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Microeconomics: Concept of Indifference Curve and Budget Line. Definition of Marginal Rate of Substitution. Mathematical Derivation of Budget Line, relationship of budget line with indifference curve...
The document explains the concepts of budget line, indifference curve, and marginal rate of substitution in microeconomics, detailing their equations and relationships. It outlines how shifts and swings of the budget line occur due to changes in prices and income, and it discusses the decomposition of substitution and income effects into price effects for normal goods. Additionally, the document touches on iso-quant and iso-cost lines.
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Microeconomics: Concept of Indifference Curve and Budget Line. Definition of Marginal Rate of Substitution. Mathematical Derivation of Budget Line, relationship of budget line with indifference curve...
- 1. Microeconomics Page 1 of7 1. Define Budget Line. Derive Budget Line. Budget Line can be defined as the locus of combination of good X and good Y that a consumer can afford at the specific prices and is constraint by the level of income. Thus, budget line contains two major parts, that is, availability of resource (income), and the requirements (demand). The general equation for the budget line is: 𝑀 = 𝑥𝑃𝑥 + 𝑦𝑃𝑦 − − − − − − − − 1 Here, M represents the total income of the consumer, so it is the total availability of resources. x and y denotes the quantity of good X and good Y. also, Px and Py represents the respective price of good X and good Y. thus, xPX + yPY represents the total requirement of a consumer, which is constraint by total income, M. Slope of the budget line can be obtained by converting the equation 1 in the form of y = mx + c. So, 𝑀 = 𝑥𝑃𝑥 + 𝑦𝑃𝑦 𝑦𝑃𝑦 = 𝑀 – 𝑥𝑃𝑥 𝑦 = 𝑀 𝑃𝑦 – ( 𝑃𝑥 𝑃𝑦 ) 𝑥 𝑦 = − 𝑃𝑥 𝑃𝑦 𝑥 + 𝑀 𝑃𝑦 So, 𝑆𝑙𝑜𝑝𝑒(𝑚) = − 𝑃𝑥 𝑃𝑦 Again, for X- and Y- intercepts of the budget line converting the equation 1 in the form 𝑥 𝑎 + 𝑦 𝑏 = 1 So, 𝑀 = 𝑥𝑃𝑥 + 𝑦𝑃𝑦 Dividing both sides by M, we get 𝑀 𝑀 = 𝑥𝑃𝑥 𝑀 + 𝑦𝑃𝑦 𝑀 1 = 𝑥 ( 𝑀 𝑃𝑥 ) + 𝑦 ( 𝑀 𝑃𝑦 ) So, 𝑋 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = 𝑀 𝑃𝑥 and 𝑌 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = 𝑀 𝑃𝑦
