**INDIAN ECONOMIC SERVICES : INDIFFERENCE CURVE ANALYSIS AND UTILITY FUNCTION; DUALITY, INDIRECT UTILITY FUNCTION**

Hello Guys, we have started our online classes again from 8th November this year. Here I am documenting what all we have completed this week, and within each class what is discussed, including the previous years questions which are discussed in each class/recording. You may look at this blogpost, as the set of question bank, when you are preparing your notes. I think I will take one more week to complete this topic 1, of Paper 1.

This coming week I will be discussing Revealed preference approach, Elasticity of Demand and Supply (several questions are asked in previous years from this subtopic), and Game Theory concepts, Nash Equilibrium and Dominant Strategy Equilibrium

**Indifference Curve Analysis and Utility function (Part 1) : Rational Preferences, Diminishing MRS, Properties of Indifference Curves**

**Topics discussed**

- Consumption Set
- Consumer Preferences
- Rationality Axioms
- Utility Representation
- Indifference Curve and Marginal Utility
- Marginal Rate of Substitution
- Properties of Indifference Curve

**2. Indifference Curve Analysis and Utility functions (Part 2) (Demand functions) (IES 2011, 2015, 2017, 2018, 2019, 2020)**

**Topics Discussed**

- Optimization Principle Utility Maximization
- Budget Constraint
- First order conditions for a maximum
- Second order conditions
- Perfect Substitutes

**Past Year Questions Discussed** **in this class/recording**

**IES 2018, 9 (a)**

Consider the utility function

U = x^{α} y^{β}, α> 0, β > 0

which is to be maximized subject to the budget constraint m =p_{x}X + pyY where m = income (nominal) and Px and Py are the prices respectively per unit of the goods X and Y.

Derive the demand function for X and Y. Show that these demand functions are homogeneous of degree zero in prices and income.

**IES 2015, Q 2**

Consider the utility function as U = √q_{1} q_{2}, where q_{1} and q_{2} are two commodities on which the consumer spends his entire income of the month. Let the price per unit of q_{1} and q_{2} be ₹ 40 and ₹ 16 respectively and the monthly income of the consumer be ₹ 4,000.

Find out the optimal quantities of q_{1} and q_{2}**.**

**IES 2019, Q 2 (a)**

An individual has the utility function U = XY and her budget equation is 10X + 10 Y = 1000. Find the maximum utility that she can attend.

**IES 2011, Q 1 (b)**

What do you mean by corner solution? In the case of perfect complementary goods, where do you get the corner solution?

**IES 2020, Q 3 (a)**

Suppose the utility function for the consumer takes one of the following forms:

- U = 50 x + 20 y
- U = 20 x + 50 y
- U = 80 x + 40 y

The budget of the consumer is ₹ 10,000. The prices of good X and good Y are ₹ 50 and ₹ 20 per unit respectively. Determine the possibility of determination of the equilibrium basket in each case using diagram and comment on the nature of the solutions

**IES 2019, Q 1 (a)**

In a two-commodity framework, the marginal rate of substitution is everywhere equal to 2. The prices of the two goods are equal. Draw a diagram to identify the utility maximizing equilibrium.

**IES 2017, Q 9 (b)**

Consider the utility function u log x_{1} + x_{2} which is to be maximized subject to the budget constraint m = p_{1}x_{1} + p_{2}x_{2}, where p_{1} and p_{2} are the prices per unit of the goods x_{1} and x_{2} respectively, and m is the income of the consumer. Derive the demand for x_{1} and x_{2} and interpret your results.

**3. Normal Goods, Inferior goods, Giffen Goods, Substitutes, Complements, Income Offer curve, Price Offer curve, Engel curves (IES 2012)**

**Topic Discussed**

- Normal and Inferior Goods
- How demand changes as income changes?
- Comparative Statics
- What conditions need to be satisfied for a good to be Giffen good?
- Veblen Goods
- Income Offer curve and Engel curve of Perfect Substitutes
- Income Offer curve and Engel curve of Perfect Complements
- Income Offer curve and Engel curve for Quasi-linear Pref
- Price Offer curve and Demand curve : Perfect Substitutes
- Price Offer curve and Demand curve : Perfect Complements
- Income Offer Curves
- Both Goods can’t be Inferior
- A Good can’t be Inferior at all Income Levels
- Engel Curves
- Substitute Goods and Complement Goods

**Past Year Question(s) Discussed** **in this class/recording** :

**IES 2012, Q 1 (h)**

Show graphically in your answer-book that if a consumer buys only two goods, both cannot be inferior at the same time.

