Notes Week 3 - Course Coordinator: Dr Yaping Shan PDF

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Course Coordinator: Dr Yaping Shan...

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Introduction to Mathematical Economics Week 3 2.3 Indices and logarithms  If 𝑀 =𝑏 𝑛 We say that 𝑏 𝑛 is the exponential form of 𝑀 to base of 𝑏. The number 𝑛 is then referred to as the index, power or exponent. 𝑏 2 = 𝑏 ×𝑏 𝑏 3 = 𝑏 ×𝑏 ×𝑏 And in general 𝑏𝑛 = 𝑏⏟×𝑏 ×𝑏 ×𝑏 ×… 𝑏 𝑎 𝑡𝑜𝑡𝑎𝑙 𝑜𝑓 𝑛 𝑏′ 𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑑 𝑡𝑜𝑔𝑒𝑡ℎ𝑒𝑟

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 Negative powers are evaluated by taking the reciprocal of the corresponding power. Define 𝑏0 = 1 And 1 𝑏𝑛 where 𝑛 is any positive whole number. 𝑏 −𝑛=

Example Evaluate (a) 32 (b) (−2)6 (c) 3−4 (d) (1.723)0

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 Assume n is a positive whole number, we define 𝑏1/𝑛= 𝑛𝑡ℎ 𝑟𝑜𝑜𝑡 𝑜𝑓 𝑏 By this we mean that 𝑏1/𝑛 is a number which, when raised to the power n, produces b. 91/2= 𝑠𝑞𝑢𝑎𝑟𝑒 𝑟𝑜𝑜𝑡 𝑜𝑓 9 = 3 (𝑏𝑒𝑐𝑎𝑢𝑠𝑒 32 =9) 81/3= 𝑐𝑢𝑏𝑒 𝑟𝑜𝑜𝑡 𝑜𝑓 8 = 2 (𝑏𝑒𝑐𝑎𝑢𝑠𝑒 23 =8) 6251/4= 𝑓𝑜𝑢𝑟𝑡ℎ 𝑟𝑜𝑜𝑡 𝑜𝑓 625= 5 (𝑏𝑒𝑐𝑎𝑢𝑠𝑒 54 = 625)

 For the case 𝑏 𝑚 where m is a general fraction form 𝑝/𝑞 for some whole numbers p and q. 163/4 163 = 16× 16× 16= 4096 163/4=(4096)1/4=8 or 161/4=2 163/4=23 =8 We define 𝑏 𝑝/𝑞=(𝑏 𝑝 )1/𝑞=(𝑏1/𝑞 )𝑝

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Example Evaluate (a) 84/3 (b) 25−3/2

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2.3.2 Rules of indices 𝑅𝑢𝑙𝑒 1: 𝑏 𝑚 ×𝑏 𝑛 =𝑏 𝑚+𝑛 𝑅𝑢𝑙𝑒 2: 𝑏 𝑚 ÷𝑏 𝑛 =𝑏 𝑚−𝑛 𝑅𝑢𝑙𝑒 3: (𝑏𝑚 )𝑛 =𝑏 𝑚𝑛 𝑅𝑢𝑙𝑒 4: (𝑎𝑏)𝑛 =𝑎𝑛 𝑏𝑛

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𝑎

 ( )𝑛 = (𝑎×1 )𝑛 =𝑎𝑛 (1 𝑛 𝑎𝑏𝑛𝑛 ) = 𝑏 𝑏 𝑏 Example Simplify (a) 𝑥 1/4×𝑥 3/4

(b)

𝑥2 𝑦3 𝑥4 𝑦

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 The output, 𝑄, of any production process depends on a variety of inputs, known as factors of production. For example land, capital, labour and enterprise. For simplicity, we focus on capital and labour.  Capital, 𝐾, denotes all man-made aids to production.  Labour, 𝐿, denotes all paid work in the production process. 𝑄 =𝑓(𝐾. 𝐿) is called a production function. For example: 𝑄 =100𝐾 1/3 𝐿1/2

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 In general. A function 𝑄 =𝑓(𝐾. 𝐿) is said to be homogeneous if 𝑓(𝜆𝐾 , 𝜆𝐿) =𝜆𝑛 𝑓(𝐾, 𝐿) for some number 𝑛. The power 𝑛 is called the degree of homogeneity.  If the degree of homogeneity satisfies: 𝑛 1, the function is said to be display increasing return to scale. 8

Example Show that the following production function is homogeneous and find it degree of homogeneity 𝑄 =2𝐾 1/2 𝐿3/2 Does this function exhibit decreasing return to scale, constant return to scale or increasing return to scale?