- 2. Microeconomics Page 2 of7 Thus, the slope of the budget line is − 𝑃𝑥 𝑃𝑦 , the budget line slopes downward and budget line passes through ( 𝑀 𝑃𝑥 , 0) 𝑎𝑛𝑑 (0, 𝑀 𝑃𝑦 ). In the figure, X and Y – axes measures the quantity of good X, and the quantity of good Y respectively. AB is budget line and it passes through ( 𝑀 𝑃𝑥 , 0) 𝑎𝑛𝑑 (0, 𝑀 𝑃𝑦 ). It is downward sloping as its slope is negative, that is, 𝑆𝑙𝑜𝑝𝑒 = − 𝑃𝑥 𝑃𝑦 . Alternative Question: Define and Derive Iso-cost line by assuming hypothetical labour and capital. ( 𝑀 𝑃𝑥 , 0) (0, 𝑀 𝑃𝑦 ) 𝐴 𝐵 𝑆𝑙𝑜𝑝𝑒 = − 𝑃𝑥 𝑃𝑦 𝑀 = 𝑥𝑃𝑥 + 𝑦𝑃𝑦 O X 𝐺𝑜𝑜𝑑𝑌 𝐺𝑜𝑜𝑑 𝑋 Y
- 3. Microeconomics Page 3 of7 2. Define Indifference Curve. Show the relationship between indifference curve and budget line. Indifference curve can be defined as the locus of combinations of various quantities of good X and Good Y that provides same level of utility. It is also known as line of equal satisfaction. Relationship between indifference curve and budget line is: Let us assume the total utility derived by a customer is 𝑈(𝑥, 𝑦). Similarly, M be the total income of the consumer, x and y be any two commodities, and Px and Py be the price of good X and good Y respectively. Let Z be the total utility, so Z = 𝑈(𝑥, 𝑦) − − − − − −𝐴 Partially differentiating equation A with respect to x and y, 𝜕𝑍 𝜕𝑥 = 𝜕𝑈 𝜕𝑥 𝜕𝑥 𝜕𝑈 𝜕𝑥 𝜕𝑥 = 0 − − − −𝐴. 1 Similarly, 𝜕𝑍 𝜕𝑦 = 𝜕𝑈 𝜕𝑦 𝜕𝑦 𝜕𝑈 𝜕𝑦 𝜕𝑦 = 0 − − − −𝐴. 2 Adding equation A.1 and A.2, we get, 𝜕𝑈 𝜕𝑥 𝜕𝑥 + 𝜕𝑈 𝜕𝑦 𝜕𝑦 = 0 𝜕𝑈 𝜕𝑥 𝜕𝑥 = − 𝜕𝑈 𝜕𝑦 𝜕𝑦 𝜕𝑦 𝜕𝑥 = − ( 𝜕𝑈 𝜕𝑥 ) ( 𝜕𝑈 𝜕𝑦 ) = − 𝑀𝑈𝑥 𝑀𝑈𝑦 So, the equation of budget line is 𝑀 = 𝑥𝑃𝑥 + 𝑦𝑃𝑦. Now using Lagrange Multiple (𝜆), Let Z be total utility derived by the customer being in budget constraint, 𝑍 = 𝑈(𝑥, 𝑦) + 𝜆(𝑀 − 𝑥𝑃𝑥 − 𝑦𝑃𝑦) 𝑍 = 𝑈(𝑥, 𝑦) + 𝜆𝑀 − 𝜆𝑥𝑃𝑥 − 𝜆𝑦𝑃𝑦 − − − −1 Partially differentiating equation 1 with respect to x,
- 4. Microeconomics Page 4 of7 𝜕𝑍 𝜕𝑥 = 𝜕𝑈 𝜕𝑥 + 0 − 𝜆𝑃𝑥 𝜕𝑥 𝜕𝑥 − 0 0 = 𝜕𝑈 𝜕𝑥 − 𝜆𝑃𝑥 − − − − − −1.1 Partially differentiating equation 1 with respect to y, 𝜕𝑍 𝜕𝑦 = 𝜕𝑈 𝜕𝑦 + 0 − 0 − 𝜆𝑃𝑦 𝜕𝑦 𝜕𝑦 0 = 𝜕𝑈 𝜕𝑦 − 𝜆𝑃𝑦 − − − − − −1.2 Partially differentiating equation 1 with respect to 𝜆, 𝜕𝑍 𝜕𝜆 = 0 − 0 + (𝑀 − 𝑥𝑃𝑥 − 𝑦𝑃𝑦) ( 𝜕𝜆 𝜕𝜆 ) 0 = 𝑀 − 𝑥𝑃𝑥 − 𝑦𝑃𝑦 − − − 1.3 Solving equation 1.1 and 1.2 as it contains 𝜆. From Equation 1.1, 𝜕𝑈 𝜕𝑥 − 𝜆𝑃𝑥 = 0 −𝜆𝑃𝑥 = − 𝜕𝑈 𝜕𝑥 𝜆 = 𝜕𝑈 𝜕𝑥 × 1 𝑃𝑥 − − − − − − − 1.1.1 From Equation 1.2, 𝜕𝑈 𝜕𝑦 − 𝜆𝑃𝑦 = 0 −𝜆𝑃𝑦 = − 𝜕𝑈 𝜕𝑦 𝜆 = 𝜕𝑈 𝜕𝑦 × 1 𝑃𝑦 − − − − − − − 1.2.1 Equating equation 1.1.1 and 1.2.1, We get, 𝜕𝑈 𝜕𝑥 × 1 𝑃𝑥 = 𝜕𝑈 𝜕𝑦 × 1 𝑃𝑦 ( 𝜕𝑈 𝜕𝑥 ) ( 𝜕𝑈 𝜕𝑦 ) = 𝑃𝑥 𝑃𝑦 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 𝑀𝑈𝑥 𝑀𝑈𝑦 = 𝑃𝑥 𝑃𝑦 ,that is, if the slope of indifference curve equals to the slope of budget line, the customer is in maximum satisfaction.
