A firm has two types of customers. High-value customers have utility 2vx-Fif they buyx=0 units of...

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A firm has two types of customers. High-value customers have utility 2vx-Fif they buyx=0 units of the firm’s output for a feeF. Low-value customers have utilityvx-Fif they buyx=0 units of the product at feeF. The consumers cannot be forced to buy the firm’s output.
1. (a) What conditions must the deal (x, F) satisfy to get both types of customer to buy it?
2. (b) Suppose the firm producesxat marginal costcwhat is the deal it will choose to maximize profits per customer?
3. (c) The firm now plans to offer two deals (x,F) and (y,G). It wants the high- value customers to buy the first of these and the low-value customers to buy the second. What conditions must these deals satisfy now?
4. (d) Do any of the conditions in part (ii) not matter to the firm?
5. (e) If one third of the customers are high-value types and two thirds of the cus- tomers are low-value types write down the problem the firm faces in choosing these deals to maximize expected profits (do not solve).
6. (f) Simplify the above problem by making the substitutionG=vyandF= 2vx-vyinto the firm’s profit and dropping all of the constraints. Solve this simplified problem
Workers come in many different types, or productivities. The productivity of a worker is given by?. Where?is distributed uniformly on the interval 0=?=2. When a firm hires a worker with productivity?, the firm can produce?units of output and earnp?in revenue, wherep
1. (a) If the firm pays a wagew, which types of workers want to work for the firm?
2. (b) What is the average output of the workers hired by the firm and what is the
  
  average output of those who don’t work for it?
3. (c) If the firm is a monoposony purchaser of labour, what wage would it set to maximize its profits. What is the employment level?
4. (d) If the firm is in a perfectly competitive industry what will the wage be?

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Robert · Accepted Answer

Answer to 1: 
Utility of high-value customers: 2√ -F 
Utility of low-value customers: √ -F 
(a) 
For high-value customers to buy the output, the deal (xh, Fh) should be such that they are able to 
generate positive utility i.e. 2√  -Fh  0 or 
2√   Fh   …………… this is called individual rationality condition for high-value customers. 
Similarly, for low-value customers to buy the output, the deal (xl, Fl) should be such that they are 
able to generate positive utility i.e. √  -Fl  0 or 
√    Fl  ……………. this is called individual rationality condition for low-value customers. 
 (b) 
In case of high-value customers, the firm should charge a price equal to ‘c’ and the fixed fee ‘F’ 
should be set to exhaust all consumer’s surplus value. The utility of high value customers is given 
as; 2√  -F. Without any fee (F), utility = 2√ h 
So in case of high-value customers, price charged = c and fixed fee F = 2√  . 
Similarly, in case of low-value customers, the firm should charge a price equal to ‘c’ and fixed fee ‘F’ 
should be set to exhaust all consumer’s surplus value.

A firm has two types of customers. High-value customers have utility 2vx-Fif they buyx=0 units of the firm’s output for a feeF. Low-value customers have utilityvx-Fif they buyx=0 units of the product...

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