Question

Consider the market for a good in which there are two income groups of buyers. There are 8 buyers with income $40 and 6 buyers with income $54. All buyers have the same utility function u = 20q − q2 + m, where q denotes the amount of the good and m the money left after buying the good. Denote the price of the good by p.

(a) For each income group, determine the individual demand of a buyer.

(b) Find the market demand and draw the demand curve.  

(c) Consider an individual buyer in income group $40. For the following prices, determine how much money is left with the buyer after buying the good: (i) p = 8, (ii) p = 13.  

Answer 1

Solution:-

Given that

Consider the market for a good in which there are two income groups of buyers. There are 8 buyers with income $40 and 6 buyers with income $54. All buyers have the same utility function u = 20q − q^2 + m, where q denotes the amount of the good and m the money left after buying the good. Denote the price of the good by p.

a)

Type 1 Buyer with income $ 40 (8 Buyers)

m = money left after buying the good.

A consumer will maximize its utility subject to budget constraint

Let p be price of quantity (good) then,

B.C is

pq + m = 40

........(1)

..........(2)

pq + m = 40 .......(3)

equating (1) and (2)

20 - 2q = p

(20 - p)/2 = q

q = 40 - 0.5 p

m = 40 - pq

m = 40 - 10p + 0.5p^2

Demand function for type 1 buyers (with income 40)

Type 2 Buyers with income $ 54 (buyers)

............(1)

...........(2)

54 = pq + m ..........(3)

equating (1) and (2) gives

q = (20 - p)/2

m = 54 - 10p + 0.5p^2

They bith have the same demand curve s for q

qd = (20 - p)/2

b)

Market dd:-

When we add individual demand curves horizontally we get market demand curve.

we can see we have 8 +6 = 14 buyers in total.

when we add this demand curve horizontally for 14 buyers we get market demand curve. price will remain same but quantities will be multiplied by 14.

qd = 140 - 7p

market demand curve

c)

Type 1 Buyer with income $ 40

(i)

p = 8

m = 40 - 10p +(1/2) p^2

= 40 - 10 * 8 + (1/2) (8)^2

= 40 - 80 + (64/2)

= -40 + 32

m = -8

(ii)

p = 13

m = 40 - 10 (13) + (1/2) (13)^2

m = 40 - 130 + (1/2) * 169

m = -90 + 84.5

m = -5.5

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