Derivatives of Exponential and Logarithmic Functions Chapter 13

Derivative of Natural Exponent Rule 13.1 The derivative of a natural exponential function is the natural exponential function itself: Rule 13.2 Example 13.1

Continuous Growth Consider the continuous growth formula: By how much will the value of your deposit grow if you wait for one more year, given that you have already waited for x years? After x years the value of your deposit is , and the marginal increase can be approximated by differential:

Derivative of the Natural Logarithmic Function Rule 13.3 Rule 13.4: Rate of growth

Cobb-Douglas Production Function Consider the following Cobb-Douglas production function: where K is capital, L is labor, and A is a technological parameter. Take logs of both sides: What is the interpretation of and ? Differentiate both sides: It follows that , which is the elasticity of output with respect to capital input.

13.5 Discrete Growth Absolute change in y, an economic variable that grows over time (e.g. GDP), is denoted as . Proportionate change in y is defined as The absolute change is expressed in terms of its proportion in terms of the initial value . Example. Suppose . The absolute change will be , and the proportionate change will be

Rate of Proportionate Change Definition. Consider an economic variable y whose proportionate change is given by . Suppose that y=y(x), for example, time. The rate of proportionate change of y is defined as . The rate of proportionate change is often called rate of growth. Note. Rate of proportionate change is equal to the proportionate change itself in case . However, in case is large (long) enough, the rate of proportionate change is likely to result in the biased estimates of growth since the growth gets evenly distributed across all the subperiods of , ignoring the compounding effects.

Example: UK GDP Growth Growth rate between 1995 and 1996: 2.46%=34.37%/14, which means we distribute total cumulative growth over 14 years without accounting for compounding.

UK GDP Growth If the UK GDP grows by 2.46% every year relative to the base year value of 946,780, it means that the proportionate rate of growth is declining every year, which we do not observe in real world data. Let us assume that the UK GDP grew by an equal proportionate amount each year: Setting y=1296,390, a=964,780, and x=14, we can solve for r, the compound growth rate: r=2.13%<2.46%: compounding effects decrease the yearly growth rate significantly.

Logs and Growth Rates Suppose an economic variable y=y(t), i.e. it evolves over time. Taking logs and differentiating this function, we obtain: which is exactly the rate of proportionate change, or the growth rate in y. The growth rate of a variable y=y(t) is thus easily obtained by differentiating the log of this variable.

13.6 Continuous Growth Consider a proportionate growth formula: It has difference quotient in it, which is equal to the slope of chord JK. The growth rate r computed in this fashion is thus the average slope of function y=f(x) between J and K. However, function y=f(x) can be anything between these two points.

Instantaneous Rate of Growth Take the discrete growth formula to the limit: Definition. The rate of growth r defined as is called the instantaneous rate of growth of variable y. Note. The instantaneous rate of growth depends only on the point of its measurement , so it is uniquely defined at any point on the function y=y(x)’s graph.

Instantaneous Rate of Growth: Examples Consider a linear function y=a+bx. For a linear function, the instantaneous rate of growth declines with time. Consider a continuously compounded growth: The instantaneous rate of growth is given by: The instantaneous rate of growth in case of continuous compounding is equal to r!

13.8 Semi-Log Graphs Recap: Remember that in order to deal with plotting fast-growing functions, we use semi-log graphs where we have ln(y) rather than just y on the vertical axis. What is the slope of a curve on the semi-log graph? The answer is given by , which is exactly the instantaneous growth rate! Note. For the natural logs only, the rate of absolute change is equal to the rate of proportionate change . Increasing slope of a semi-log graph indicates an increasing rate of growth, decreasing slope corresponds to decreasing rate of growth. Increasing slope of the original graph tells us nothing about what is happening to the growth rate.

Log-Linear Functions Log the continuously compounded growth formula: This is a semi-log graph of the function , and it is a straight line. Such functions are called log-linear functions. The rate of proportionate change, or the growth rate, is constant at all times for the log-linear functions.

13.10 Log Scales and Elasticity Consider the following demand function: . Take logs of both sides: Differentiate both sides: Rearranging, one obtains: , or the price elasticity of demand is equal to . The variable on the horizontal axis is also logged!