Question

Explain consumer's equilibrium by utility analysis. OR When the price of a commodity decreases from Rs. 5 per unit to Rs. 4 per unit, its demand increases from 100 units to 110 units. Calculate elasticity of demand.

Hint:

For elasticity calculations, always use the initial price and quantity as the base (P and Q in the formula \(\frac{\Delta Q}{\Delta P} \times \frac{P}{Q}\)). A negative sign in the formula is often used to make the result a positive number, as price and quantity demanded are inversely related.

Answer 1

Step 1: Understanding the Concept:
Consumer's Equilibrium refers to a situation where a consumer, with their given income and market prices, spends their money on goods and services in such a way that they get the maximum possible satisfaction (utility). At this point, they have no incentive to change their consumption pattern. Utility analysis, also known as Cardinal Utility Analysis, assumes that utility can be measured in cardinal numbers (utils).

Step 2: Equilibrium in Case of a Single Commodity:
A consumer purchasing a single commodity will be at equilibrium when the marginal utility of the commodity in terms of money is equal to its price. The condition is: \[ \frac{MU_x}{P_x} = MU_m \] Where: \begin{itemize} \item \(MU_x\) is the Marginal Utility of good X. \item \(P_x\) is the Price of good X. \item \(MU_m\) is the Marginal Utility of Money (the utility of one rupee, assumed to be constant). \end{itemize} If \(\frac{MU_x}{P_x} > MU_m\), the consumer will buy more of X. If \(\frac{MU_x}{P_x} < MU_m\), they will buy less. Equilibrium is reached only when they are equal.

Step 3: Equilibrium in Case of Two or More Commodities (Law of Equi-Marginal Utility):
When a consumer is buying two or more commodities, the equilibrium condition is that the ratio of the marginal utility to the price must be the same for all commodities consumed. This is known as the Law of Equi-Marginal Utility or Gossen's Second Law. The condition for two goods, X and Y, is: \[ \frac{MU_x}{P_x} = \frac{MU_y}{P_y} = MU_m \] This means that the consumer gets the same marginal utility from the last rupee spent on each good. If the ratio is higher for good X than for good Y, the consumer will shift expenditure from Y to X until the ratios become equal.

Step 4: Final Answer:
Consumer's equilibrium under utility analysis is achieved when the marginal utility per rupee spent is equal for all goods purchased and is also equal to the marginal utility of money. This ensures that the consumer is maximizing their total satisfaction.
Solution (Calculation of Elasticity of Demand):

Step 1: Understanding the Concept and Formula:
The question asks to calculate the Price Elasticity of Demand (\(E_d\)). The percentage method is the most appropriate here. The formula is: \[ E_d = (-) \frac{\text{Percentage Change in Quantity Demanded}}{\text{Percentage Change in Price}} \] \[ E_d = (-) \frac{\frac{\Delta Q}{Q} \times 100}{\frac{\Delta P}{P} \times 100} = (-) \frac{\Delta Q}{\Delta P} \times \frac{P}{Q} \] Where: \begin{itemize} \item P = Initial Price \item Q = Initial Quantity \item \(\Delta P\) = Change in Price (\(P_1 - P\)) \item \(\Delta Q\) = Change in Quantity (\(Q_1 - Q\)) \end{itemize}

Step 2: Identifying the Given Values:
\begin{itemize} \item Initial Price (P) = ₹ 5 \item New Price (\(P_1\)) = ₹ 4 \item Initial Quantity (Q) = 100 units \item New Quantity (\(Q_1\)) = 110 units \end{itemize}

Step 3: Calculating the Changes:
\begin{itemize} \item Change in Price (\(\Delta P\)) = \(P_1 - P = 4 - 5 = -1\) \item Change in Quantity (\(\Delta Q\)) = \(Q_1 - Q = 110 - 100 = 10\) \end{itemize}

Step 4: Substituting the Values into the Formula and Calculating:
\[ E_d = (-) \frac{10}{-1} \times \frac{5}{100} \] \[ E_d = -(-10) \times \frac{5}{100} \] \[ E_d = 10 \times \frac{5}{100} \] \[ E_d = \frac{50}{100} = 0.5 \] Step 5: Final Answer and Interpretation:
The price elasticity of demand is 0.5.
Since \(E_d < 1\), the demand is inelastic. This means that the percentage change in quantity demanded (10%) is less than the percentage change in price (20%).