The law states that marginal utility derived from the consumption of a commodity declines as more units of that commodity are consumed

It can be seen from the above schedule that total utility increases at a diminishing rate, which leads to fall in marginal utility.

The given statement is refuted. When Marginal Utility is positive till point Q as shown in figure then total Utility increases at a diminishing rate and when Marginal Utility is negative after point Q, total Utility decreases.

In case of free chocolates, consumer will carry on the consumption till his total utility is maximum. It means till the additional chocolates gives positive satisfaction, consumer will keep on having chocolates. Let us understand this with the help of the figure Consumer will stop the consumption at the point of satiety (Point 'Q'), i.e., where marginal utility is equal to zero.

(i) When MU decreases, TU increases at a diminishing rate. (As shown in figure till consumption level OQ).
(ii) When MU is zero, TU is maximum at P.
(iii) When MU is negative, TU starts diminishing.

Let P_x = P_y = Rs.10
Money Income = 40
(i) Bundles available to consumer are: (0,0), (0,1), (0,2), (0,3), (0,4), (1,0), (1,1), (1,2), (1,3), (2,0), (2,1), (2,2), (3,0), (3,1) and (4, 0).
(ii) (0,4), (1,3), (2,2), (3,1) and (4,0) cost exactly Rs.40. All the other bundles cost less than Rs.40.

(i) To define consumer equilibrium, we use Indifference Curve map and the budget line.
Two Conditions Cor Consumer Equilibrium
(a) Necessary Condition
Marginal Rate of Substitution Market Rate of Exchange \(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)
Or MRS _x,y =\(\frac { { P }_{ x } }{ { P }_{ y } } \)
(Market Rate of Exchange) MRE
Or MRS _x,y =MRE\(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)
If MRS _x,y>MRE\(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)
(a) It means that to obtain one more unit of X, the consumer is willing to sacrifice more unit of Y as compared to what is required in the market.
(b) In other words, consumers willingness to pay for commodity X is higher than what market values for commodity X.
(c) This induces the consumer to buy more of X and less of Y, which result in fall in MRS till it become equal to price.
(ii) If MRS _x,y< MRE\(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)
(a) It means that to obtain one more unit of X, the consumer is willing to sacrifice less unit of Y as compared to what is required in the market.
(b) In other words, consumer's willingness to pay for commodity X is lower than what market values for commodity X.
(c) This induces the consumer to buy less of X and more of Y, which result in rise in MRS till it become equal to price.

(b) Sufficient Condition MRS Diminishing (Convex) at X, Y a point of equilibrium i.e., when MRS _x,y=MRE \(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)
(ii) The consumer will reach equilibrium when the budget line is tangential to the higher possible Indifference Curve, i.e., where necessary and sufficient conditions satisfy. In the above diagram, the consumer will reach equilibrium at point E where budget line RS is tangential to the highest possible IC₂.
(iii) The consumer cannot move to Indifference Curve, i.e., IC₃ as this is beyond his money income.
(iv) Even on IC₂, all the other points except E are beyond his means.
(v) Hence, at point E, the consumer is in equilibrium where his satisfaction maximizes, given his income and prices of goods X and Y. In equilibrium at E, the slope of Budget line = the slope of Indifference, Therefore, MRS xy is eq to the ratio of the prices of two goods \(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)

(i) To define consumer equilibrium, we use Indifference Curve map and the budget line.
Two Conditions Cor Consumer Equilibrium
(a) Necessary Condition
Marginal Rate of Substitution Market Rate of Exchange \(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)
Or MRS _x,y =\(\frac { { P }_{ x } }{ { P }_{ y } } \)
(Market Rate of Exchange) MRE
Or MRS _x,y =MRE\(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)
If MRS _x,y>MRE\(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)
(a) It means that to obtain one more unit of X, the consumer is willing to sacrifice more unit of Y as compared to what is required in the market.
(b) In other words, consumers willingness to pay for commodity X is higher than what market values for commodity X.
(c) This induces the consumer to buy more of X and less of Y, which result in fall in MRS till it become equal to price.
(ii) If MRS _x,y< MRE\(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)
(a) It means that to obtain one more unit of X, the consumer is willing to sacrifice less unit of Y as compared to what is required in the market.
(b) In other words, consumer's willingness to pay for commodity X is lower than what market values for commodity X.
(c) This induces the consumer to buy less of X and more of Y, which result in rise in MRS till it become equal to price.

