Uses of Elementary CalculusArising in EconomicsKCB∗Contents1 Continuous compounding of interest 22 Continuously variable interest rates 43 First order linear differential equations 94 The Cobb–Douglas Model of Production 115 Constant Returns to Scale 136 Euler’s Theorem for homogeneous functions 147 The basic Solow Model of capital accumulation 178 Economic Growth in a Solow model 19∗When I took calculus, the books were chock full of examples and applications from physicsand engineering, but they had very few applications to economics or other social sciences. As aresult, many students believed that calculus, and more generally, mathematics, is useful only inthe natural sciences and engineering. Current textbooks are better in this regard, but nonethelessmany of my undergraduates are delighted to discover that they can use their mathematical aptitudein other ways. I have jotted down a few examples for use by calculus teachers who are looking forsomething different to show their students. These examples are mostly well-known to economists.11 Continuous compounding of interestWhen a dollar is invested at the annual rate of interest of r, after a year it is worth(1 + r). But most financial institutions compound interest more frequently thanannually. For instance, with monthly compounding, the interest rate is divided by12, and after the first month, the amount r/12 is credited to your savings account,and after two months the 1 +r12has earned another month’s interest (1 +r12)(r/12),so you have (1 +r12)2= (1 +r12) + (1 +r12)(r/12) in your account. By the end ofthe year, you have (1 +r12)12. You might think that if interest is compounded daily,the annual rate would be divided by 365, so the total return would be (1 +r365)365,but banking is an industry where it was once acceptable to use a 360-day year, soyour bank may have payed (1 +r360)360instead.If interest rate r is compounded n-fold annually, at the end of t years you haveC(r, t, n) = (1 +rn)ntfor every dollar invested.Let’s examine the properties of the function C(r, t, n).• If r1t1= r2t2, 0, writer2r1=t1t2= a, then C(r1, t1, n) = C(r2, t2, an).To see this, writeC(r1, t1, n) = (1 +r1n)nt1= (1 +r2an)ant2= C(r2, t2, an).• For r > 0, and t > 0, the limit limn→∞C(r, t, n), if it exists, depends only onthe product rt.Let r1t1= r2t2, wherer2r1=t1t2= a > 0. Thenlimn→∞C(r1, t1, n) = limn→∞C(r2, t2, an) = limn→∞C(r2, t2, n).• The limit limn→∞C(r, t, n) exists for each r, t.In fact, we shall compute the limit in Proposition 1.Continuous compounding is the limit of n-fold compounding as n → ∞.Proposition 1limn→∞C(r, t, n) = limn→∞(1 +rn)nt= ert.2Proof : Now1 +rnnt=eln1+rnnt= ent ln1+rnso we first find limn→∞nt ln1 +rn. Using Taylor’s Theorem1 +rn= ln(1) + ln0(1)rn+ R(n) =rn+ R(n),where the remainder R(n) satisfies R(n)/(rn) = nR(n)/r −−−−→n→∞0. Thuslimn→∞nt1 +rn= limn→∞ntrn+ R(n)= limn→∞rt + tnR(n) = rt.Since the exponential function is continuous,limn→∞(1 +rn)nt= limn→∞ent1+rn= elimn→∞nt1+rn= ert.The next proposition explains why a bank prefers to use a 360-day year forcalculating interest on deposits.Proposition 2 For r > 0, t > 0, and n > 0, the compounded return C(r, n, t) is anincreasing function of n. That is, more frequent compounding leads to a higherreturn.Proof : Again let us rewrite C asC(r, n, t) =1 +rnnt=eln1+rnnt= ent ln1+rn.While the economic interpretation of n is as the number of subdivisions per an-num, there is no mathematical reason that in the formula above that n needs tobe an integer. In fact we can go ahead and differentiate with respect to n. By thechain rule,ddnent ln1+rn= ent ln1+rnddnnt ln1 +rn= tent ln1+rn ln1 +rn+ n11 +rn−rn2!= tent ln1+rn ln1 +rn−rn1 +rn!> 0.3The last inequality follows from Lemma 3 immediately below. Since dC/dn > 0,the function C is an increasing function of n.Lemma 3 For x > 0,ln(1 + x) >x1 + x.Proof : Note that ln(1 + x) = ln(1 + x) − ln(1). By the Fundamental Theorem ofCalculus,ln(1 + x) − ln(1) =Z1+x1ddtln(t) dt.Nowddtln(t) = 1/t, and for 1 6 t < 1 + x we have 1/t > 1/(1 + x), soZ1+x11tdt >Z1+x111 + xdt.But 1/(1 + x) is independent of t, soZ1+x111 + xdt =11 + x(1 + x) − 1=x1 + x.Putting it all togetherln(1 + x) =Z1+x11tdt >Z1+x111 + xdt =x1 + xgives the desired conclusion.Proof of Lemma 3 based on convex analysis: It is well known that the logarithmfunction is strictly concave (its second derivative is strictly negative) and thatstrictly concave functions lie below their tangent lines. That is, if f is strictlyconcave and differentiable, f (y) < f (z) + f0(z)(y − z) for all y , z. Letting y = 1and z = 1 + x, for the log function we get ln(1) < ln(1 + x) − ln0(1 + x) · x for x , 0,which rearranges to give the conclusion.2 Continuously variable interest ratesFor this we’ll just consider a unit length of time, so t ∈ [0, 1]. Let r(t) denote theinstantaneous rate of interest at time t. By this I mean that the value at time4t = 1 of a dollar invested at time t = 0 is gotten by taking the limit as the numberof compounding periods tends to infinity of the following simple procedure.Over the interval between time t = (k − 1)/n and t = k/n, the investment earnsat a rate of interest equal to r(k − 1)/nif the rate is set prospectively or r(k/n)if the rate is set retrospectively. But since the length of the compounding periodis 1/n, the interest rate per compounding period must be divided by n. Thus theratio of the value at time t = (k − 1)/n and t = k/n is just1 +r(k/n)nin the retrospective case. Therefore the value at time t = 1 of a single dollarinvested at time t = 0 isnYk=11 +r(k/n)n.I claim that if r is continuous, then the limit islimn→∞nYk=11 +r(k/n)n= eR10r(t) dt.Note that this generalizes the continuous compounding case of the previoussection, which corresponds to the function r being constant.Note too that the continuous varying interest rate r delivers the same returnas continuous compounding of a constant rate equal to its simple time averageR10r(t) dt. Thus continuous time compounding of an instantaneous rate of returnconverts the complex geometric averaging of growth rates to a simple time aver-age rate of growth.Formally we have the following.Proposition 4 (Continuously variable rate of return) Let r : [0, 1] → R be