Question

The demand function (\(Q^D_x\)) and supply function \((Q^S_x)\) are given as:
\(Q^D_x=f(P_x,I)\ and\ Q^S_x=g(P_x,A)\) where I (Income) and A (Advertisement expenses) are the exogenous factors affecting quantity demanded and supplied, respectively. Further, \(\frac{\partial f}{\partial P_x}\lt0,\frac{\partial g}{\partial P_x}\gt0\) but \(\frac{\partial f}{\partial I}\) and \(\frac{\partial g}{\partial A}\) may have any sign. Considering that there exists an equilibrium \((Q_x^D=Q_x^S=Q)\), which of the following is/are CORRECT?

• A. \(e_{P_xA}=(\frac{\partial g}{\partial A}\frac{A}{Q})/(\frac{\partial f}{\partial P_x}\frac{P_x}{Q}-\frac{\partial g}{\partial P_x}\frac{P_x}{Q})\)
• B. \(\frac{dP_x}{dA}=(\frac{\partial g}{\partial A})/(\frac{\partial f}{\partial P_x}-\frac{\partial g}{\partial P_x})\)
• C. \(e_{P_xI}=(\frac{\partial g}{\partial I}\frac{I}{Q})/(\frac{\partial f}{\partial P_x}\frac{P_x}{Q}-\frac{\partial g}{\partial P_x}\frac{P_x}{Q})\)
• D. The sign of \(\frac{dP_x}{dA}\) does not depend on \(\frac{\partial g}{\partial A}\).

Answer 1

The Correct Option is A, B

To solve the problem, let's first comprehend the demand and supply functions presented:

• The demand function is \(Q^D_x = f(P_x, I)\), which indicates that quantity demanded is a function of price (\(P_x\)) and income (\(I\)).
• The supply function is \(Q^S_x = g(P_x, A)\), indicating quantity supplied depends on price (\(P_x\)) and advertisement expenses (\(A\)).

Key derivatives are provided: \(\frac{\partial f}{\partial P_x} < 0\) (Price has an inverse relationship with demand) and \(\frac{\partial g}{\partial P_x} > 0\) (Price has a direct relationship with supply).

Equilibrium is when \(Q^D_x = Q^S_x = Q\). We need to evaluate the given statements based on these insights:

The first statement is that the cross elasticity \(e_{P_xA}\) can be expressed as: \[ e_{P_xA} = \frac{\left(\frac{\partial g}{\partial A}\right)\left(\frac{A}{Q}\right)}{\left(\frac{\partial f}{\partial P_x}\right)\left(\frac{P_x}{Q}\right) - \left(\frac{\partial g}{\partial P_x}\right)\left(\frac{P_x}{Q}\right)} \] This is a mathematical interpretation of the elasticity of price concerning advertisement expenditure (A). It reflects how the quantity changes as we alter advertisement expenses, provided the equilibrium price changes accordingly. This expression is correct.

The second statement is the derivative of price concerning advertisement expenditure, expressed as: \[ \frac{dP_x}{dA} = \frac{\frac{\partial g}{\partial A}}{\frac{\partial f}{\partial P_x} - \frac{\partial g}{\partial P_x}} \] This derivative shows the rate of change of price with respect to advertisement expenses, holding the equilibrium condition (quantity demanded equals quantity supplied). This expression is also correct, as it is derived from the equilibrium dynamics.

The third statement about \(e_{P_xI}\), the cross elasticity with respect to income, is incorrect since the question only involves analyzing the advertisement impact (\(A\)) and not income (\(I\)).

The fourth statement claims that the sign of \(\frac{dP_x}{dA}\) does not depend on \(\frac{\partial g}{\partial A}\). This is incorrect because the numerator of \(\frac{dP_x}{dA}\) is \(\frac{\partial g}{\partial A}\). Hence, the sign is directly dependent on \(\frac{\partial g}{\partial A}\).

The correct options are the first and the second, as correctly interpreted equations provided for elasticity and derivative relate to advertisement expenses in equilibrium.