Outline

I. Summation Operation and Descriptive Statistics Basic Mathematical Tools II. Properties of Linear Functions III. Proportions and Percentages Read Wooldridge, Appendix A IV. Special Functions V. Differential Calculus

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 2 Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

I. Summation Operation and Descriptive Statistics Summation Operation

• Summation operator () involves the sum of many numbers. • Property s.1: For any constant c, n • Given a sequence of n numbers cnc {xi; i=1, …, n} i1

• The sum of these numbers • The sum of n constants (c) equals the product of n and c n  xi = x1 + x2 + …. + xn i1

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 3 I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 4

I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Summation Operation Summation Operation

• Property s.2: • Property s.3: If {(xi,yi): i=1, …,n} is a set of n pairs of numbers and a and b are constants, then nn nnn cxii c x ii11 ()axii by a x i  b y i iii111

The sum of c times xi equals c time the sum of xi.

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 5 I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 6

I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

Summation Operation Summation Operation

• Notes that the sum of ratios is not the ratio of the sums. • Note that the sum of the squares is not the square of the sum.

n nn x x 22 ()x n x  i ii ii1 ii11   n i1 yi  yi i1

• Example: n = 2 2 2 2 • Example: n = 2 x1 + x2  (x1 + x2 )

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 7 8 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Summation Operation and Descriptive Statistics: Sample average Descriptive Statistics

• When the xi are a sample of data on a particular variable, we • Given a sequence of n numbers {xi; i=1, …, n}, the average call this the sample average or sample mean. or mean can be written as 1 n • Sample average is an example of a descriptive statistic. x   xi n i 1 • Sample average is a statistic that describes the central tendency of the set of n points. • Average is computed by adding them up and dividing by n

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 9 10 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

Descriptive Statistics: Sample median Descriptive Statistics: Sample median

• Other measure of central tendency is sample • Steps in finding sample median median. Step 1: order the values of the xi from smallest to largest.

• Example: Given numbers, {‐4, 8, 2, 0, 21, ‐10, 18} Step 2: if n is odd, the sample median is the middle number of – Sample mean = 35/7 = 5 the ordered observations. – Sample median = 2 • Ordered sequence {‐10, ‐4, 0, 2, 10, 18, 21} Step 3: if n is even, the median is defined to be the average of the two middle values.

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 11 12 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Descriptive Statistics: Sample median Summation Operation and Descriptive Statistics

• If 21 is changed to 42 • Numbers {‐4, 8, 2, 0, 42, ‐10, 18} • Deviations – Sample mean = 56/7 = 8 – Sample median = 2 • Deviations can be found by taking each observation and subtracting off • Ordered sequence {‐10, ‐4, 0, 2, 10, 18, 42} the sample average

• Sample median: dxxii Good point: it is less sensitive than sample average to changes in the extreme values in a list of numbers. Examples are median housing values or median income.

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 13 14 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

Summation Operation and Descriptive Statistics Deviations and Demean Sample

Example: n =5 • Properties d1: Given {x ; i=1, …, n}, i x = 6, x = 1, x = ‐2, x = 0, x = 5 The sum of the deviations equal zero. 1 2 3 4 5 nn x  ? dxxii0 Demean sample is {4, ‐1, ‐4, ‐2, 3} ii11 n  xxi 0 i1

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 15 16 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Descriptive Statistics: Algebraic Fact Descriptive Statistics: Algebraic Fact

• Properties d2: Given {xi; i=1, …, n}, • Properties d3: Given {(xi,yi): i=1, …,n}, the sum of squared deviations is the sum of squared xi minus It can be shown that n times the squared of sample mean. nn ()()xiixy y xy ii () y ii11 nn 222 nn ()xxii x  nx ()  ()xxyii xynxy ii () ii11 ii11

• Show!

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 17 18 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

Summary: Summation and Deviation Problem A.1

Summation Deviation A.1 The following table contains monthly housing s.1 d.1 expenditures for 10 families. 2 s.2 d.2 2 (i) Find the average monthly housing expenditure. [ans.] s.3 d.3 (ii) Find the median monthly housing expenditure. [ans.]

