SolowLectureNotes-1 - 1 Math preparation In this chapter we...

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1 Math preparation In this chapter we deal with the movement of the macroeconomic variables, such as GDP, over time. There are two ways of dealing with time in macroeconomics: discrete time and continuous time . Discrete-time models treat the time as integers: t = 0 , 1 , 2 , ... . Continuous-time models treat the time as real numbers. Here we will consider a continuous-time model. In a continuous-time model, we view a particular macroeconomic variable as a function of time . For example, when we consider GDP, instead of calling it as one number Y , we consider it as a function of time t , Y ( t ). In this way, we can capture the movement of GDP over time. For example, sometimes we want to see how much GDP changes per unit of time. Mathematically, the change of GDP can be viewed as the slope of Y ( t ) with respect to t . And we know that the slope can be expressed mathematically as a derivative . Therefore, the change of GDP per unit of time, measured at time t = dY ( t ) dt . Since we use this expression a lot, we use the short-cut expression by defining ˙ Y ( t ) dY ( t ) /dt . In usual mathematics, the use of is more common as a shortcut for a derivative (i.e. Y ( t ) dY ( t ) /dt ), but a convention in growth economics is to use the ˙ instead. For example, if the GDP is a linear function of t , that is, Y ( t ) = at (where a is a constant number), then the change of GDP per unit of time is equal to ˙ Y ( t ) = dY ( t ) /dt = a . If GDP is an exponential function of t , that is, Y ( t ) = exp( bt ) (where b is a constant), then the change of GDP per unit of time is equal to ˙ Y ( t ) = dY ( t ) /dt = b exp( bt ). Note that there is a economic meaning of ˙ Y ( t ). When ˙ Y ( t ) = 0, it means that the GDP is not changing over time. When ˙ Y ( t ) > 0, it means that the GDP is increasing over time. When ˙ Y ( t ) < 0, it means that the GDP is decreasing over time. Since we analyze economic growth in this chapter, we will have many occasions where we look at the growth rate. The growth rate of a variable is defined as the change of the variable divided by the level of the variable. For example, the growth rate of Y ( t ), g Y , is defined by g Y change of Y ( t ) level of Y ( t ) = ˙ Y ( t ) Y ( t ) . For example, when Y ( t ) = at , g Y = ˙ Y ( t ) /Y ( t ) = a/ ( at ) = 1 /t . When Y ( t ) = exp( bt ), g Y = ˙ Y ( t ) /Y ( t ) = b exp( bt ) / exp( bt ) = b . Therefore, when Y ( t ) is an exponential function of t , the growth rate of Y ( t ) is constant. There is another, very convenient, way of looking at the growth rate. Before explaining it, let’s refresh our memory of calculus a little bit. 1
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Chain rule: Suppose that a variable y is a function of a variable x : y = g ( x ). Also suppose that a variable z is a function of y : z = f ( y ). Therefore, z is eventually a function of x : z = f ( g ( x )). The right hand side, f ( g ( x )), is called a composite function . There is a rule called chain rule in calculus that allows us to di ff erentiate z with respect to x , that is, to take a derivative of a composite function. The formula is: d dx f ( g ( x )) = df ( y ) dy · dg ( x ) dx , where y = g ( x ) .
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