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At first, the population growth rate is fast, but it begins to slow as the population grows until it levels off to the maximum growth rate, after which it begins to decrease (figure 2). The equation for figure 2 is the differential of equation 1.1 ( Verhulst's 1838 growth model ): [ 13 ]
The standard logistic function is the logistic function with parameters =, =, =, which yields = + = + = / / + /.In practice, due to the nature of the exponential function, it is often sufficient to compute the standard logistic function for over a small range of real numbers, such as a range contained in [−6, +6], as it quickly converges very close to its saturation values of 0 and 1.
For example, with an annual growth rate of 4.8% the doubling time is 14.78 years, and a doubling time of 10 years corresponds to a growth rate between 7% and 7.5% (actually about 7.18%). When applied to the constant growth in consumption of a resource, the total amount consumed in one doubling period equals the total amount consumed in all ...
Originally developed for growth modelling, it allows for more flexible S-shaped curves. The function is sometimes named Richards's curve after F. J. Richards , who proposed the general form for the family of models in 1959.
The exchange rate disconnect puzzle: The exchange rate disconnect puzzle, also one of the so-called real exchange rate puzzles, concerns the weak short-term feedback link between exchange rates and the rest of the economy. In most economies, the exchange rate is the most important relative price, so it is surprising, and thus far unexplained ...
When calculating or discussing relative growth rate, it is important to pay attention to the units of time being considered. [ 2 ] For example, if an initial population of S 0 bacteria doubles every twenty minutes, then at time interval t {\displaystyle t} it is given by solving the equation:
If we ignore the problem of how consumption is distributed, then the rate of utility is a function of aggregate consumption. That is, U = U ( C , t ) {\displaystyle U=U(C,t)} . To avoid the problem of infinity, we exponentially discount future utility at a discount rate ρ ∈ ( 0 , ∞ ) {\displaystyle \rho \in (0,\infty )} .
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics.Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, or other computational methods.