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 19 20 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Housing Problem A.1 continue Solution A.1 (i) Family Expenditures 1 300 • (iii) If monthly housing expenditures were measured in 2 440 (i) $566. 3 350 , rather than in dollars, what hundreds of dollars 4 1100 n would be the average and median expenditures? 1 5 640 x x [ans.]   i 6 480 n i1 7 450 • (iv) Suppose that family number 8 increases its monthly 8 700 housing expenditure to $900 dollars, but the 9 670 expenditures of all other families remain the same. 10 530 Sum 5,660 Compute the average and median housing Mean 566 expenditures. [ans.]

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 21 22 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

Solution A.1 (ii) Solution A.1 (iii) (ii) 505 (iii) Steps in finding sample mean • $566 and $505 (in dollars), respectively • Step 1: order the values of the xi from smallest to largest. • 5.66 and 5.05 (in hundreds of dollars), {300, 350, 440, 450, 480, 530, 640, 670, 700, 1100,} respectively. • Step 3: if n is even, the median is defined to be the average of the two middle values. The two middle numbers are 480 and 530; when these are averaged, we obtain 505, or $505.

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 23 24 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat II. Properties of Linear Functions Solution A.1 (iv) Housing Housing Family Expenditures Expenditures (iv) 1 300 300 A linear function can be written as • 2 440 440 The average increases to y = 0 + 1x $586 from $566. 3 350 350 4 1100 1100 • y and x are variables; • while the median is 5 640 640 • 0 and 1 are parameters; unchanged ($505). 6 480 480 – 0 is called the intercept; 7 450 450 – 1 is called the slope. {300, 350, 440, 450, 8 700 900 480, 530, 640, 900, 9 670 670 • We say that y is a linear function of x. 670, 1100,} 10 530 530 Sum 5,660 5,860 Mean 566 586 I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 25 26 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat II. Properties of Linear Functions Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

Linear Functions Example A.2 Linear Housing Expenditure Functions

• Relationship between monthly housing (dollar) expenditure • y = 0 + 1x and monthly income (dollar) y =  x 1 – housing = 0 + 1income  denotes “change”. – housing = 164 + 0.27income

• The change in y is always 1 times the change in x, x. • Interpret: 1 = 0.27 or slope – When family income increases by 1 dollar, housing expenditure will go up by 0.27 dollar or 27 cents • In other words, the marginal effect of x on y is constant and equals to  . 1 • What if family income increases by $300??

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 27 28 II. Properties of Linear Functions Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat II. Properties of Linear Functions Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Example A.2 Linear Housing Expenditure Example A.2 Linear Housing Expenditure Functions Functions

– housing = 0 + 1income – housing = 164 + 0.27income

• Interpret: 0 = 164 or intercept – When income=0, housing expenditures equal $164. – A family with no income spends $164 on housing.

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 29 30 II. Properties of Linear Functions Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat II. Properties of Linear Functions Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

Example A.2 Linear Housing Expenditure Functions Linear functions • housing = 164 + 0.27income • MPC and APC • A linear function can have more than two variables. –  is the marginal propensity to consume (MPC). 1 • y = 0 + 1x1 + 2x2 1 = 0.27 – The average propensity to consume can be written as –  housin g 164 0 is called the intercept (the value of y when x1=0 and 0.27 x =0) income income 2 – 1 and 2 are slopes. • Note that (1) APC is not constant (2) APC>MPC (3) APC gets closer to MPC as income increases.

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 31 32 II. Properties of Linear Functions Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat II. Properties of Linear Functions Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Linear functions Linear functions

• 1 is the slope of the relationship in the direction of x1 • y = 0 + 1x1 + 2x2 y • The change in y, for given changes in x1 and x2 is 12 .... if x 0 x1 y = 1 x1 + 2 x2

• 1 is how y changes with x1, holding x2 fixed. We called the • If x2 does not change (x2 =0), then partial effect of x1 on y. y = 1 x1 if x2 =0. • The notion of ceteris paribus. • If x1 does not change (x1 =0), then Note that partial effect involves holding some factors fixed. y = 2 x2 if x1 =0.

