Question

2. (3 points total; 1 point each) Before working this problem, read the box on page 99 of the textbook. The rate of growth over time of some variable X, %AX, is just (new X-old X)/(old X)-AXX. Calculus! The rate of growth over time of X is also equal to the derivative of In(X) with regard to time Let t stand for time. Remember the chain rule Remember the derivative of In(X) 1/X And remember: a derivative is just change, so dX/dt is the change in X over the change int. And thug we havedoa da) ax 1 a.Aover And thus we have, growth rate of x Use the rules of natural logs and calculus to show that the rate of growth over time of a product oftwo variables, Xy, is the sum of the rates of growth of the variables x and y. a. Use the rules of natural logs and calculus to show that the rate of growth over time of a quotient of two variables, xły, is the difference between the rates of growth of the variables x and y. b. Use the rules of natural logs and calculus to show that the rate of growth over time of a variable x raised to a constant power b, x, is the constant b times the rate of growth of the variable x. c.

Answer 1

a.From the definition of growth rate, we have 1 + xyc = xt+1yt+1 xtyt Taking ln of both sides ln (1 + xyc) = ln xt+1yt+1 xtyt = ln xt+1 xt + ln yt+1 yt = ln (1 + ^x) + ln (1 + ^y) Recall that ln (1 + g) g for small g. Thus, the above equation is approximately xyc = ^x + ^y

b.From the definition of growth rate, we have 1 + dx y = xt+1 yt+1 = xt yt = xt+1 yt+1 yt xt = xt+1 xt = yt+1 yt Taking ln of both sides ln 1 + dx y ! = ln xt+1 xt ln yt+1 yt = ln (1 + ^x) ln (1 + ^y) Recall that ln (1 + g) g for small g. Thus, the above equation is approximately dx y = ^x

c. Let us suppose that y = x^b.Then %Δy = b(%Δx).

For example, if y = x², then the growth rate of y is twice the growth rate of x. If then the growth rate of y is half the growth rate of x (remembering that a square root is the same as a power of ½).

If the value of Y at time 0 equals Y₀ and if Y grows at the constant rate g (where g is an “annualized” or per year growth rate), then at time t (measured in years),Y_t = e^gtY₀.

e^gt = Y_t/Y₀,

which also means

gt = ln(Y_t/Y₀),

where ln() is the natural logarithm. You do not need to know exactly what this means; you can simply calculate a logarithm using a scientific calculator or a spreadsheet. Dividing by t we get the average growth rate

g = ln(Y_t/Y₀)/t.