Question

Explain consumer's equilibrium with the help of indifference curves.
OR
When the price of a commodity increases from Rs. 10 per unit to Rs. 11 per unit, demand shrinks from 100 units to 90 units. Calculate elasticity of demand.

Hint:

For consumer equilibrium, the key idea is "tangency." The point where the budget line just touches the highest indifference curve is the solution. For elasticity calculations, set up your variables (P, Q, \(\Delta P\), \(\Delta Q\)) clearly before plugging them into the formula to avoid errors.

Answer 1

Step 1: Understanding the Concept:
Consumer's Equilibrium is a point of maximum satisfaction for a consumer. It is a situation where a consumer spends their limited income on different goods in such a way that their total utility is maximized. At this point, the consumer has no tendency to change their consumption pattern. Indifference curve analysis (or Ordinal Utility Analysis) explains this equilibrium.

Step 2: Tools and Conditions for Equilibrium:
The equilibrium is determined using two tools: \begin{enumerate} \item Indifference Map: This represents the consumer's preferences, with higher indifference curves (ICs) showing higher levels of satisfaction. \item Budget Line: This represents the consumer's income and the prices of goods, showing all affordable combinations. \end{enumerate} The consumer reaches equilibrium when two conditions are met: \begin{enumerate} \item The budget line is tangent to the highest possible indifference curve. At this point, the slope of the IC equals the slope of the budget line. \[ \text{MRS}_{xy} = \frac{P_x}{P_y} \] (Where MRS is the Marginal Rate of Substitution, and \(P_x, P_y\) are prices of goods X and Y). \item The indifference curve must be convex to the origin at the point of tangency. This implies a diminishing MRS. \end{enumerate}

Step 3: Explanation with Diagram:
\begin{center} \begin{tikzpicture}[scale=0.9] \draw[->] (0,0) -- (7,0) node[right] {Good X}; \draw[->] (0,0) -- (0,5) node[above] {Good Y}; \draw[thick, color=red] (0,4) -- (6,0) node[midway, above, sloped] {Budget Line (AB)}; \draw[color=blue, domain=0.8:6] plot (\x, {6/\x}) node[right] {$IC_1$}; \draw[color=blue, domain=1.2:6] plot (\x, {10/\x}) node[right] {$IC_2$}; \draw[color=blue, domain=2:6] plot (\x, {16/\x}) node[right] {$IC_3$}; \fill (3, 2) circle (2pt) node[above right] {E (Equilibrium)}; \draw[dashed] (3, 2) -- (3, 0) node[below] {$X^*$}; \draw[dashed] (3, 2) -- (0, 2) node[left] {$Y^*$}; \end{tikzpicture} \end{center} In the diagram, the consumer can afford any point on the budget line AB. Points on \(IC_1\) are attainable but offer less satisfaction. Points on \(IC_3\) offer more satisfaction but are unaffordable. The optimal choice is point E, where the budget line is tangent to the highest possible indifference curve, \(IC_2\). At this point, the consumer maximizes their satisfaction by purchasing \(X^*\) of Good X and \(Y^*\) of Good Y.
Solution (Calculation of Elasticity of Demand):

Step 1: Understanding the Concept and Formula:
The question asks for the Price Elasticity of Demand (\(E_d\)), which measures how responsive the quantity demanded is to a price change. We use the percentage method. \[ E_d = (-) \frac{\text{Percentage Change in Quantity Demanded}}{\text{Percentage Change in Price}} = (-) \frac{\Delta Q}{\Delta P} \times \frac{P}{Q} \] Where P = Initial Price, Q = Initial Quantity, \(\Delta P\) = Change in Price, and \(\Delta Q\) = Change in Quantity.

Step 2: Identifying the Given Values:
\begin{itemize} \item Initial Price (P) = Rupees 10 \item New Price (\(P_1\)) = Rupees 11 \item Initial Quantity (Q) = 100 units \item New Quantity (\(Q_1\)) = 90 units \end{itemize}

Step 3: Calculating the Changes:
\begin{itemize} \item Change in Price (\(\Delta P\)) = \(P_1 - P = 11 - 10 = 1\) \item Change in Quantity (\(\Delta Q\)) = \(Q_1 - Q = 90 - 100 = -10\) \end{itemize}

Step 4: Substituting the Values and Calculating:
\[ E_d = (-) \frac{-10}{1} \times \frac{10}{100} \] \[ E_d = -(-10) \times \frac{10}{100} \] \[ E_d = 10 \times \frac{10}{100} \] \[ E_d = \frac{100}{100} = 1 \] Step 5: Final Answer and Interpretation:
The price elasticity of demand is 1.
Since \(E_d = 1\), the demand is unitary elastic. This means the percentage change in quantity demanded (-10%) is exactly equal to the percentage change in price (+10%).