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 33 34 II. Properties of Linear Functions Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat II. Properties of Linear Functions Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

Example A.2: Demand for Compact Discs Example A.2: Demand for Compact Discs

• y = 0 + 1x1 + 2x2

• quantity = 120 –9.8price + 0.03income price: dollars per disc income: measured in dollars

• Interpretation  –9.8 is the partial effect of price on quantity. Holding income fixed, when the price of compact discs increases by one dollar, the quantity demanded falls by 9.8.

 Interpret 2

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 35 36 II. Properties of Linear Functions Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat II. Properties of Linear Functions Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Problem A.2 Problem A.2 continue

• A.2 Suppose the following equation describes the • (ii) What is the average number of classes relationship between the average number of classes missed for someone who lives five miles missed during a semester (missed) and the distance from school (distance, measured in miles): away? [ans.]

missed = 3 + 0.2distance. • (iii) What is the difference in the average number of classes missed for someone who (i) Sketch this line, being sure to label the axes. How do you lives 10 miles away and someone who lives 20 interpret the intercept in this equation? [ans.] miles away? [ans.]

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 37 38 II. Properties of Linear Functions Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat II. Properties of Linear Functions Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

Solution A.2 (i) Solution A.2 (ii)

(i) (ii) missed = 3 + 0.2distance distance = 5 missed = 3 + 0.2distance • This is just a standard linear equation with intercept equal to 3 and slope equal to .2. • • The intercept is the number of missed classes 3 + .2(5) = 4 classes. for a student who lives on campus.

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 39 I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 40

II. Properties of Linear Functions Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat II. Properties of Linear Functions Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Solution A.2 (iii) Problem A.3

(iii) A.3 In Example A.2, quantity of compact disks was related to price and income by missed = 3 + 0.2distance Difference quantity = 120 –9.8price +.03 income. distance = 20: missed = 3 + 0.2(20) distance = 10: missed = 3 + 0.2(10) What is the demand for CDs if price = 15 and income = 200? What does this suggest about • 10(.2) = 2 classes. using linear functions to describe demand curves? [ans.]

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 41 42 II. Properties of Linear Functions Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat II. Properties of Linear Functions Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

Solution A.3 III. Proportions and Percentages

A.3 quantity = 120 – 9.8price + 0.3income • Proportions and percentages play an important role in applied economics. • If price = 15 and income = 200, quantity = 120 – 9.8(15) + .03(200) • Examples in the form of percentages = –21, – inflation rates, – unemployment rates, and • This is nonsense. – entrance acceptance rates. • This shows that linear demand functions generally cannot describe demand over a wide range of prices and income.

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 43 44 II. Properties of Linear Functions Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat III. Proportions and Percentages Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Proportions and Percentages Proportions and Percentages

• A proportion is the decimal form of percent. Example: Admission Rate A percentage is simply obtained by multiplying a proportion by 100. • Proportion: • When using percentages, we often need to convert them to – the proportion of applicants who are admitted to MABE is 0.50. decimal form. • Percentage: For example: find interest income – 50 percent of the applications are admitted to MABE program. – if the annual return on time deposit is 7.6% and we save 30,000 baht at the beginning of the year, – 7.6% = 0.76 (percentage = proportion) • If there are 200 applicants, how many students are admitted? – our interest income is 30,000*0.076 = 2280

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 45 46 III. Proportions and Percentages Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

Proportions and Percentages Proportions

• Find changes in various quantities. • A report by popular media can be incorrect. • Let x be annual income Which one is correct? x0 is the initial value x1 is the subsequent value. • In Thailand, the percentage of high school dropout is .20. The proportionate change in x in moving from x0 to x1 is • Her percentage in econometric exam is .75. xx x 10 x00x This is sometimes called the relative change.

• Note that to get the proportionate change we simply divide the change in x, or x, by its initial value.

