Consumer equilibrium in two commodity and multi commodity

In two commodity

In reality consumer consumes different commodity at a time, but for shake of our convenience let us suppose that consumer consumes only two commodities x and y at a time. Px and Py represent price of commodity x and y respectively. Similarly, MUx and MUy represents marginal utility derived from commodity x and y respectively.

   In case of consumptions of two commodities the consumer will be in equilibrium when marginal utilities from two commodities are proportional to their respective prices and these proportional marginal utility are again equal to constant marginal utility of money i.e.

MUx/Px=MUy/Py=MUm…………….(1)

 Proof:

For consumption of commodity x only the equilibrium consumer is MUx=Px

Or MUx/Px=1

MUx/Px=MUm……………..(2)

For consumption of commodity y only equilibrium of consumer is given by

MUy=Py

Or MUy/Py=MUm…………….(3)

From equation (2) and (3) we can write as

MUx/Px=MUy/Py=MUm

N^th commodity model

 Consumer consumes multiple numbers of commodity models at a time. Let’s take He consumes x, y & z at a time. Px, Py and Pz are the prices of the commodity x, y and z respectively. MUx, MUy and MUz be the marginal utility derived from it.

     In case of N^th commodity model, the consumer will be in equilibrium when the marginal utility derived from total commodity consumed is equal to the prices of the commodity and those proportional marginal utility are equal to the constant marginal utility of money.

MUx/Px=MUy/Py=MUz/Pz=MUm

Proof:

For consumption of commodity x MUx=Px

Or MUx/Px=1

MUx/Px=MUm………(1)

For consumptions of commodity y

MUy=Py

MUy/Py=1

MUy/Py=MUm……………(2)

For consumption of commodity z

MUz=Pz

MUz/Pz=1

MUz/Pz=MUm………………..(3)

Now combining (1),(2) and (3)

MUx/Px=MUy/Py=MUz/Pz=MUm