Question 12.1.3: Let D = D(p,m) denote the demand for a commodity as a functi......
Let D = D(p,m) denote the demand for a commodity as a function of price p and incomem. Suppose that price p and incomemvary continuously with time t, so that p = p(t) and m = m(t). Then demand can be determined as a function D = D(p(t),m(t)) of t alone.
Find an expression for \dot{D}/D, the relative rate of growth of D.
Using (12.1.1) we obtain
{\frac{\mathrm{d}z}{\mathrm{d}t}}=f_{1}^{\prime}(x,y){\frac{\mathrm{d}x}{\mathrm{d}t}}+f_{2}^{\prime}(x,y){\frac{\mathrm{d}y}{\mathrm{d}t}} (12.1.1)
{\dot{D}}={\frac{\partial D(p,m)}{\partial p}}{\dot{p}}+{\frac{\partial D(p,m)}{\partial m}}{\dot{m}}where we have denoted time derivatives by “dots”. The first term on the right-hand side gives the effect on demand that arises because the price p is changing, and the second term gives the effect of the change in m. We can write the relative rate of growth of D as
{\frac{\dot{D}}{D}}={\frac{p}{D}}{\frac{\partial D(p,m)}{\partial p}}{\frac{\dot{p}}{p}}+{\frac{m}{D}}{\frac{\partial D(p,m)}{\partial m}}{\frac{\dot{m}}{m}}={\frac{\dot{p}}{p}}{\mathrm{El}}_{p}D+{\frac{\dot{m}}{m}}{ \mathrm E}{\mathrm l}_{m}DSo the relative rate of growth is found by multiplying the relative rates of change of price and income by their respective elasticities, then adding.