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 47 48 III. Proportions and Percentages Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat III. Proportions and Percentages Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Proportions Percentages

• The percentage change in x in going from x to x is • For example, if income goes from $30,000 per year to $36,000 0 1 per year, what is the proportionate change? x %100*x x0 x0 = 30,000 (initial value: income in 1994) x1 = 36,000 (subsequent value: income in 1995 • It is simply 100 times the proportionate change. proportionate change = 6,000/30,000 = 0.2 xx x 10 • %x is read as the percentage change in x. x00x

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 49 50 Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat III. Proportions and Percentages Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

Percentages and Proportions Percentage Point Change vs. Percentage Change

• Percentage Point Change vs. Percentage Change • Suppose the real GDP in 2010 and 2011 are 300 and 315 trillion baht, respectively. What are the proportionate change • Variable in interest is itself a percentage! and the percentage change in real GDP in 2011?

xx x • How to find percentage change? – 10 proportionate change = 15/300 = 0.05 Let x be unemployment rate. Suppose xx00 x is 4% and x 2002 – percentage change = 100*(0.05) = 5% %x 100* x is 5%, x 2003 0 what is the percentage point change? What is the percentage change? • True or False: The percentage change in GDP is 0.05. 1) percentage point change = x = 5%‐4% = 1% 2) percentage change = 100*x/x0 = 1%/4% = 25% x %x 100* x0

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 51 52 Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat III. Proportions and Percentages Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Percentages Proportions and Percentages

• Unemployment rate has increased by one percentage point. • Example: During General Chavalit administration (1997), the • But unemployment rate has increased by 25 percent! value added tax was increased from 7% to 10%.

• Summary • Who is correct? – The percentage point change is the change in the percentages. – Supporters – The percentage change is the change relative to the initial value (x2002) This is simply a three percentage point increase! or an increase of three satang on the baht. xx10 x 100* 100* – Opponents xx00 This is a 43% increase in value added tax! x %x 100* x0

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 53 54 Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat III. Proportions and Percentages Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

Problem A.4 Solution A.4 (i)

• A.4 Suppose the unemployment rate in the (i) United States goes from 6.4% in one year to • The percentage point change is 5.6 – 6.4 = –.8, 5.6% in the next.

(i) What is the percentage point decrease in the • or an eight‐tenths of a percentage point unemployment rate? [ans.] decrease in the unemployment rate.

(ii) By what percent has the unemployment rate fallen? [ans.]

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 55 56 III. Proportions and Percentages Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Solution A.4 (ii) Problem A.5

• A.5 Suppose that the return from holding a particular (ii) firm’s stock goes from 15% in one year to 18% in the • The percentage change in the unemployment following year. rate is – The majority shareholder claims that “the stock 100[(5.6 – 6.4)/6.4] = –12.5%. return only increased by 3%,”

– while the chief executive officer claims that “the return on the firm’s stock has increased by 20%.”

– Reconcile their disagreement. [ans.]

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 57 58 III. Proportions and Percentages Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat III. Proportions and Percentages Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

A. quadratic IV. Some Special Functions and B. Logarithm Solution A.5 Their Properties C. Exponential

A.5 • Linear function • The CEO is referring to the change relative to the – y = 0 + 1x1 initial return of 15%. – Interpret: • One unit change in x results in a same change in y, regardless of the starting value of x. • The majority shareholder is referring to the • Marginal effect of x on y is constant percentage point increase in the stock return, • – To be precise, the shareholder should specifically refer to a Example: utility function –the notion of diminishing marginal 3 percentage point increase. returns is not consistent with a linear relationship.

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 59 60 III. Proportions and Percentages Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Some Special Functions and A. quadratic A. quadratic B. Logarithm Quadratic functions B. Logarithm Their Properties C. Exponential C. Exponential

• Many economic phenomena requires the use of nonlinear • One way to capture diminishing returns is to include a functions quadratic term to a linear function. 2 y = 0 + 1x + 2x • Nonlinear function is characterized by the fact that the change in y for a given change in x depends on the starting • Given 1>0 and 2 <0 value of x. The relationship between y and x has the parabolic shape. Let • Quadratic functions 0=6 1=8 and 2 = ‐2 • Natural Logarithm • Exponential Function y = 6 + 8x ‐ 2x2

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 61 62 IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

A. quadratic A. quadratic B. Logarithm B. Logarithm Graph: Quadratic functions C. Exponential Quadratic functions C. Exponential

