Partial Derivatives and Total Differentials

Partial Differentiation

For a function of several variables: \[y=f\left(x_{1}, x_{2}, \cdots, x_{n}\right)\]

If \(x_1\) changes by \(\Delta x_1\) but all other variables remain constant: \[\frac{\Delta y}{\Delta x_{1}}=\frac{f\left(x_{1}+\Delta x_1, x_{2}, \cdots, x_{n}\right)-f\left(x_{1}, x_{2}, \cdots, x_{n}\right)}{\Delta x_{1}}\]

Partial derivative of \(y\) with respect to \(x_i\): \[\frac{\partial y}{\partial x_{i}}= f_i = \lim_{\Delta x_{i} \rightarrow 0} \frac{\Delta y}{\Delta x_{i}}\]

Partial Derivatives

Gradient Vector

Gradient: vector of all partial derivatives of a function

\[\nabla f(x_1, x_2, \cdots, x_n) = [f_1, f_2, \cdots, f_n]^T\]

Example

\[y=f\left(x_{1}, x_{2}\right)=3 x_{1}^{2}+x_{1} x_{2}+4 x_{2}^{2}\]

\[\frac{\partial y}{\partial x_{1}} = f_1 = \hspace{8em}\]

\[\frac{\partial y}{\partial x_{2}} = f_2 = \hspace{8em}\]

Example

\[y=f(u, v)=(u+4)(3 u+2 v)\]

\[\frac{\partial y}{\partial u} = f_u = \hspace{8em}\]

\[\frac{\partial y}{\partial v} = f_v = \hspace{8em}\]

Production Function

\[Q = A K^{\alpha} L^{1-\alpha}\]

Marginal product of capital (MPK): \[\frac{\partial Q}{\partial K} = Q_K = \hspace{8em}\]

Marginal product of labor (MPL): \[\frac{\partial Q}{\partial L} = Q_L = \hspace{8em}\]

Differentials

Note that, \[\Delta y \equiv\left[\frac{\Delta y}{\Delta x}\right] \Delta x\]

Then for infinitesimal changes, \[d y \equiv\left[\frac{d y}{d x}\right] d x \quad \text{or} \quad d y=f'(x) d x\]

We will call \(d y\) and \(dx\) the differentials of \(y\) and \(x\), respectively.

Derivative as a Ratio

Given that, \[d y \equiv\left[\frac{d y}{d x}\right] d x \quad \text{or} \quad d y=f'(x) d x\]

We can think of \(f'(x)\) as a ratio of two quantities, \(dy\) and \(dx\).

Elasticity

An important quantity that economists love to calculate is the elasticity of a function.

Elasticity is defined as: \[\varepsilon = \frac{\text{Percentage change in } y}{\text{Percentage change in } x} = \frac{dy/y}{dx/x}\]

We can calculate this as: \[\varepsilon = \frac{dy}{dx} \cdot \frac{x}{y}\]

Elasticity

Elasticity: \[\varepsilon = \frac{dy}{dx} \cdot \frac{x}{y}\]

• \(|\varepsilon|>1\), elastic

• \(|\varepsilon|=1\), unit elasticity

• \(|\varepsilon|<1\), inelastic

Example

\[C = a + bY\]

Total Differential

For a function of \(n\) variables \[y=f\left(x_{1}, x_{2}, \cdots, x_{n}\right)\]

Total differential: \[d f=\frac{\partial f}{\partial x_{1}} d x_{1}+\frac{\partial f}{\partial x_{2}} d x_{2}+\cdots+\frac{\partial f}{\partial x_{n}} d x_{n}=\sum_{i=1}^{n} f_{i} d x_{i}\]

Note: we use \(\partial\) to tell partial derivatives apart from total derivatives. In particular, \[\frac{\partial f}{\partial x_{i}} = \left.\frac{d f}{d x_{i}} \right\vert_{\text{other variables are constant}}\]

Total Differential

Consider a savings function: \[S=S(Y, i)\] where \(S\) is savings, \(Y\) is national income, and \(i\) is the interest rate.

Total differential: \[d S=\frac{\partial S}{\partial Y} d Y+\frac{\partial S}{\partial i} d i\]

Example

\[y = 5 x_1^2 + 3x_2\]

Total Derivative

Total differential: \[d f= f_1 d x_{1}+ f_2 d x_{2}+\cdots+f_n d x_{n}\]

We can divide the total differential by \(dx_1\) to get the total derivative of \(f\) with respect to \(x_1\): \[\frac{d f}{dx_1} = f_1 + f_2 \cdot \frac{d x_{2}}{d x_{1}}+\cdots+f_n \cdot\frac{d x_{n}}{d x_{1}}\]

Total Derivative

Given the function \[y = f(x_1, x_2)\]

We are interested in how \(y\) changes with respect to \(x_1\), but \(x_2\) also depends on \(x_1\): \[x_2 = g(x_1)\]

We know that \(dy = f_1 dx_1 + f_2dx_2\). Dividing both sides by \(dx_1\), \[\frac{dy}{dx_1} = f_1+f_2 \cdot g'(x_1) = \frac{\partial y}{\partial x_1} + \frac{\partial y}{\partial x_2} \cdot \frac{d x_2}{dx_1}\]

A Variation on the Theme

For a function \[y=f\left(x_{1}, x_{2}, w\right), \quad x_{1}=g(w), \quad x_{2}=h(w)\]

The total derivative of \(y\) is given by \[\frac{d y}{d w}=\frac{\partial f}{\partial x_{1}} \frac{d x_{1}}{d w}+\frac{\partial f}{\partial x_{2}} \frac{d x_{2}}{d w}+\frac{\partial f}{\partial w}\]

Example

Let a production function be \[Q(t) = A(t) K(t)^{\alpha} L(t)^{1-\alpha}\] where \[K(t) = K_0 + at \qquad L(t) = L_0 + bt\]

Another Variation on the Theme

If a function is given, \[y=f\left(x_{1}, x_{2}, u, v\right)\] with \(x_{1}=g(u, v)\) and \(x_{2}=h(u, v)\).

Then, holding \(v\) constant, the partial total derivative with respect to \(u\) is \[\frac{\S y}{\S u}=\frac{\partial y}{\partial x_{1}} \frac{\partial x_{1}}{\partial u}+\frac{\partial y}{\partial x_{2}} \frac{\partial x_{2}}{\partial u}+\frac{\partial y}{\partial u}\]

Homework Questions

• Exercise 7.4: 1 (a), (d), 2 (a), (b), 3, 5, 7

• Exercise 8.1: 1 (a), 4, 6

• Exercise 8.2: 3 (a), 4, 5, 6, 7 (b), (f)

• Exercise 8.4: 2, 4

Textbook reference: 7.4, 8.1, 8.2, 8.4

Reminder: Quiz 2 on Monday.