Notes I: Introduction, Useful Mathematical Concepts, and Measurement. Julio Garín Intermediate Macroeconomics Fall 2018

Transcription

1 Notes I: Introduction, Useful Mathematical Concepts, and Measurement Julio Garín Intermediate Macroeconomics Fall 2018

2 Introduction Intermediate Macroeconomics Intro, Math, and Measurement 2 / 60

3 Don t Take Shortcuts Read Claremont McKenna s Statement of Academic Integrity. Don t take nothing from nobody. Miley Cyrus Intermediate Macroeconomics Intro, Math, and Measurement 3 / 60

4 Introduction Distinction between micro and macro is NOT clear. Economics is micro. Macro just studies these at the aggregate level. The emphasis is on the intertemporal nature of economic decisions. Some of the questions we are going to address: How can unemployment and job openings coexist? How can we explain consumption and leisure behavior? Why does the economy experience periodic deviations from its growth path i.e. business cycles? What are the implications? Can policymakers dampen the fluctuations? If so, should they? Is fiscal policy or monetary policy more effective? WTH happened in ? Intermediate Macroeconomics Intro, Math, and Measurement 4 / 60

5 Intermediate Macroeconomics Principles provided you with concepts, definitions, and graphical arguments. We will analyze formal models. Why? Real world is very complex. Consider an increase in government spending. This should increase demand for goods. What if households anticipate future tax increase? They will reduce the demand for goods! Which of these two forces dominates? Is it something too much or too little? We like to think about what if? A model allows us to formalize these ideas and helps us to answer these questions. Intermediate Macroeconomics Intro, Math, and Measurement 5 / 60

6 Okay, got it, but do we really need more math? What do we want? 1. We want our models to be based on optimizing behavior by individuals. 2. We also want our models to be equilibrium models decisions are consistent with what is going on. 3. Since many economic decisions involve intertemporal tradeoffs, we need to think about dynamic models. This deeper, more formal approach will require that we utilize more math and it is the basis of modern macroeconomics. Hence, the answer is yes. Intermediate Macroeconomics Intro, Math, and Measurement 6 / 60

7 That said, it is true that modern macroeconomics uses mathematics and statistics to understand behavior in situations where there is uncertainty about how the future will unfold from the past. But a rule of thumb is that the more dynamic, uncertain and ambiguous is the economic environment that you seek to model, the more you are going to have to roll up your sleeves, and learn and use some math. That s life. Thomas Sargent, 2011 Nobel Prize Winner Intermediate Macroeconomics Intro, Math, and Measurement 7 / 60

8 Math Review Intermediate Macroeconomics Intro, Math, and Measurement 8 / 60

9 Outline of Topics and Relevant Material Notation. Exponents, logs, and growth rates. Calculus. Optimization. 1. Appendix A GLS: Mathematical Appendix. 2. Chapter 3 GLS: Brief History of Macroeconomic Thought. Intermediate Macroeconomics Intro, Math, and Measurement 9 / 60

10 Notation We have two types of variables. 1. Exogenous. 2. Endogenous. We also have parameters: fixed values governing mathematical relationships. Time is discrete. We are going to use the summation notation. Intermediate Macroeconomics Intro, Math, and Measurement 10 / 60

11 Properties of Exponents and Logs Exponents. x a x b = xa x b = (x a ) b = 1. Product rule: ln(αβ) = 2. Quotient rule: ln(α/β) = 3. Power rule: ln(x α ) = Remember: ln = log e. Since log x x = 1, what is ln(e α ) equal to? Fun facts: ln(1 + α) α and exp(α) 1 + α for small α. Growth rate: g x t ln x t ln x t 1 Intermediate Macroeconomics Intro, Math, and Measurement 11 / 60

12 Calculus: Notation and Preliminaries Consider a function y = f (x). First derivative: how y changes as x changes. Derivative is itself a function. Notation: dy dx, f (x), and f x (x). Second derivative: derivative of a derivative. Notation: dy 2 d 2 x, f (x), and f xx (x). Don t confuse a derivative with a derivative evaluated at a point. f (x) = f (x o ). Intermediate Macroeconomics Intro, Math, and Measurement 12 / 60