• When 1>0 and 2 <0, the maximum of the function occurs at  x*  1 (2 2 )

• This is called the turning point. 2 If y = 6 + 8x ‐ 2x (so 1=8 and 2 = ‐2), the largest value of y occur at x* = 8/4 = 2 y* = 6+8(2)‐2(22) = 14

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 63 64 IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat A. quadratic A. quadratic B. Logarithm B. Logarithm Quadratic functions C. Exponential Quadratic functions C. Exponential

• 2 <0. This implies a diminishing marginal effect of x on y • A diminishing marginal effect of x on y is the same as saying that – the slope of the function decreases as x increases. – Suppose we start at a low value of x and then increase x by some amount, say c • Calculus: the derivative of the quadratic function: 2 • This has a larger effect on y than if we start at a higher value of x y = 0 + 1x + 2x and increase x by the same amount. (See graph) • For “small” changes in x, the approximate slope of the quadratic function is – Once x>x*, an increase in x actually decreases y. y slope 2 x x 12

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 65 66 IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

A. quadratic A. quadratic Example A.4: A quadratic wage B. Logarithm B. Logarithm function C. Exponential Quadratic functions C. Exponential

• wage hourly wage • y =  +  x +  x2 • exper years in the workforce 0 1 2 • When 1<0 and 2>0, the graph of the quadratic function wage = 5.25 + .48exper – 0.008exper2 has U‐shape.

Interpretation: 1) there is an increasing marginal return. 1) Since 2<0, this implies the diminishing marginal effect of exper on wage. 2) exper* = .48/[2(.008)] = 30. This is turning point. 2) the minimum of the function is at the point – exper has a positive effect on wage up to the turning point. 3) The marginal effect of exper on wage depends on years of experience. st 1 1 year (x0=0, x1=1, x=1) wage = .48 – 2*0.008(0) = .48 th x*  5 year (x4=4, x5=5 x=1) wage = .48 – 2*0.008(4) = .416 4) At 30 years, an additional year of experience would lower the wage. (2 2 ) th 32 year (x31=31, x32=32 x=1) wage = .48 – 2*0.008(31) = ‐.016

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 67 68 IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat A. quadratic A. quadratic B. Logarithm B. Logarithm Natural Logarithm C. Exponential Natural Logarithm C. Exponential • Properties 1) The log function is defined only for positive values of x (x>0). • The natural logarithm, an important nonlinear function, plays an See graph of a log function important role in econometric analysis. for 01 log(x) >0 y = log(x) 2) When y= log(x), the effect of x on y never becomes negative. • Other common symbols include log (x) and ln(x). y 1 e y = log(x)  – Most calculators use ln(x). x x – Different symbols are useful when we use logarithm with different bases. • The relationship between x and y displays diminishing returns. • The slope of the function gets closer and closer to zero as x gets large

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 69 70 IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

A. quadratic A. quadratic B. Logarithm Natural Logarithm B. Logarithm Graph: log function C. Exponential C. Exponential

• What is the difference in slopes between the quadratic and the log function? Ans. The marginal effect of x on y of the log function never becomes negative.

• Some useful algebraic facts:

l.1) log(x1x2) = log(x1) + log(x2)

l.2) log(x1/x2) = log(x1) –log(x2) l.3) log(xc) = clog(x) for any constant c.

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 71 72 IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat A. quadratic A. quadratic Natural Logarithm B. Logarithm B. Logarithm C. Exponential Natural Logarithm C. Exponential

• How good is the approximation? • Let x0 and x1 be positive values.