(b) Sufficient Condition MRS Diminishing (Convex) at X, Y a point of equilibrium i.e., when MRS _x,y=MRE \(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)
(ii) The consumer will reach equilibrium when the budget line is tangential to the higher possible Indifference Curve, i.e., where necessary and sufficient conditions satisfy. In the above diagram, the consumer will reach equilibrium at point E where budget line RS is tangential to the highest possible IC₂.
(iii) The consumer cannot move to Indifference Curve, i.e., IC₃ as this is beyond his money income.
(iv) Even on IC₂, all the other points except E are beyond his means.
(v) Hence, at point E, the consumer is in equilibrium where his satisfaction maximizes, given his income and prices of goods X and Y. In equilibrium at E, the slope of Budget line = the slope of Indifference, Therefore, MRS xy is eq to the ratio of the prices of two goods \(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)

(i) To define consumer equilibrium, we use Indifference Curve map and the budget line.
Two Conditions Cor Consumer Equilibrium
(a) Necessary Condition
Marginal Rate of Substitution Market Rate of Exchange \(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)
Or MRS _x,y =\(\frac { { P }_{ x } }{ { P }_{ y } } \)
(Market Rate of Exchange) MRE
Or MRS _x,y =MRE\(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)
If MRS _x,y>MRE\(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)
(a) It means that to obtain one more unit of X, the consumer is willing to sacrifice more unit of Y as compared to what is required in the market.
(b) In other words, consumers willingness to pay for commodity X is higher than what market values for commodity X.
(c) This induces the consumer to buy more of X and less of Y, which result in fall in MRS till it become equal to price.
(ii) If MRS _x,y< MRE\(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)
(a) It means that to obtain one more unit of X, the consumer is willing to sacrifice less unit of Y as compared to what is required in the market.
(b) In other words, consumer's willingness to pay for commodity X is lower than what market values for commodity X.
(c) This induces the consumer to buy less of X and more of Y, which result in rise in MRS till it become equal to price.

(b) Sufficient Condition MRS Diminishing (Convex) at X, Y a point of equilibrium i.e., when MRS _x,y=MRE \(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)
(ii) The consumer will reach equilibrium when the budget line is tangential to the higher possible Indifference Curve, i.e., where necessary and sufficient conditions satisfy. In the above diagram, the consumer will reach equilibrium at point E where budget line RS is tangential to the highest possible IC₂.
(iii) The consumer cannot move to Indifference Curve, i.e., IC₃ as this is beyond his money income.
(iv) Even on IC₂, all the other points except E are beyond his means.
(v) Hence, at point E, the consumer is in equilibrium where his satisfaction maximizes, given his income and prices of goods X and Y. In equilibrium at E, the slope of Budget line = the slope of Indifference, Therefore, MRS xy is eq to the ratio of the prices of two goods \(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)

(i) To define consumer equilibrium, we use Indifference Curve map and the budget line.
Two Conditions Cor Consumer Equilibrium
(a) Necessary Condition
Marginal Rate of Substitution Market Rate of Exchange \(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)
Or MRS _x,y =\(\frac { { P }_{ x } }{ { P }_{ y } } \)
(Market Rate of Exchange) MRE
Or MRS _x,y =MRE\(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)
If MRS _x,y>MRE\(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)
(a) It means that to obtain one more unit of X, the consumer is willing to sacrifice more unit of Y as compared to what is required in the market.
(b) In other words, consumers willingness to pay for commodity X is higher than what market values for commodity X.
(c) This induces the consumer to buy more of X and less of Y, which result in fall in MRS till it become equal to price.
(ii) If MRS _x,y< MRE\(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)
(a) It means that to obtain one more unit of X, the consumer is willing to sacrifice less unit of Y as compared to what is required in the market.
(b) In other words, consumer's willingness to pay for commodity X is lower than what market values for commodity X.
(c) This induces the consumer to buy less of X and more of Y, which result in rise in MRS till it become equal to price.