13 Derivative Rules: Examples 1. f (x) = x α. 2. f (z) = βz (1 α). 3. f (c) = γ ln(φc). 4. f (k) = (k α ) ( βk 1 γ). 5. f (g(c)) = (c 2 c 3 ) α. 6. f (g(x)) = (x α ) γ. Intermediate Macroeconomics Intro, Math, and Measurement 13 / 60

14 Calculus: Multivariate Derivatives Let s assume there are two variables: y = f (x, z). Partial derivative is change in x for a change in x (or z), holding z (or x) constant. Notation: y First derivative: x f x (x, z). Second derivative: Cross derivative: ( y x x ( ) y x z ) 2 y x 2 f xx (x, z). 2 y x z f xz(x, z). Intermediate Macroeconomics Intro, Math, and Measurement 14 / 60

15 Multivariate Derivatives: Examples For the following functional forms: 1. f (x, z) = ln x + z α. 2. f (k, n) = k α n (1 α). 3. f (x, z) = βx α + 3xz + γz θ. Calculate, for each function: 1. First and second partial derivatives. 2. Find the cross partial derivatives. Intermediate Macroeconomics Intro, Math, and Measurement 15 / 60

16 Calculus: Optimization Pick x to either maximize or minimize f (x). First order condition (FOC): x needs to satisfy f (x ) = 0 Second order condition (SOC): Sign of f (x ) tells you whether you have a maximum or a minimum. Multivariate optimization works in the same way: FOCs set partial derivatives with respect to (wrt) each choice variable equal to zero. Intermediate Macroeconomics Intro, Math, and Measurement 16 / 60

17 Optimization: Examples For the following functions: 1. f (x) = x f (c) = ln c 2c. Calculate: i FOCs. ii SOCs. iii Is this a maximum or a minimum? Intermediate Macroeconomics Intro, Math, and Measurement 17 / 60

18 Calculus: Constrained Optimization The problem now is to optimize f (x, z) subject to an inequality constraint that x and z must satisfy. Our approach: 1. Assume constraint holds with equality. 2. Eliminate one of the choice variables. 3. Perform an unconstrained optimization. Intermediate Macroeconomics Intro, Math, and Measurement 18 / 60

19 Constrained Optimization: Example Let s solve this problem: max U = ln x + ln z x,z subject to p x x + p z z y. Intermediate Macroeconomics Intro, Math, and Measurement 19 / 60

20 Measurement Intermediate Macroeconomics Intro, Math, and Measurement 20 / 60

21 Outline of Topics and Relevant Material GDP and its components. Data. Real vs. nominal. CPI and inflation. 1. Chapter 1 GLS: Macroeconomic Data. Intermediate Macroeconomics Intro, Math, and Measurement 21 / 60

22 Preliminaries While the GDP and the rest of the national income accounts may seem to be arcane concepts, they are truly among the great inventions of the twentieth century Paul A. Samuelson and William D. Nordhaus Why do we care about economic activity? Why do we care about income? It is closely related with higher standards of living. GDP simultaneously measures: 1. Total income of everyone in the economy. 2. Total expenditure on the economy s output of goods and services. 3. Total production of goods and services. Intermediate Macroeconomics Intro, Math, and Measurement 22 / 60

23 Gross Domestic Product and its Components Intermediate Macroeconomics Intro, Math, and Measurement 23 / 60

24 GDP I Gross Domestic Product (GDP) is the market value of all final goods and services produced within a country in a given period of time. You cannot compare burritos and markers. GDP uses market prices. It its comprehensive. GDP includes all items produced in the economy and sold legally in markets. For instance, it includes the value of housing services provided by the economy s stock of housing. What about transactions within the family? GDP only includes final goods. Intermediate goods that increase inventories add to GDP. Intermediate Macroeconomics Intro, Math, and Measurement 24 / 60

25 GDP II GDP includes both tangible goods and intangible services. GDP only includes goods and services currently produced. What about new cars? What about used ones? GDP measures the value of production within the geographic confines of a country. What about an American producing in Canada? What about an Uruguayan producing in America? GDP measures the value of production that takes place within a specific interval of time. Usually a year or a quarter. When GDP is reported at a quarterly frequency, it usually represents GPD at an annual rate. Intermediate Macroeconomics Intro, Math, and Measurement 25 / 60