• The difference in logs can be used to approximate proportionate changes. • Example, for a small change: x0=40 and x1=41 – Exact percentage change xx10 x log(xx10 ) log( ) = 100*(41‐40)/40 = 2.5% xx for small changes in x. 00 – Approximate percentage change = 100*[log(41)‐log(40)] = 2.47% • Note that log(x) = log(x1)‐log(x0)

• Approximate percent change is • Example, for a large change: x0=40 and x1=60 – Exact percentage change = 100*(60‐40)/40 = 50% 100log(x) ≈ %x for small changes in x. – Approximate percentage change = 100*[log(60)‐log(40)] = 40.55%

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 73 74 IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

A. quadratic A. quadratic B. Logarithm B. Logarithm Natural Logarithm C. Exponential Natural Logarithm C. Exponential

• Elasticity of y with respect to x can be written as • Constant Elasticity Model

yx% y log(y) = 0 + 1log(x)  x yx% • The slope or elasticity is approximately equal to – It is the percentage change in y when x increases by 1%  log(y )   1  log(x ) • If y is a linear function, y = 0 + 1x, then the elasticity is yx x x  11 • xy y01 x It is the elasticity of y with respect to x – It depends on the value of x.

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 75 76 IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat A. quadratic A. quadratic B. Logarithm B. Logarithm Example A.5: Constant Elasticity C. Exponential C. Exponential Demand Function Natural Logarithm

• Log‐level model   • Let q quantity demanded (unit) log(y) = 0 + 1x log(y) = 1x

p price (dollar) 100log(y) = 1001x

%y = (1001)x log(q) = 4.7 –1.25log(p) • The slope or the semi‐elasticity of y with respect to x is

– The price elasticity of demand = ‐1.25 %y 100 1 – Interpret x • A 1% percent increase in price leads to a 1.25% fall in the quantity • The semi‐elasticity is the percentage change in y when x increases by one demanded. unit. It is equal to 1001.

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 77 78 IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

A. quadratic A. quadratic B. Logarithm B. Logarithm Example A.6: Logarithmic C. Exponential C. Exponential Wage Equation Natural Logarithm • Level‐log function

• Hourly wage and years of education are related by y = 0 + 1log(x) y = 1 log(x) log(wage) = 2.78 + .094 educ 100y = 1100log(x)

• Using approximation, %(wage) = 100(.094)educ 100y =  %x %(wage) = 9.4educ 1 y = (1/100)(%x) • Interpret: • Interpret – One more year education increases hourly wage by about 9.4% – 1/100 is the unit change in y when x increases by 1%

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 79 80 IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat A. quadratic A. quadratic Example A.7: Labor Supply B. Logarithm B. Logarithm C. Exponential Summary C. Exponential Function

• Let hours hours worked per week wage hourly wage

hours = 33 + 45.1log(wage) hours = (45.1/100)(%wage) hours = 0.451(%wage)

• Interpret 1% increase in wage increases the weekly hours worked by about 0.45, or slightly less than one‐half hour.

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 81 82 IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

A. quadratic A. quadratic B. Logarithm B. Logarithm Exponential Function C. Exponential Exponential Function C. Exponential

• We write the exponential function as • The exponential function is related to the log function. y = exp(x) – Other notation can be written as • For example, given log(y) that is a linear function of x, y = ex – How to find y itself as a function of x? • Facts – exp(0) =1; – exp(1) = 2.7183 – exp(x) is defined for any value of x. – exp(x) is always greater than 0

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 83 84 IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat A. quadratic A. quadratic B. Logarithm B. Logarithm Graph: Exponential Function C. Exponential Exponential Function C. Exponential

• The exponential function is the inverse of the log function in the following sense:

log[exp(x)] = x, for all x exp[log(x)] = x, for x>0

 In other words, the log “undoes” the exponential, and vice versa.

 The exponential function is sometimes called the anti‐log function.

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 85 86 IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

A. quadratic B. Logarithm Exponential Function C. Exponential Summary: Logarithm and Exponential • Given a function,

log (y)= 0 + 1x Logarithm Exponential

• It is equivalent to the function l.1 log(x1x2) = log(x1) + log(x2) e.1 exp(x1+ x2) = exp(x1)∙exp(x2) y = exp(0 + 1x)

• If 1>0, then x has an increasing marginal effect on y. l.2 log(x1/x2) = log(x1) –log(x2) e.2 exp(x1‐ x2) = exp(x1)/exp(x2)

• Some algebraic facts: c c x1+ x2 x1 x2 l.3 log(x ) = clog(x) e.3 exp[clog(x)] = x e.1) exp(x1+ x2) = exp(x1)∙exp(x2)or e = e ∙e x1‐ x2 x1 x2 e.2) exp(x1‐ x2) = exp(x1)/exp(x2)or e = e /e e.3) exp[clog(x)] = xc or eclog(x) = xc

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 87 88 IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Problem A.6 Solution A.6 (i)

A.6 Suppose that Person A earns $35,000 per (i) year and Person B earns $42,000. 100[42,000 – 35,000)/35,000] = 20%. (i) Find the exact percentage by which Person B’s salary exceeds Person A’s. [ans.]

