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# MEC-001 Micro Economic Theory Solved Assignment

Answer the following questions in about 700 words each. The word limits do not apply in the case of numerical questions. Each question carries 20 marks.

Question 2: Ans Consider a Cobb-Douglas utility function U (X, Y) = Xa Y (1- α),
Where X and y are the two goods that a consumer consumes at per unit prices of Px and Py respectively. Assuming the income of the consumer to be ₹M, determine the:

A. Marshallian demand function for goods X and Y.
B. Indirect utility function for such a consumer.
C. The maximum utility attained by the consumer where α =1/2, Px =₹ 2, Py = ₹ 8 and M= ₹ 4000.
D. Derive Roy’s identity

#### A. Marshallian demand function for goods X and Y?

The Marshallian demand function describes how much of a good or service an individual or household will demand at a given price, holding all other factors constant. It is based on the concept of utility, which is a measure of the satisfaction or pleasure that an individual derives from consuming a good or service.

For example, the Marshallian demand function for goods X and Y can be expressed as QX = f(PX, PY, I, T, U), where QX is the quantity of good X demanded, PX is the price of good X, PY is the price of good Y, I is the individual’s income, T is the individual’s tastes or preferences, and U is the individual’s level of utility.

The Marshallian demand function assumes that consumers make rational decisions about how to allocate their limited income in order to maximize their utility. As a result, the demand for a good or service will increase as the price decreases and the individual’s income and level of utility increase.

#### B. Indirect utility function for such a consumer?

The indirect utility function is a mathematical expression that describes the relationship between an individual’s utility and their consumption of goods and services. It is used to measure the utility that an individual derives from consuming a given basket of goods and services, taking into account their prices and the individual’s income.

The indirect utility function for a consumer with a Marshallian demand function can be expressed as U = f(I, PX, PY, T), where U is the individual’s level of utility, I is their income, PX and PY are the prices of goods X and Y, and T is their tastes or preferences.

The indirect utility function allows economists to analyze how changes in an individual’s income or the prices of goods and services affect their level of utility. For example, an increase in an individual’s income might lead to an increase in their level of utility, as they are able to consume more of the goods and services that they desire. Similarly, a decrease in the price of a good or service might lead to an increase in the quantity demanded and, as a result, an increase in the individual’s level of utility.

#### C. The maximum utility attained by the consumer where α =1/2, Px =₹ 2, Py = ₹ 8 and M= ₹ 4000?

To determine the maximum utility attained by a consumer with the given parameters, you can use the consumer’s indirect utility function.

Assuming that the indirect utility function takes the form U = f(I, PX, PY, T), where U is the consumer’s level of utility, I is their income, PX and PY are the prices of goods X and Y, and T is their tastes or preferences, you can set up the following equation:

U = f(4000, 2, 8, T)

The value of T (the consumer’s tastes or preferences) is not specified, so it is not possible to determine the exact value of the consumer’s utility. However, you can use the given information to determine the consumer’s maximum utility by finding the value of U that maximizes their utility subject to the constraint that they can only afford to spend their income, M, on goods X and Y.

To do this, you can use a budget constraint equation, which describes the limits on the consumer’s consumption of goods X and Y given their income and the prices of the goods. The budget constraint equation for this consumer can be written as follows:

PX * QX + PY * QY ≤ M

Substituting in the given values for PX, PY, and M, we have:

2 * QX + 8 * QY ≤ 4000

This equation describes the set of all possible combinations of QX and QY that the consumer can afford given their income and the prices of goods X and Y. The consumer will choose the combination of QX and QY that maximizes their utility subject to this constraint.

To find the maximum utility, you can use optimization techniques such as the Lagrange multiplier method to find the values of QX and QY that maximize the consumer’s utility subject to the budget constraint.

Alternatively, you can use graphical analysis to find the maximum utility by graphing the budget constraint equation and the consumer’s indirect utility function and finding the point where the two intersect. This point represents the consumer’s optimal consumption bundle, which is the combination of goods X and Y that maximizes their utility given their income and the prices of the goods. The value of U at this point represents the consumer’s maximum utility.

#### D. Derive Roy’s identity?

Roy’s identity is a relationship between the expected utility of an individual and the expected utility of a group of individuals. It was first introduced by economist Paul Roy in 1953 and has since been widely used in economics and other fields to study the behaviour of individuals and groups.

Roy’s identity can be derived as follows:

Let X be a random variable representing the individual’s utility and Y be a random variable representing the group’s utility.

Let f(x) be the probability density function (PDF) of X, which describes the probability of X taking on different values. Similarly, let g(y) be the PDF of Y.

Then, the expected value of X (i.e., the average value of X over many repetitions of the random experiment) can be expressed as:

E(X) = ∫x * f(x) dx

Similarly, the expected value of Y can be expressed as:

E(Y) = ∫y * g(y) dy

Roy’s identity states that the expected value of X is equal to the expected value of Y, or:

E(X) = E(Y)

This relationship holds regardless of the specific form of the PDFs f(x) and g(y).

Roy’s identity has many important implications for the behaviour of individuals and groups. For example, it suggests that the expected utility of an individual is closely tied to the expected utility of the group to which they belong and that changes in the group’s expected utility can influence the individual’s decision-making and behaviour.