- 5. Microeconomics Page 5 of7 In the figure, OX and OY measures quantity of good X and quantity of good Y respectively. AB budget line is drawn and it is downward sloping. IC, IC1, IC2 are the indifference curves. The indifference curve IC is tangent to budget line AB, that is, slope of IC equals to slope of the budget line. IC2 is higher than IC, so IC2 derives higher utility. So, the consumer wills to achieve higher utility at IC2 but he has no sufficient income, so he is constrained by total income. On other hand, the consumer can be at IC1 but it derives lower satisfaction than IC, thus he discards IC1. Hence, the consumer maximizes his satisfaction at IC. Alternative question Define Iso-Quant. Show the relationship between iso-quant curve and iso-cost line. 3. What is MRS? MRS = 2, what does it mean? Marginal Rate of Substitution (MRS) is the amount of good (Say good Y) that a consumer wills to give up to consume some extra unit of other good (Say good X) while maintaining same level of utility. 𝑀𝑅𝑆 = − ∆𝑦 ∆𝑥 = − 𝑀𝑈𝑥 𝑀𝑈𝑦 𝑀𝑅𝑆 = − ∆𝑦 ∆𝑥 = −2 1 , so it means that the consumer wills to give up 2 units of good Y to consume one extra unit of good Y. Alternative Question What is MRTS? MRTS = 3, what does it mean? 4. What are the causes of shift and swing of budget line? The causes of swing of budget line are: 1. Rightward swing is due to decrease in price of good X. 2. Leftward swing is due to increase in price of good X. 3. Upward swing is due to decrease in price of good Y. 4. Downward swing is due to increase in price of good Y. The causes of shift in budget line are: 1. Rightward/Outward/Upward shift is due to increase in income of consumer. 2. Leftward/Inward/downward shift is due to decrease in income of consumer. Alternative Question What are the causes of shift and swing of iso-cost line? ( 𝑀 𝑃𝑥 , 0) (0, 𝑀 𝑃𝑦 ) 𝐴 𝐵 𝑆𝑙𝑜𝑝𝑒 = − 𝑃𝑥 𝑃𝑦 = − 𝑀𝑈𝑥 𝑀𝑈𝑦 O X 𝐺𝑜𝑜𝑑𝑌 𝐺𝑜𝑜𝑑 𝑋 IC IC1 IC2 Y
- 6. Microeconomics Page 6 of7 5. Illustrate the Decomposition of substitution and income effects into Price Effect for normal goods. In the figure, OX and OY are the X- and Y- Axes. X-axis measures good X and Y-axis measures good Y. Initially, the AB budget line is constructed, and an indifference curve (IC1) is drawn tangent to budget line AB. Thus, the consumer consumes Q1 quantity of good X. Suppose, the price of good X decreases. With the decrease in the price of good X, the consumer can purchase more units of good X that previous. So, the budget line AB swung outward and formed new budget line AE. The new indifference curve (IC2) is formed tangent to new budget line AE. Thus, the consumer consumes Q2 unit of good X. Thus, the increase in the consumption of good X from Q1 to Q2 is called price effect. Now, let us suppose that income of consumer that decreases, the budget line has shifted inward to CD. The new budget line CD is parallel to AE. The line CD is tangent to IC1 at point W. thus, a consumer consumers Q3 units of Good X. Hence, the shift of Consumption from Q1 to Q3 is called substitution effect because the consumer has substituted some amount of good Y to consume some extra amount of good X. It is substitution effect as combination V(QX,QY) and W(QX,QY) lies on same Indifference curve i.e. both the combinations gives same amount of utility. Finally, the increase in the consumption of Good X from Q2 to Q3 is income effect. It is income effect because combination Z (QX, QY) is tangent to AE, and W (QX, QY) tangent to CD, AE is parallel to CD and AE is above CD represents the increase in income. Thus, shift of consumer’s choice from combination W (QX, QY) to Z (QX, QY) is an income effect. Price effect = OQ2 − OQ1 = 𝐐𝟐𝐐𝟏 Substitution effect = 𝐎𝐐𝟑 − 𝐎𝐐𝟏 = Q3Q1 Income effect = 𝐎𝐐𝟐 − 𝐎𝐐𝟑 = Q2Q3 Price Effect = Substitution Effect + Income Effect Q 2 Z Q 3 W O X Y ED C B A IC 2 IC 1 Q 1 V Quantity of good X QuantityofgoodY Substitution Effect Price Effect Income Effect
- 7. Microeconomics Page 7 of7 Price Effect = 𝐎𝐐𝟑 − 𝐎𝐐𝟏 + 𝐎𝐐𝟐 − 𝐎𝐐𝟑 Price Effect = OQ2 − OQ1 = 𝐐𝟐𝐐𝟏 Price Effect is also called Total Effect. Alternative Question Illustrate the Decomposition of substitution and income effects into Price Effect for inferior goods. Illustrate the Decomposition of substitution and income effects into Price Effect for giffen goods.