**4**. **Duality, Indirect Utility function, Expenditure function (IES 2013, 2014, 2018, 2010)**

**Topic Discussed**:

- Properties of IOF
- Perfects Compliments
- Perfect Substitutes
- Max Function
- Roy’s Identity
- Expenditure Function
- Properties of Expenditure Function
- Sheppard’s Lemma

**Past Year Questions Discussed** **in this class/recording** :

**IES 2013, Q 1 (j)**

Given utility function U= q_{1}q_{2} and budget constraint Y = p_{1}q_{1} + p_{2}q_{2}, derive the indirect utility function.

**IES 2014, Q 1 (a)**

Is the following statement true or false?

Explain.

‘’If a consumer’s utility function is of the form = x_{1}^{1/3} x_{2}^{1/3}, she faces prices p_{1} and p_{2} and her income is I, then her indirect utility function is V = I^{3} / (3p_{1} p_{2}).”

**IES 2018, Q 2 (a)**

A price-taking consumer consumes two goods X and Y. Let x and y denote the quantities of goods X and Y respectively, and let P_{X} and P_{Y} be the respective prices of the two goods. Assume that (i) the consumer’s budget is given by M, ∞ > M> o; and (ii) P_{X} and P_{Y} are finite and positive.

(a) Let the consumer’s utility function be given by

U (x, y) = min [x, y]

Define Indirect Utility Function and derive this consumer’s Indirect Utility Function.

**IES 2018, Q 1 (e)**

Explain with a diagram why the compensated demand curve is vertical if the consumer’s utility function is of the form:

V (x, y) = min [x, y]

**5. Price, Income and Substitution Effects (IES 2010, 2011, 2012, 2016, 2017)**

**Topic Discussed:**

- Income and Substitution Effect of a fall in price of X
- Slutskian Approach
- Comparison of Slutskian and Hicksian Methods
- Effects of a price change for inferior goods
- Effect of a price change for a Giffen goods
- Giffen’s Paradox
- Inferior Goods

**Past Year Questions Discussed** **in this class/ recording :**

**IES 2011, Q 2**

Define income effect, substitution effect and price effect of any change in price. Show that the price effect can be decomposed into the income effect and the substitution effect.

**IES 2017, Q 1 (b)**

Define the method of Compensating Variation of lncome and the method of Cost Difference. Why is the latter method superior to the former one?

**IES 2012, Q 2**

Separate income effect from substitution effect for a price change using (i) Hicks’ method (ii) Slutsky’s method. Hence explain the difference between the two compensated demand curves.

**IES 2012, Q 1 (e)**

Suppose you have a demand function for milk of the form x_{1} = 100 + m/ 100 p_{1} and your weekly income (m) is ₹ 12,000 and the price of milk (p_{1}) is ₹ 20 per litre. Now suppose the price of milk falls from ₹ 20 to ₹ 15 per litre, then what will be the substitution effect?

**6. Slutsky equation and Demand Curve**

**Topic Discussed**:

- Slutsky Substitution effect
- Perfect Complements
- Perfect Substitutes
- Quasilinear Preference
- Rebating a Tax
- Compensated Demand Curve
- Normal Good

**7. Consumer Surplus Part 1 and Part 2 (IES 2010, 2011, 2014)**

**Topic Discussed**

- Consumer Surplus
- Producer Surplus
- Calculation of Equilibrium Price
- Calculation of Consumer Surplus and Producer Surplus

**Past Years Question**s **Discussed** **in this class/ recording :**

**IES 2011, Q 9**

If D = 250 – 50p and S = 25p + 25 are the demand and supply functions respectively, calculate the equilibrium price and the quantity. Hence calculate both consumer’s and producer’s surpluses under equilibrium.

**IES 2015, Q 1(e)**

Define consumer’s and producer’s surplus Given the demand function Pc = 113 – q^{2} and the supply function p = (q + 2)^{2} under perfect competition, find out the consumers’ surplus and producers’ surplus.

**IES 2014, Q 1(c)**

Other things equal, what happens to consumer surplus if the price of a good falls? Why? lllustrate using a demand curve.

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