Introduction to Economic Growth Art Slides Introduction to Economic Growth Third Edition by Charles I. Jones, Dietrich Vollrath

MATHEMATICAL REVIEW Art Slides Charles I. Jones Dietrich Vollrath APPENDIX A MATHEMATICAL REVIEW Charles I. Jones Dietrich Vollrath 2

Derivatives The derivative of some function f(x) with respect to x reveals how f() changes when x changes by a small amount. implies that a function is increasing For example, for a function f=5x the value of is equal to 5 everywhere. In general, if exists it is equal to the slope of the graph of f(x) at some point.

Time Derivatives and Growth It is often convenient to consider functions of time, e.g. f=f(t). Let the capital stock at time t be equal to . Time derivative of K is then equal to, by definition, Setting implies The time derivative is approximately equal to the value of investment into capital stock K between years t-1 and t.

Growth Rates Consider the following expression: It is easy to see that this fraction is equal to the growth rate of capital K. Indeed, We subtract 1 here in order to get the percentage figure. For example, if and then The growth rate is 1%.

Growth Rates and Time Derivatives Let us rewrite the growth rate expression as follows: The limit of the nominator is exactly the time derivative of K: The denominator is the value of capital at the start of the time period lasting . As a result, we can claim

Log Function A logarithmic function is the one that translates positive numbers into the real line: x is translated into the power y of some base b such that b raised into that power y results in x.

Natural Logarithms When the base of a logarithmic function is a special irrational number e, we write and we call it natural logarithms. We prefer using natural logarithms because they possess a useful property:

Rules for Logarithms The important rule in the growth context is this one: If y=y(t) then . In other words, differentiating the log function results in the growth rate.

Take Logs and Derivatives Consider a Cobb-Douglas production function: Take logs of both sides: Take derivatives of both sides:

Ratios and Growth Rates Suppose some ratio is constant: is constant. Take logs and derivatives of both sides: If the ratio of two variables is constant, the growth rate of those two variables must be the same.

Exponential Growth Suppose a variable is growing exponentially: For instance, y(t) could be the country’s per capita income. Taking logs, In the previous time period, at t-1, this equation will look like Subtract both sides of the second equation from the first one: The rate of exponential growth g is equal to the change in the log of the underlying variable y.

Exponential Growth and Percentage Change Compute the percentage change in y between t-1 and t: It can be demonstrated that for small values of x the following is true: Setting x=g, one obtains: The percentage change and change in logs can be thought of as approximately equivalent for small values of growth rates.

Integrals as Sums Imagine a production function written as: In case output is the weighted sum of a continuum of inputs ranging from 1 to 10, we can write an infinite sum as integral: Suppose that for each type of input i exactly 100 units are used:

Simple Integration Rules C here is any real number constant, and so are b and a.

Simple Differential Equations There is really one key differential equation in this course: That is, the growth rate of a variable x is equal to g. How would a graph of x(t) look like? Remember that growth rates are actually log derivatives: Now we need to integrate both sides.

Exponential Growth Equation Integrating both sides of this equation results in: Taking the exponential of both sides results in:

Initial Conditions and Constant We solved the differential equation up to an unknown constant: To find out what is equal to, we should employ information on initial conditions. For example, suppose . Substituting t=0 into the right-hand side of the original solution, In other words, the unknown constant is equal to x(0).

Exponential Growth

Continuous Compounding Suppose your bank provides you with a compound 5% interest every year: where t is the number of years, and 100 is the value of your initial deposit. Suppose now that the interest is compounded every instant. In this case, the growth of your deposit is described by Notice that at t=1 (after one year) This is how continuously compounded interest is roughly equivalent to the discretely compounded interest.

Maximization of a Function