question_answer

A consumer consumes only two goods. Explain consumer's equilibrium with the help of utility analysis.

Or

A consumer consumes only two goods A and B and is in equilibrium. Show that when price of good B falls demand for B rises. Answer this question with the help of utility analysis.

 

Answer:

The consumer?s equilibrium in this scenario can be explained by the Law of Equi-Marginal Utility. According to this, a consumer allocates his expenditure between two commodities in such a manner that the utility derived from each additional unit of the rupee spent on each of the commodities is equal to the marginal utility of money. This is algebraically described as follows:
\[M{{U}_{x}}/{{P}_{x}}=M{{U}_{y}}/{{P}_{y}}=M{{U}_{n}}/{{P}_{n}}=M{{U}_{m}}\]
where,
\[M{{U}_{M}}\] represents the marginal utility of money
\[M{{U}_{x}}\] represents the marginal utility of good x
\[M{{U}_{y}}\] represents the marginal utility of good y
Let us suppose that \[M{{U}_{M}}\] is Rs 10 and the price of both the goods i.e., \[{{P}_{x}}\] and \[{{P}_{y}}\] is same at Rs 5. \[M{{U}_{x}}\] and \[M{{U}_{y}}\] for different units of goods consumed is tabulated below.
Units	\[M{{U}_{x}}\]	\[M{{U}_{y}}\]
1	57	62
2	55	58
3	\[\]	53
4	47	51
5	42	\[\]
6	35	45
7	30	40
From the schedule, we can conclude that the consumer attains equilibrium when he consumes 3 units of good x and 5 units of good y. At this consumption bundle, the consumer?s equilibrium is featured by:
i.e.,       \[\left. \begin{align}   & \frac{M{{U}_{x}}}{{{P}_{x}}}=\frac{M{{U}_{y}}}{{{P}_{y}}}=M{{U}_{M}} \\  & \frac{50}{5}=\frac{50}{5}=10\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\ \end{align} \right\}\,\,\text{Consumer }\!\!'\!\!\text{ s}\,\,\text{Equilibrium}\]
Or
In this situation, the consumer?s equilibrium is attained at that point, where Q the utility derived from each additional unit of the rupee spent on each of the goods is equal.
That is Marginal Utility of a Rupee spent on the good A (i.e.,\[M{{U}_{A}}/{{P}_{A}}\]) is equal to the Marginal Utility of a Rupee spent on the good B (i.e.,\[M{{U}_{B}}/{{P}_{B}}\]), which in turn is equal to the Marginal Utility of Money (\[M{{U}_{M}}\]) - That is, \[M{{U}_{A}}/{{P}_{A}}=M{{U}_{B}}/{{P}_{B}}=M{{U}_{M}}\]
If price of the good B falls, then the value of the fraction (i.e.,\[M{{U}_{B}}/{{P}_{B}}\]) increases.
Mathematically, this implies: \[M{{U}_{B}}/{{P}_{B}}>M{{U}_{A}}/{{P}_{A}}=M{{U}_{M}}\]
In such a situation, the demand for good B rises and consumer would increase his consumption of good B. He will continue to increase the consumption of good B until the equality between the marginal utilities of each of the goods become equal to the marginal utility of money. At this situation, the equilibrium is restored. That is,  \[M{{U}_{A}}/{{P}_{A}}=M{{U}_{B}}/{{P}_{B}}=M{{U}_{M}}\].