(b) Sufficient Condition MRS Diminishing (Convex) at X, Y a point of equilibrium i.e., when MRS _x,y=MRE \(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)
(ii) The consumer will reach equilibrium when the budget line is tangential to the higher possible Indifference Curve, i.e., where necessary and sufficient conditions satisfy. In the above diagram, the consumer will reach equilibrium at point E where budget line RS is tangential to the highest possible IC₂.
(iii) The consumer cannot move to Indifference Curve, i.e., IC₃ as this is beyond his money income.
(iv) Even on IC₂, all the other points except E are beyond his means.
(v) Hence, at point E, the consumer is in equilibrium where his satisfaction maximizes, given his income and prices of goods X and Y. In equilibrium at E, the slope of Budget line = the slope of Indifference, Therefore, MRS xy is eq to the ratio of the prices of two goods \(\left[ \frac { { P }_{ x } }{ { P }_{ y } } \right] \)

According to the two commodity consumer equilibrium a consumer gets maximum satisfaction, when  
\(\frac { M{ U }_{ x } }{ { P }_{ x } } =\frac { M{ U }_{ y } }{ { P }_{ y } } \)
Conditions of Consumer's Equilibrium In case of Two Commodities
Necessary Condition
Marginal utility of last rupee spent on each commodity is same. Suppose there are two commodities, X and Y respectively. So, for commodity X, the condition is, marginal utility in terms of Money= Price of X
\(\frac { Marginal\ utility\ of\ a\ product\ in\ utils[M{ U }_{ X }] }{ Marginal\ utility\ of\ one\ rupee[M{ U }_{ M }] } = Price\ of\ X\)
\(\frac { M{ U }_{ x } }{ { P }_{ x } } =M{ U }_{ m }\) -----(1)
Similarly,forcommodityY,the condition is,
\(\frac { M{ U }_{ y } }{ { P }_{ y } } =M{ U }_{ m }\) -----(2)
Putting equation (2)in (1),we get
\(\frac { M{ U }_{ x } }{ { P }_{ x } } =\frac { M{ U }_{ y } }{ { P }_{ y } } \)
If \(\frac { M{ U }_{ x } }{ P_{ y } } >\frac { M{ U }_{ y } }{ { P }_{ y } } \)
It means, marginal utility from the last rupee spent on commodity X is more than marginal utility from the last rupee spent on commodity Y. So, to attain the equilibrium, consumer must increase the MU quantity of X, which decrease the \(\frac { M{ U }_{ x } }{ P_{ x } } \)and decrease the quantity of Y which will increase the \(\frac { M{ U }_{ y } }{ { P }_{ y } } \)Increase in quantity of X and decrease in quantity of Y will continue till.\(\frac { M{ U }_{ x } }{ { P }_{ x } } =\frac { M{ U }_{ y } }{ { P }_{ y } } .\)
If, \(\frac { M{ U }_{ x } }{ P_{ x } } <\frac { M{ U }_{ y } }{ { P }_{ y } } \)
It means, marginal utility from the last rupee spent on commodity X is less than marginal utility from the last rupee spent on commodity Y. So, to attain the equilibrium the consumer must decrease the quantity of X, will increase the \(\frac { M{ U }_{ x } }{ P_{ x } } \) and decrease the quantity of Y, which will decrease the \(\frac { M{ U }_{ y } }{ { P }_{ y } } .\)  Decrease in quantity of X and increase in quantity of Y continues till \(\frac { M{ U }_{ x } }{ { P }_{ x } } =\frac { M{ U }_{ y } }{ { P }_{ y } } \)
Sufficient Condition: Expenditure on commodity X + Expenditure on commodity Y = Money Income. In other words, law of Diminishing Marginal utility should be applicable.