(ii) Now use the difference in natural logs to find the approximate percentage difference. [ans.]

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 89 90 IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

Solution A.6 (ii) Problem A.7

(ii) • A.7 Suppose the following model describes the • The approximate proportionate change is relationship between annual salary (salary) and the number of previous years of labor market log(42,000) – log(35,000) =.182, experience (exper):

so the approximate percentage change is %18.2. log(salary) = 10.6 + .027exper.

•  [Note: log( ) denotes the natural log.] (i) What is salary when exper = 0? when exper = 5? (Hint: You will need to exponentiate.) [ans.]

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 91 92 IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Problem A.7 continue … Solution A.7 (i)

(ii) Use equation (A.28) to approximate the (i) log(salary) = 10.6 + .027exper percentage increase in salary when exper increases by five years. [ans.] • When exper = 0, log(salary)= 10.6; therefore, salary = (iii) Use the results of part (i) to compute the exp(10.6) = $40,134.84. exact percentage difference in salary when exper = 5 and exper = 0. Comment on how this • When exper = 5, compares with the approximation in part (ii). salary = exp[10.6 +.027(5)] [ans.] = $45,935.80.

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 93 94 IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

Solution A.7 (ii) Solution A.7 (iii)

(iii) log(salary) = 10.6 + .027exper (ii) exper=0 log(salary) = 10.6 + .027exper salary = exp(10.6) = $40,134.84. 100log(y)= 1001x (A.28) exper=5 log(salary) = 10.6 + .027exper salary = exp[10.6 +.027(5)] = $45,935.80. • The approximate proportionate increase is .027(5) = .135, • Exact percentage increase 100[(45,935.80‐40,134.84)/40,134.84) = 14.5%, so the approximate percentage change is 13.5%. so the exact percentage increase is about one percentage point higher.

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 95 I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 96

IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Problem A.8 Solution A.8

A.8 Let A.8 grthemp = –.78(salestax). grthemp denote the proportionate growth in employment, at the county level, from 1990 to 1995, and • Since both variables are in proportion form, we can multiply the equation through by 100 to turn each variable into let percentage form. salestax denote the county sales tax rate, stated as a • Slope = –.78. proportion. – So, a one percentage point increase in the sales tax rate (say, from 4% to 5%) reduces employment growth by –.78 percentage points. Interpret the intercept and slope in the equation • Intercept = .043 – When salestax = 0, the proportionate growth in employment is .043. grthemp = .043 –.78salestax. [ans.]

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 97 98 IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat IV. Some Special Functions and Their Properties Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

V. Differential Calculus Differential Calculus

 Let y = f(x) for some function f • Example

 For small changes in x y = log(x) dy 1  dx x

• For a small change, evaluated at the initial point x0,  df/dx is the derivative of the function f, evaluated at the 1 x initial point x0. y x log(x ) , x x0 0 which is the approximation of the proportionate change in x.

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 99 100 V. Differential Calculus Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat V. Differential Calculus Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Differential Calculus Differential Calculus

• Other Important Rules • Let y = f(x) DC.5 Let y =cf(x) = c DC.1 Let f(x) = c = 0or = 0 – The derivative of a constant times any function is that same constant times the derivative of the function, – The derivative of a constant c is zero. – Example: y =cf(x) = xc = c = cxc‐1 DC.2 Let f(x) = log(x) = or = – The derivative of the log function of x is one over x. DC.6 Let y = f(x)+g(x) = +

