, an ocean-going vessel, an accident killed two men. The Occupational Safety and Health...
-
2. YOU BE THE JUDGE WRITING PROBLEM An off-duty, out-of-uniform police officer and his son purchased some food from a 7-Eleven store and were still in the parking lot when a carload of teenagers...
Study smarter with the SolutionInn App