Mathematics in economics Lecture 8 Mgr. Jiří Mazurek, Ph.D. Mathematics in Economics/PMAT Lecture 8 Application of calculus in economics Consumer surplus and producer surplus Consumer surplus (CS) and producer surplus (PS) under perfect competition is given as follows: • • 1. Lecture 8 Consumer surplus and producer surplus Problem 1 • • 1. Find CS and PS if and . Solution: In the equilibrium demand equals supply: From this quadratic equation we obtain two roots: Q = 5 and Q = -10. However, the negative root is implausible, so QE=5. Lecture 8 Consumer surplus and producer surplus Problem 1 – cont. • • 1. Now, we can found CS and PS with the use of aforementioned integral formulas: Note, that results must be positive. Lecture 8 Consumer surplus and producer surplus • • 1. PE QE Cena za jednotku Přebytek výrobce (QE, PE) Množství P = D(Q) Q P P = S (Q) Přebytek spotřebitele A triangular area (upper arrow) is CS, lower arrow depicts PS. Lecture 8 A total revenue as an integral of an intensity • • 1. In some cases, a total revenue (TR) is given as a sum of an intensity (or a flow) f (t) during a given period of time: Example: Find the total revenue for the intensity From 1 to 15 years. Solution: = 324996 CZK. Lecture 8 Infinite number series • • 1. An infinite number series is given and denoted as follows: It is an infinite sum of numbers ai. • If the sum above (a series) has a finite sum, it is called convergent. • Otherwise (when the sum is infinite or does not exist) a series is called divergent. Lecture 8 Infinite number series • • 1. The most important problems are: • Is a series convergent or divergent? • • If a series is convergent, what is its sum? A historical note: recall the paradox of Achilleus and a turtle. In ancient Greece, it was assumed that a sum of an infinite series is always infinite. Lecture 8 Infinite number series Introduction example • • 1. Consider a division of a pizza so that in the first step ½ of pizza is eaten. In the second step ½ of what remained Is eaten, etc. This situation can be rewwritten as the infinite series: Unsurprisingly, although the series is infinite, its sum is 1. • Lecture 8 Infinite number series Another examples • • 1. Find the following sum (Grandi series): This sum does not exist. Find the following sum (harmonic series): The sum is infinite. Lecture 8 Infinite number series A necessary condition for convergence • • 1. A necessary condition for convergence can be stated as follows: However, this condition is not sufficient, as harmonic series fullfils the condition above, but its sum is infinite. Convergence or divergence is decided upon a suitable convergence criterion. Lecture 8 Infinite number series Criterions (tests) of convergence • • 1. The most common criteria or tests of convergence include: • Comparison test • • Ratio test • • Root test • • Integral test Lecture 8 Infinite number series Dirichlet series • • 1. Dirichlet series is convergent if . This series is often used in a comparison test. Example: proove that is convergent. Solution: Because each number of the given series is smaller than corresponding number of the series . However, this new series is convergent according to Dirichlet test. Hence, the given series is also convergent. Lecture 8 Infinite number series Ratio test • • 1. This test is used in a case when a series includes a factorial. Example: Decide whether the series is convergent. Conclusion: convergent. A decision table: If L < 1 … convergent, if L > 1 … divirgent, L= 1 … cannot be decided. Lecture 8 Infinite number series Root test • • 1. The root test is used when the series includes exponential factor. Example: Decide, whether the series is convergent. Solution: Because L < 1, the series is convergent. Lecture 8 Infinite number series Root test • • 1. The root test is used when the series includes exponential factor. Example: Decide, whether the series is convergent. Solution: Because L > 1, the series is divergent. Lecture 8 Infinite number series Integral test • • 1. The integral test can be considered the most universal. It is usually used when other tests fail. Example: Decide, whether the harmonic series is convergent. Solution: Because the integral is infinite, also the original series is infinite, so the series is divergent. Lecture 8 Infinite number series Leibniz test • • 1. Up to date, we focused on series with positive elements. Some series might be alternating: changing plus and minus Sings from element to element. For alternating series we can use Leibniz test: Alternating series is convergent if and . Lecture 8 Infinite number series Leibniz test - Example • • 1. Example: Decide, whether the series is convergent. Solution: We chceck whether both conditions are fullfilled: Both conditions are met, hence, the series is convergent. However, the series is not absolutely convergent. Lecture 8 Infinite number series Operations with series • • 1. Convergent series might be added or multiplied. Operations on convergence series include: Also, adding or removing a finite number of elements does not change convergence. Lecture 8 Infinite number series Problems to solve - 1 • • 1. Decide, whether the series is convergent: Lecture 8 Infinite number series Problems to solve - 2 • • 1. Decide, whether the series is convergent: Lecture 8 Infinite number series Problems to solve - 2 • • 1. Decide, whether the series is convergent: Lecture 8 What is next? • • 1. Next time, we will address the geometric series and their application in economics. The last two topics are infinite function series and differential equations. The last lecture is devoted to problem solving. Lecture 8 Thank you for your attention • • 1.