Marginal rate of substitution

The marginal rate of substitution is that rate at which consumer gives up or sacrifices some units of commodities y for additional unit of commodity x. in other words the MRS is that rate where consumer substitute some units of y by an additional unit of x so that consumer derives the level of satisfaction as much as he could derive previously. The marginal rate of substitution is also defined as the ratio of change in unit of commodity y to change in commodity x. therefore MRSxy=∆y/∆x.

The MRS at any point on IC is the slope of IC at that point.

Therefore MRSxy=∆y/∆x=slope of IC

Likewise it can be also expressed as MRSxy=∆y/∆x=-MUx/MUy

Where

MUx=marginal utility of x

MUY=marginal utility of y

MRSxy=(-MUx/MUy)<0

Which means that marginal rate of substitution diminishes. The diminishing MRs represents the convexity of IC. In other words as MRSxy diminishes the IC is convex to the origin.

In the figure we have a downward sloping IC consisting three points of combination A, B, C. when consumer moves from A to B the unit of x increases by MB and that of Y reduces by AM ie ∆y=AM. As unit of x increases by ∆x with the marginal utility MUx_, there is gain in satisfaction equal to +∆x.MUx. similarly if unit of y reduces by ∆y with its marginal utility the loss in total satisfaction=-∆y.MUy.

The consumer derives equal level of satisfaction from both combination A and B only when total gain in satisfaction on= total loss in satisfaction ie +∆x.MUx=-∆y.MUy

Or ∆x/∆y=-MUy/MUx

Or ∆y/∆x=-MUx/MUy

Or ∆y/∆x=MRs=-MUx/MUy

Geometrical proof of diminishing MRs

From the above figure, initially when there is no MRs, the MRs at point B=∆y/∆x=AM/BM

The MRS at point c=∆y/∆x=BN/NC

BN=NC and BN<AM=>BN<NC<AM/BM

This clearly implies that the MRs diminishes along the ic as we move from left to right. Ultimately it insures convexity of IC.

Indifference curve

Indifference curve is famous technique to deal with the consumer behavior in case of consumption of two goods x and y. this technique is based on the ordinal approach. An indifference curve is the locus of various combinations of two goods x and y each combination yielding same or equal level of satisfaction to the consumer. Indifference curve is that curve which consists of different combination of two goods x and y at which consumer is able to derive same or equal level of satisfaction from each combination so that consumer will be indifferent to choose any one of the combinations available to him.

Indifference curve analysis can be preceded with the help of some assumptions:

·       Rationality- the consumer is a rational person who aims at satisfaction or utility maximization subject to the budgetary constraint.

·       Two commodities- there are two commodities x and y in consumption pattern which are not perfect substitutes but they can be substituted to some extent.

·       Satiable- particular want of a consumer is satiable (satisfied).

·       Total utility is a function of individual units of commodity- as unit consumption of individual commodities rises, total utility also rises.

·       Consistency- consumer is assumed to be consistent in his choice. For eg, if the consumer prefers A then B (i.e. A is better than B) than he doesn’t prefer B to A(i.e. B better than A) if both commodities are available in another situation. This behavior of consumer is said to be consistence.

·       Transitivity- if A>B;B>C; then A>C. this behavior of consumer in choice is said to be transitivity.

·       Non-satiety- consumer always prefers more of the commodity to less of the commodity. This behavior of consumer is said to be Non-satiety.

·       Marginal rate of substitution diminishes- the unit of that consumer will be ready to give up more units of x is called marginal rate of substitution.

Derivation

Indifference curve can be constructed with the help of hypothetical indifference schedule. An indifference schedule shows the different combination of two goods x and y at which consumer derives same or equal level of utility or satisfaction so that the consumer is indifference to choose any one of the combination available to him.

Indifference schedule

Combination of x1	units of x1	Units of Y1	MRSxy
A	1	20
B	2	15	5:1
C	3	11	4:1
D	4	8	3:1
E	5	6	2:1

in the above indifference schedule the consumer consumes 1 unit of x and 20 units of y as represented by combination A. in combination B as unit of x increases from 1-2 units consumption of y reduces from 20-15 units. The MRSxy is 5:1. Here MRS is defined as the units of commodity y that the consumer is willing to give up or sacrifice for additional unit of x. as unit of x increases from 1-2 there is gain in satisfaction (+∆x MUx) as unit of y reduces from 20-15 units there is loss in satisfaction (-∆y MUy). It is diminishing MRS due to which gain equals loss in satisfaction. As the gain in satisfaction is cancelled by the equal amount of loss the consumer is neither better off nor worse off. He is as before. Thus we can say that consumer derives same or equal level of satisfaction from 2 combinations a and b as a result, consumer is indifference to choose a or b.

           Similarly, in combination c, d, and e as unit of x increases to 3, 4, and 5 the units of y reduces to 11, 8 and 6 with MRSxy 4:1, 3:1 and 2:1 respectively. From all the combination a, b, c, d and e consumer derives same level of satisfaction so the consumer is indifferent to choose any one of the combinations.

    When the above 5 combinations of x and y goods are plotted on the graph measuring unit of x along x-axis and that of y along y-axis, we may obtain a downward slopping convexes shaped indifference curve as shown by the following figure.

Derivation of demand curve from law of diminishing marginal utility

The downward sloping demand curve which represents inverse association between price and quantity demand can be derived with the help of law of diminishing marginal utility. The MU curve implies that as unit consumption of X goes on rising, the MU from X goes on diminishing. Every point on the MU curve represents consumers equilibrium where MUx=Px. The changing MUx and Px brings consumer into equilibrium in each point of the MUx cuve.

     The following figure demonstrates the derivation of downward slopping demand curve with the help of principle of diminishing marginal utility. The figure has two parts. Upper panel represents derivation of MU curve and lower panel the derivation of demand curve.

This figure has two parts. Part A represents diminishing marginal utility and part B represents demand curve. Initially when a consumer consumes OX1 unit of commodity he derives OMU3 marginal utility at OP3 unit of price. Now when the price falls to OP2 MU is greater than price so to make equilibrium MU should be reduced. Consumer should increase the consumption of X. when he reaches OX2 he obtains the equilibrium at OMU2=OP2. When price falls to OP1 the MUX is less than price, so to make equilibrium MU should be increased and this can be done only when the quantity demanded is increased. When the consumption is increased at OX3 he attains the equilibrium at OMU1=OP1.

    In this way the demand curve can be explained with the help of diminishing marginal utility curve.

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