– The derivative of the sum of two functions is the sum of the derivatives DC.3 Let f(x) = exp(x) = exp(x) or = exp(x) – The derivative of the exponential function of x is the exponential function of x DC.7 Let y = z(f(x)) = DC.4 Let f(x) = xc = cxc‐1 or = cxc‐1 – Chain Rule

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 101 102 V. Differential Calculus Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat V. Differential Calculus Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

Chain Rule Differential Calculus

Find the derivatives of the following functions: DC.7 Let y = f(z(x)) = = – y =  +  x+  x2 Example: 0 1 2   y = exp( 0 + 1x) – y = 0 + 1/x z = 0 + 1x

– y = 0 + 1 x = = – y =  +  log(x) = exp 0 1 = exp x – y = exp(0 + 1x)

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 103 104 V. Differential Calculus Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Summary: Calculus Differential Calculus

• Notion of a partial derivative DC.1 Let f(x) = c = 0 DC.5 Let y =cf(x) = c • Suppose that DC.2 Let f(x) = log(x) = DC.6 Let y = f(x)+g(x) = + y = f(x1, x2)

DC.3 Let f(x) = exp(x) = exp(x) DC.7 Let y = z(f(x)) = • Two partial derivatives y 1) The partial derivative of y with respect to x1 = Example DC.7: y = exp(0 + 1x) x1 (where x2 is treated as a constant) c c‐1 DC.4 Let f(x) = x = cx x = = y 2) The partial derivative of y with respect to x = 2 x = exp = exp x 2 (where x1 is treated as a constant)

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 105 106 V. Differential Calculus Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat V. Differential Calculus Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

Differential Calculus Differential Calculus

• We can approximate the change in y as • y = f(x1,…, xn)

x2,…, xn fixed

• Example Example:

y = 0 + 1x1 + 2x2 log() = 0 + 1educ + 2exper + 3tenure + 4age log() = .514 + .078educ +.002exper +.0088tenure +.0033age 1) What is the partial derivative of y with respect to x1 ?

2) What is the partial derivative of y with respect to x2? 1) What is the partial derivative of y with respect to x1 ?

2) What is the partial derivative of y with respect to x4?

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 107 108 V. Differential Calculus Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Example A.8 wage function Differential Calculus with interaction

• Example wage = 3.1 + .41educ + .19exper ‐ .004exper2 + .007educexper • Find the partial effect of exper on wage: wage/exper = .19 ‐ .008exper +.007educ

• At the initial values (exper0=5, educ0=12), the approximate change in wage is 1) What is the partial derivative of y with respect to x1 ? wage/exper = 23.4 cents per hour 2) What is the partial derivative of y with respect to x ? 2 • Given exper0=5, educ0=12; and exper1=6, educ1=12, the exact change in wage is wage = 23 cents per hour

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 109 110 V. Differential Calculus Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat V. Differential Calculus Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat

Basic Mathematical Tools Differential Calculus Problem A.9

A.9 Suppose the yield of a certain crop (in bushels per acre) • Minimizing and maximizing functions. Let f(x1, x2, …, xk) is the differentiable function of k variables. is related to fertilizer amount (in pounds per acre) as

* * * • A necessary condition for x1 , x2 , …, xk to optimize f over all possible values of xj is yield = 120 +.19 fertilizer

(i) Graph this relationship by plugging in several values for fertilizer. [ans.] • Notes 1) All of the partial derivatives of f must be zero when they are evaluated at (ii) Describe how the shape of this relationship compares * the xh . with a linear function between yield and fertilizer. [ans.] 2) These are called the first order conditions. * 3) We hope to solve above equations for the xh .

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 111 112 V. Differential Calculus Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat V. Differential Calculus Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Solution A.9 (i) Solution A.9 (ii) (i) • The relationship between yield and fertilizer is graphed below. (ii)

yield 122 • Compared with a linear function, the function yield = .120 + .19 has a diminishing effect. fertilizer

121 • The slope approaches zero as fertilizer gets large. – The initial pound of fertilizer has the largest effect, and each additional pound has an effect smaller than the previous pound.

120 0 50 100 fertilizer

I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 113 I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 114

V. Differential Calculus Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat V. Differential Calculus Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat