At a company, 20 employees are maing contributions for a retirement gift. Each p

ID: 1189076 • Letter: A

Question

At a company, 20 employees are maing contributions for a retirement gift. Each person is choosing howmany dollars to contribute from the interval [0,10]. The payoff to person i is bi*xi-xi, where bi>0 is the 'warm glow' he receives from each dollar he contributes, and he incurs a personal cost of 1.

A. Assume bi<1 for all i. Find all Nash equilibria. How much is collected.

B. Assume bi>1 for all i. Find all Nash equilibria. How much is collected.

C.Assume bi=1 for all i. Find all Nash equilibria. How much is collected.

Now suppose the manager of these 20 employees has announced that she will contribute d>0 for each dollar that an employee contributes. The warm glow effect to employee i from contributing a dollar is now bi*(1+d) because each dollar contributed actually results in a total contribution of 1+d. Assume bi=0.1 for i =1,...,5; bi=0.2 fir i=6,...,10; bi=0.25 for i=11,...,15; and bi=0.5 for i = 16,..., 20.

D. What value must the manager hoose for d in order to get her employees to contribute $100?

E. What value must the manager choose or d inorder to raise $750 in total from both her employees and her own matching contribution?

Explanation / Answer

a) U= b.x – x

Now, when b<1

b-1 <0

x(b-1) <0

Hence, for every dollar spent the utility falls further.

Thus in equilibrium in order to maximize his utility each employee will spend $0 because in that case utility from the contribution will be 0, in all other cases the utility will be negative.

Hence, nash eq is $0

Total contribution = 20*0= 0

b) U= b.x – x

Now, when b>1

b-1 >0

x(b-1) >0

Hence, for every dollar spent the utility increases more and more.

Thus in equilibrium in order to maximize his utility each employee will spend $10 (the maximum possible amount) because in that case utility from the contribution will be maximum, in all other cases the utility will be lesser.

Hence, nash eq is $10 for each employee

Total contribution = 20*10= $200

c) U= b.x – x

Now, when b=1

b-1 =0

x(b-1) =0

Hence, for every dollar spent the utility remains the same.

Thus in equilibrium in order to maximize his utility each employee can spend any amount between $0 and $10because in each case utility from the contribution will be 0.

Hence, nash eq are all contributions from $0 to $10.

Total contribution (20*0, 20*10) = (0,200)