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Section 12.3 Partial elasticity
Objectives
Measure how a multivariable function changes when only one input varies.
Interpret partial derivatives visually using traces with one variable held constant.
Compute partial derivatives by treating other variables as fixed and applying single-variable rules.
Recall single-variable elasticity:
\(\rdv{y}{x} = \dfrac{\dd{y}/y}{\dd{x}/x} = \DS\dv{y}{x}\cdot\frac{x}{y}\)
Define partial elasticity of \(z=f(x,y)\) with respect to \(x\) as
\begin{equation*}
\rpdv{z}{x} = \frac{\rd_x z}{\rd x} = \frac{\mathrm{d}_x{z}/z}{\dd{x}/x} = \pdv{z}{x}\cdot\frac{x}{z}
\end{equation*}
and with respect to \(y\) as
\begin{equation*}
\rpdv{z}{y} = \frac{\rd_y z}{\rd y} = \frac{\mathrm{d}_y{z}/z}{\dd{y}/y} = \pdv{z}{y}\cdot\frac{y}{z}
\end{equation*}
A rate law for a chemical reaction might be \(r = k [A]^{\alpha} [B]^{\beta}\text{,}\) where \([A]\) and \([B]\) are concentrations of reactants. Then the partial elasticity of the rate with respect to \([A]\) is
\begin{equation*}
\rpdv{r}{[A]} = \alpha
\end{equation*}
and the partial elasticity of the rate with respect to \([B]\) is
\begin{equation*}
\rpdv{r}{[B]} = \beta\text{.}
\end{equation*}
For example, if
\(r = k [A]^{2} [B]^{3}\text{,}\) then the partial elasticity with respect to
\([A]\) is 2 and the partial elasticity with respect to
\([B]\) is 3. This means that a 1% increase in
\([A]\) will lead to a 2% increase in the reaction rate, while a 1% increase in
\([B]\) will lead to a 3% increase in the reaction rate.
Utility function in economics: \(z = U(x,y)\text{,}\) where \(x\) and \(y\) are quantities of two goods. Then the partial elasticity of utility with respect to good 1 is
\begin{equation*}
\rpdv{U}{x} = \pdv{U}{x}\cdot\frac{x}{U}
\end{equation*}
and the partial elasticity of utility with respect to good 2 is
\begin{equation*}
\rpdv{U}{y} = \pdv{U}{y}\cdot\frac{y}{U}\text{.}
\end{equation*}
Cobb-Douglas production function:
\(z = A K^{\alpha} L^{\beta}\)
Here,
\(K\) and
\(L\) are inputs (capital and labor), and
\(A\) is total factor productivity.
Partial elasticities:
\(\DS\rpdv{z}{K} = \alpha\) and
\(\DS\rpdv{z}{L} = \beta\)
Suppose two goods have quantities demanded given by \(q_1=f(p_1,p_2)\) and \(q_2=g(p_1,p_2)\text{,}\) where \(p_1\) and \(p_2\) are their prices. Then the cross-price elasticity of demand for good 1 with respect to the price of good 2 is
\begin{equation*}
\rpdv{q_1}{p_2} = \pdv{q_1}{p_2}\cdot\frac{p_2}{q_1}
\end{equation*}
and the own-price elasticity of demand for good 1 is
\begin{equation*}
\rpdv{q_1}{p_1} = \pdv{q_1}{p_1}\cdot\frac{p_1}{q_1}\text{.}
\end{equation*}
