Convex problem set -- Who can solve this? I already solvedit

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Carlomario1990
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Convex problem set -- Who can solve this? I already solvedit

Postby Carlomario1990 » Wed Mar 16, 2016 5:22 pm UTC

1. Consider a firm hiring an employee for the sales department. Let e denote the effort level of the employee and suppose that marginal productivity of effort in terms of sales is equal to £50, i.e. every additional unit of effort increases sales by £50. Let’s assume that the cost of effort level e for the employee is an increasing and convex function denoted below:

C(e) = 0 if e<_ 50
= 1/2(e-50)^2 if e>50

That is, the employee is willing to exert between 0 and 50 units of effort for no extra compensation but she will put more than 50 units of effort only if she is compensated for doing so. We assume that when the employee is indifferent she exerts the highest level of effort, i.e. she is indifferent between 0 and 50 so, she exerts 50 units of effort, as she does not have any additional compensation. Moreover, the firm’s profit is the difference between the revenue from the employee’s production minus her wage.
a. Suppose the wage for a sales job that does not require any extra effort is a fixed wage of £500 per week.
i. What is the effort level that the employee would employ?
ii. What is the profit of the firm in this case?
b. Now suppose the firm pays the employee a fixed wage of £500 but also a 20% commission on sales.
i. How do we denote employee’s payoff function?
ii. What is the effort choice that maximises the employee’s payoff?
iii. What is the profit of the firm if the employee exerts the optimal amount of effort?

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gmalivuk
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Re: Convex problem set -- Who can solve this? I already solv

Postby gmalivuk » Wed Mar 16, 2016 9:36 pm UTC

Carlomario1990 wrote:1. Consider a firm hiring an employee for the sales department. Let e denote the effort level of the employee and suppose that marginal productivity of effort in terms of sales is equal to £50, i.e. every additional unit of effort increases sales by £50. Let’s assume that the cost of effort level e for the employee is an increasing and convex function denoted below:

C(e) = 0 if e<_ 50
= 1/2(e-50)^2 if e>50

That is, the employee is willing to exert between 0 and 50 units of effort for no extra compensation but she will put more than 50 units of effort only if she is compensated for doing so. We assume that when the employee is indifferent she exerts the highest level of effort, i.e. she is indifferent between 0 and 50 so, she exerts 50 units of effort, as she does not have any additional compensation. Moreover, the firm’s profit is the difference between the revenue from the employee’s production minus her wage.
a. Suppose the wage for a sales job that does not require any extra effort is a fixed wage of £500 per week.
i. What is the effort level that the employee would employ?
ii. What is the profit of the firm in this case?
b. Now suppose the firm pays the employee a fixed wage of £500 but also a 20% commission on sales.
i. How do we denote employee’s payoff function?
ii. What is the effort choice that maximises the employee’s payoff?
iii. What is the profit of the firm if the employee exerts the optimal amount of effort?

If you already solved this, you should put your solution between [spoiler] tags.

Otherwise everyone's gonna think you just want us to do your homework for you.
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Carlomario1990
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Joined: Wed Mar 16, 2016 5:18 pm UTC

Re: Convex problem set -- Who can solve this? I already solv

Postby Carlomario1990 » Thu Mar 17, 2016 12:11 am UTC

Trust me . I ahve solved it. Im not entirely familair with this website. So dont know how to add it.

i can add it tommorow morning. In the mean time, if you know how to do, do it so that we can compare it.
Goodnight.

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Deva
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Re: Convex problem set -- Who can solve this? I already solv

Postby Deva » Thu Mar 17, 2016 12:20 am UTC

Click Edit at the top of your post. Add this:

Code: Select all

[spoiler]
a.
i.
ii.
b.
i.
ii.
iii.
[/spoiler]


Looks like this:
Spoiler:
a.
i.
ii.
b.
i.
ii.
iii.

Deviate as desired. Describes additional tags here.
Changes its form depending on the observer.

Carlomario1990
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Re: Convex problem set -- Who can solve this? I already solv

Postby Carlomario1990 » Thu Mar 17, 2016 9:48 am UTC

Part B:
i) Employee payoff function: = 500+0.2(50)(e) - (0.5)(e-50)^2
ii) If i draw graph, optimal E=60, if i differentiation, optimal e = 61.11 which is i think is incorrect.
iii) Profit function of the firm = 50e - (500+0.2(50)(60)
= 1900
iv) 2000 > 1900, so the the company should not make the 20% bonus commission available/.


Anyone can confirm what i do wrong in the diffferentiation part?
If i plot the graph, i get the right value, but with differentiatiton, i get the wrong value.

Demki
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Re: Convex problem set -- Who can solve this? I already solv

Postby Demki » Thu Mar 17, 2016 10:56 am UTC

Carlomario1990 wrote:Anyone can confirm what i do wrong in the diffferentiation part?
If i plot the graph, i get the right value, but with differentiatiton, i get the wrong value.

We can't check your work if you don't show it.


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