26 The Components of GDP To understand how the economy is using its scarce resources, economists study the composition of GDP among various types of spending. GDP is divided into five components: GDP t = Remember: GDP is equivalently also a measure of both expenditure and income. Intermediate Macroeconomics Intro, Math, and Measurement 26 / 60

27 A Closer Look into GDP Components Consumption, C, is spending by households on goods and services, with the exception of purchases of new housing. Investment, I, is the purchase of goods that will be used in the future to produce more goods and services. It includes spending on capital equipment, inventories, and structures, including household purchases of new housing. Government purchases, G, includes spending on goods and services by local, state, and federal government. It includes the salaries of government workers as well as expenditures on public works. It does not include transfer payments. Net exports, NX, equal the foreign purchases of domestically produced goods (exports, X ) minus the domestic purchases of foreign goods (imports, IM.) Intermediate Macroeconomics Intro, Math, and Measurement 27 / 60

28 GDP as Value Added Table: Example from the Oil Industry Stage of Production Value of Sales Value Added Oil Drilling $ Refining Shipping Retail Sale Total Value Added 3.47 Intermediate Macroeconomics Intro, Math, and Measurement 28 / 60

29 What Counts and What Doesn t? For each question, identify which component GDP these are counted toward and how GDP changes. 1. I build a new home in the Village. 2. Super King Market has beer that has been sitting on the shelf for 12 months. It s getting stale, so it goes on sale. CNC students flock the store and purchase all of it. 3. You buy Apple stock on E-trade. 4. Salaries paid to soldiers. 5. Social Security benefits for the retired. 6. You buy an Australian surfboard. Intermediate Macroeconomics Intro, Math, and Measurement 29 / 60

30 Intermediate Macroeconomics Intro, Math, and Measurement 30 / 60

31 It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts. Sherlock Holmes Intermediate Macroeconomics Intro, Math, and Measurement 31 / 60

32 Where and How? Google is your friend: St. Louis Fed: Federal Reserve Economic Data (FRED). Bureau of Labor Statistics (BLS). U.S. Bureau of Economic Analysis (BEA). Census Bureau. Example: GDP. Consumption. Investment. Saving: For now, just use Microsoft Excel. Arrange the data in columns. First column should always have the time periods. Select cell B2 and: View Freeze Panes Freeze Panes. Intermediate Macroeconomics Intro, Math, and Measurement 32 / 60

33 Looking at the Data Intermediate Macroeconomics Intro, Math, and Measurement 33 / 60

34 GDP and its Components Intermediate Macroeconomics Intro, Math, and Measurement 34 / 60

35 Expenditure Shares of U.S. GDP Share of GDP (%) Consumption Investment Governmnet Net Exports Time Intermediate Macroeconomics Intro, Math, and Measurement 35 / 60

36 Within Consumption % of C Services Goods Intermediate Macroeconomics Intro, Math, and Measurement 36 / 60

37 Within Consumption % of C Services Durables Nondurables Intermediate Macroeconomics Intro, Math, and Measurement 37 / 60

38 Within Fixed Investment Residential Nonresidential Intermediate Macroeconomics Intro, Math, and Measurement 38 / 60

39 Within Nonresidential Structures Equipment Intellectual property products Intermediate Macroeconomics Intro, Math, and Measurement 39 / 60

40 Fixed Investment (As % of Investment) Intermediate Macroeconomics Intro, Math, and Measurement 40 / 60

41 Change in Private Inventories % of GDP Intermediate Macroeconomics Intro, Math, and Measurement 41 / 60

42 Within Government Expenditures Share of GDP.5.4 State and Local Federal Intermediate Macroeconomics Intro, Math, and Measurement 42 / 60

43 Within Government Expenditures 1.0 State and Local Federal Intermediate Macroeconomics Intro, Math, and Measurement 43 / 60

44 Net Exports Components Share of GDP Exports Imports Net Exports Intermediate Macroeconomics Intro, Math, and Measurement 44 / 60

45 Expenditure Approach Table: GDP and its Components (Annualized 2016Q2) Total Per Person Share of GDP (Billions of $) (in $) (Percent) GDP 18,438 57, Consumption 12,688 39, Investment 2,972 9, Government Purchases 3,267 10, Net Exports , Intermediate Macroeconomics Intro, Math, and Measurement 45 / 60

