Question

There is an economy with 50 agents. Of these agents, ten have income m = 10, ten have m = 20, ten have m = 30, ten have m = 40 and ten have m = 50. Each agent has utility function u(x1, x2) = x1^0.5 + x20.5 over goods x1 and x2. The price of good 2 equals 1. The price of good 1 is to be determined.

1. Derive each agents demand curve for good 1.

2. Derive the market demand for good 1.

Answer 1

The utility function is given as

u(x1, x2) = x1^{​​​​​0.5} + x2^{​​​​​0.5} = √x1 + √x2

The MRS or Marginal Rate of Substitution is

MRS = MU1/MU2

Marginal utility of x1 is

MU1 = du/dx1 = 1/(2√x1)

Marginal utility of x2 is

MU2 = du/dx2 = 1/(2√x2)

Hence, MRS = MU1/MU2

or, MRS = √x2/√x1........(1)

Price of good 1 is p1 and price of good 2 is p2. Income of agents is m.

We are told that, p2 = 1

The budget constraint of agents is

p1.x1 + p2.x2 = m

or, p1.x1 + p2 = m..........(2)

Now, at optimal consumption,

MRS = p1/p2

or, √x2/√x1 = p1/1

or, x2/x1 = (p1)²

or, x2 = (p1)².x1.........(3)

Putting this in equation (2) we get,

p1.x1 + (p1)².x1 = m

or, x1 = m/(p1 + p1^{​​​​​2}​​​​)...........(4)

Now, we will put the values of m for different agents.

1. Ten of the 50 agents have income m = 10.

Hence, each agent's demand curve is

x1A = 10/(p1 + p1^{​​​​​2}​​​​)

• Ten of these 50 agents have income m = 20.

Hence, each agent's demand curve is

x1B = 20/(p1 + p1^{​​​​​2}​​​​)

• Ten of these 50 agents have income m = 30.

Hence, each agent's demand curve is

x1C = 30/(p1 + p1^{​​​​​2}​​​​)

• Ten of these 50 agents have income m = 40.

Hence, each agent's demand curve is

x1D = 40/(p1 + p1^{​​​​​2}​​​​)

• Ten of these 50 agents have income m = 50.

Hence, each agent's demand curve is

x1E = 50/(p1 + p1^{​​​​​2}​​​​)

2. Hence, the market demand is the weighted sum of all the demand curves. There are 5 types eqch having 10 agents. Hence, the market demand curve for good 1 is

X = 10.x1A + 10.x1B + 10.x1C + 10.x1D + 10.x1E

or, X = 10.[x1A + x1B + x1C + x1D + x1E]

Putting the values of x1A, x1B,......, x1E from part (1) we get,

X = [10/(p1 + p1^{​​​​​2}​​​​)]×(10+20+30+40+50)

or, X = 1500/(p1 + p1^{​​​​​2}​​​​)

The maeket demand for good 1 is

X = 1500/(p1 + p1^{​​​​​2}​​​​)