bubbles930
New member
- Joined
- Apr 24, 2020
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- 14
Hey guys, apologies in advanced! This is a long one. I think I have worked out some of the solutions but am terribly stuck with question c. If anyone could show me the way, I would be SO GRATEFUL. Note, I have highlighted my thoughts in red text.
Consider an economy with three consumers (A, B and C) and m units of some (infinitely divisible) good to share amongst them.
A benevolent Social Planner wishes to maximise the sum of their utilities.
Let α denote the amount allocated to consumer A, β the amount allocated to B and σ the amount allocated to C.
Each consumer likes the good – and the more the better – but each envies the others’ consumption.
Each consumer's utility therefore depends on the consumption levels of all consumers. In particular, consumers A, B and C have respective utility functions:
??(?, ?, ?) = ? − ?(? + ?)²
??(?, ?, ?) = ? − ?(? + ?)²
??(?, ?, ?) = ? − ?(? + ?)²
where ? > 0 is a constant. Thus, the Social Planner must choose ? ≥ 0, ? ≥ 0 and ? ≥ 0 to maximise ?? + ?? + ?? subject to the constraint ? + ? + ? = ?.
a. Use Lagrange’s Method to solve the Social Planner's problem.
Here I used the lagrange method and got the following solution,
?=?=? and thus,
? = m/3
? = m/3
? = m/3
I thought about subbing this into one of the utility functions 'to solve the social planners problem' but im not sure.....
Let’s check that the substitution method gives the same answer.
i. Use the constraint to eliminate ?? from the objective function. The result will be a function of ? and ?. Show that the solution values for ? and ? that you obtained in (a) satisfy the first-order conditions for a critical point of this function.
Here I think the question is asking me to rearrange the budget constraint, to give ? = m - ? - ?. I then thought about subbing this into the utility functions (when the utility functions are all added together). This seemed to work.
ii. One advantage of the substitution approach is that we know how to check Second Order Conditions (SOCs). Show that the SOCs for a local maximum are satisfied at the critical point you found in (i). c. What is the Social Planner’s maximised welfare as a function of m and k? Show that the Social Planner would prefer to throw away some of the good (that is, reduce the value of m) if ? > 3/(8?) and provide an economic intuition for this fact.
THIS NEXT QUESTION IS THE ONE I NEED HELP ON PLEASE!
c. What is the Social Planner’s maximised welfare as a function of m and k? Show that the Social Planner would prefer to throw away some of the good (that is, reduce the value of m) if ?? > 3/(8??) and provide an economic intuition for this fact.
Consider an economy with three consumers (A, B and C) and m units of some (infinitely divisible) good to share amongst them.
A benevolent Social Planner wishes to maximise the sum of their utilities.
Let α denote the amount allocated to consumer A, β the amount allocated to B and σ the amount allocated to C.
Each consumer likes the good – and the more the better – but each envies the others’ consumption.
Each consumer's utility therefore depends on the consumption levels of all consumers. In particular, consumers A, B and C have respective utility functions:
??(?, ?, ?) = ? − ?(? + ?)²
??(?, ?, ?) = ? − ?(? + ?)²
??(?, ?, ?) = ? − ?(? + ?)²
where ? > 0 is a constant. Thus, the Social Planner must choose ? ≥ 0, ? ≥ 0 and ? ≥ 0 to maximise ?? + ?? + ?? subject to the constraint ? + ? + ? = ?.
a. Use Lagrange’s Method to solve the Social Planner's problem.
Here I used the lagrange method and got the following solution,
?=?=? and thus,
? = m/3
? = m/3
? = m/3
I thought about subbing this into one of the utility functions 'to solve the social planners problem' but im not sure.....
Let’s check that the substitution method gives the same answer.
i. Use the constraint to eliminate ?? from the objective function. The result will be a function of ? and ?. Show that the solution values for ? and ? that you obtained in (a) satisfy the first-order conditions for a critical point of this function.
Here I think the question is asking me to rearrange the budget constraint, to give ? = m - ? - ?. I then thought about subbing this into the utility functions (when the utility functions are all added together). This seemed to work.
ii. One advantage of the substitution approach is that we know how to check Second Order Conditions (SOCs). Show that the SOCs for a local maximum are satisfied at the critical point you found in (i). c. What is the Social Planner’s maximised welfare as a function of m and k? Show that the Social Planner would prefer to throw away some of the good (that is, reduce the value of m) if ? > 3/(8?) and provide an economic intuition for this fact.
THIS NEXT QUESTION IS THE ONE I NEED HELP ON PLEASE!
c. What is the Social Planner’s maximised welfare as a function of m and k? Show that the Social Planner would prefer to throw away some of the good (that is, reduce the value of m) if ?? > 3/(8??) and provide an economic intuition for this fact.
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