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 Cobb-Douglas production function: 𝑄 =𝐴𝐾 𝛼 𝐿𝛽  Cobb-Douglas production function are homogeneous of degree 𝛼 +𝛽.  Cobb-Douglas production function exhibit decreasing return to scale, if 𝛼 +𝛽 < 1 constant return to scale, if 𝛼 +𝛽 = 1 increasing return to scale, if 𝛼 +𝛽 > 1

2.3.3 Logarithms  If 𝑀 =𝑏 𝑛 then log 𝑏 𝑀 =𝑛. where 𝑛 is called the logarithm of 𝑀 to base 𝑏. Example Evaluate (a) log 3 9 (b) log 4 2 (c) log7 1 ⁄7

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 The rules of logarithm are as follows: 𝑅𝑢𝑙𝑒 1: log 𝑏 (𝑥 ×𝑦) =log 𝑏 𝑥 +log 𝑏 𝑦 𝑅𝑢𝑙𝑒 2: log 𝑏 (𝑥 ÷𝑦) =log 𝑏 𝑥 −log𝑏 𝑦 𝑅𝑢𝑙𝑒 3: log 𝑏 𝑥 𝑚 =𝑚log 𝑏 𝑥 Example Use the rules of logarithms to express each of the following as a single logarithm: (a) log 𝑏 𝑥 +log 𝑏 𝑦 −log 𝑏 𝑧

(b) 2log 𝑏 𝑥 −3log 𝑏 𝑦

 log 𝑏 (𝑥 +𝑦) ≠ log 𝑏 𝑥 +log 𝑏 𝑦 11

Example Find the value of x which satisfy (a) 200(1.1)𝑥 = 20000

(b) 5𝑥 =2(3)𝑥

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2.4 The exponential and natural logarithm functions Example Sketch the graphs of the function (a) 𝑓(𝑥 ) =2𝑥 (b) 𝑔(𝑥 ) =2−𝑥

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 There is a whole class of functions, each corresponding to a different base, 𝑏. Of particular interest is the case when 𝑏 takes the value 2.718281828459…  The number is written as 𝑒 and the function 𝑓(𝑥 ) =𝑒 𝑥 is referred to as the exponential function. Example Evaluate the expression 1 𝑚 (1+ ) 𝑚 When m=1,10,100 and 1000.

 𝑒= lim (1+ 𝑚→∞

1 𝑚

)𝑚 15

Example The percentage, y, of households possessing refrigerators, t years after they have been introduced in a developed country is modelled by 𝑦 =100− 95𝑒 −0.15𝑡 (1) Find the percentage of households that have refrigerators (a) At their launch (b) After 1 year (c) After 10 years (d) After 20 years (2) What is the market saturation level? (3) Sketch a graph of 𝑦 against 𝑡.

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 We call logarithms to base 𝑒 natural logarithms. Rather than writing log 𝑒 𝑀 we simply put ln 𝑀.  The three rules of logs can then be stated as 𝑅𝑢𝑙𝑒 1: ln(𝑥 ×𝑦) = ln 𝑥 +ln 𝑦 𝑅𝑢𝑙𝑒 2: ln(𝑥 ÷𝑦) = ln 𝑥 −ln 𝑦 𝑅𝑢𝑙𝑒 3: ln 𝑥 𝑚 = 𝑚ln 𝑥

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Example Use the rules of logs to express (a) ln

𝑥 √𝑦

in terms of ln 𝑥 and ln 𝑦

(b) 3 ln 𝑝 +ln 𝑞 −2 ln 𝑟 as a single logarithm.

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Example An economy is forecast to grow continuously so that the gross national product (GNP), measured in billions of dollars, after t years is given by 𝐺𝑁𝑃= 80𝑒 0.02𝑡 After how many years is GNP forecast to be $88 billion? What does the model predict about the value of GNP in the long run?

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