Question

A certain restaurant in town is known for refusing to give separate bills to customers. After a group has ordered and eaten together at this restaurant, the group is presented with a single bill for the entire amount that the group has eaten. It has been suggested that the restaurant does this because, with a single bill, those who dine in groups will be more likely to simply divide the charge equally, each person paying the same amount irrespective of who ordered the most, and that diners, knowing they will ultimately divide the charge equally, will order more than they would have ordered had each expected to pay only for his own order. Analyze this situation using the following model.

There are 3 diners in a group, denoted i = 1, 2, 3. Each has a utility function of the form ui (xi , ci) = ai ln (xi) - ti , where xi represents the amount of food ordered and eaten by i and ti is the amount that i has to pay. The price of food is px = 1. Assume that a1 = 1, a2 = 2 and a3 = 3.

a. How much food will each individual order knowing that the bill will be shared and how much will they order in total?  

b. How much food will each individual order if he or she has to pay for her own order and how much will they order in total?

c. Provide an intuition for why the bill is larger in (a) than in (b).

(this is edited w; variables now)

Answer 1

part a) and part b)

U1(x1,c1) = 1.ln(x1) - t1

U2(x2,c2) = 2.ln(x2) - t2

U3(x3,c3) = 3.ln(x3) - t3

Let the amount of food ordered by them individually be 5,10 and 15 units respectively.let ti = 0.1 * units of food ordered

Hence their utilities derived in that case would be,

U1(x1,c1) = 1.ln(5) - 0.5 = 1.11

U2(x2,c2) = 2.ln(10) - 1 = 3.605

U3(x3,c3) = 3.ln(15) - 1.5 = 6.624

So, food ordered in total = 5+10+15 = 20 units, Cost paid = 3 , Utility Derived = 1.11 + 3.605 + 6.624 = 11.34 units

In case they order together they will share the same bill of 3 units among themselves, which makes it 1 unit per head,

so in that case, the food ordered by them would be equal to 1/0.1 = 10 units each, so all of them together would order 30 units of food which shows that they would order more together as compared to when they go alone .

part c) The bill would be larger in case of a because each individual will try to gain his or her desired amount of utility so the person who consumes more in order to derive his utility will order more pushing the cost and amount of total food ordered for the entire group.