46 GDP Billions of $ 16,000 Nominal GDP 12,000 8,000 4, Time Intermediate Macroeconomics Intro, Math, and Measurement 46 / 60

47 Real and Nominal Variables Intermediate Macroeconomics Intro, Math, and Measurement 47 / 60

48 Motivation Nominal GDP 1994: $7,476.7 Billions of dollars. Nominal GDP 2014: $17,599.8 Billions of dollars. If GDP grew then at least one of following must be true: 1. The economy is producing more goods and services. 2. The same goods and services are being sold at higher prices. As economists, we would like to isolate these two effects. Intermediate Macroeconomics Intro, Math, and Measurement 48 / 60

49 Real Vs. Nominal Money, M, is an unit of account and numeraire. Nominal price: denominated in units of money. We care about real variables. Prices denominated in terms of goods. In a single good world real output is just the quantity. In a multi-good world this is not straightforward to compute. Intermediate Macroeconomics Intro, Math, and Measurement 49 / 60

50 Real Vs. Nominal GDP We refer to nominal GDP when prices and quantities have not been separated out. Real GDP is the total output in an economy, in a given period of time, corrected for any change in prices. In the NIPA, this is equivalent to constant dollar GDP. Formally: nominal GDP = price level real GDP. Intermediate Macroeconomics Intro, Math, and Measurement 50 / 60

51 Some Formulae Nominal GDP in year t : NGDP t = n y s,t p s,t. s=1 Real GDP in year t using year B prices: Y t = n y s,t p s,b. s=1 Money is still a numeraire, but now we can make comparisons across time. Intermediate Macroeconomics Intro, Math, and Measurement 51 / 60

52 GDP Deflator We can define an implicit price index as: P t = NGDP t RGDP t 100. By construction, it is equal to 100 in the base period. This index is also known as an implicit GDP deflator. The GDP deflator is a measure of the level of prices of all new, final goods and services produced within a country in a given period of time. It can be used to take inflation out of nominal GDP. It can deflate nominal GDP from the rise that is due to increases in prices. Intermediate Macroeconomics Intro, Math, and Measurement 52 / 60

53 Chain-Weighting Issues with the previous calculation: Choice of base year is arbitrary. Relative prices change over time. To correct for this issue, NIPA uses a chain-weighting scheme for calculating real GDP. Calculate the real GDP using different base years. Obtain growth rates. Calculate a geometric average of those growth rates. From the averages obtained above, create a new series of real GDP. Intermediate Macroeconomics Intro, Math, and Measurement 53 / 60

54 Real and Nominal GDP 16,000 Nominal GDP Real GDP 12,000 8,000 4, Intermediate Macroeconomics Intro, Math, and Measurement 54 / 60

55 Deflator, Real, and Nominal GDP 16,000 Nominal GDP Real GDP Deflator , ,000 4, Intermediate Macroeconomics Intro, Math, and Measurement 55 / 60

56 CPI Intermediate Macroeconomics Intro, Math, and Measurement 56 / 60

57 CPI The consumer price index (CPI) is a measure of the overall cost of the goods and services bought by a typical consumer. Data on the prices of thousands of goods and services is collected monthly by the Bureau of Labor Statistics (BLS). Intermediate Macroeconomics Intro, Math, and Measurement 57 / 60

58 CPI and Inflation CPI ( =100) Inflation Intermediate Macroeconomics Intro, Math, and Measurement 58 / 60

59 Problems in Measuring the Cost of Living Substitution bias. When prices change over time, they do not all change proportionally. Consumers respond to these changes. The index will tend to measure a much larger increase in the cost of living than consumers actually experience. Introduction of new goods. As new goods are introduced, consumers have more choices, and each dollar is worth more. Since the basket is fixed, it does not reflect the increase in the value of a dollar that arises from the introduction of new goods. Unmeasured quality change. If the quality changes over time, and the price remains the same, the value of the currency also changes. Intermediate Macroeconomics Intro, Math, and Measurement 59 / 60

60 Two Measures of Inflation CPI Inflation GDP Deflator Inflation Intermediate Macroeconomics Intro, Math, and Measurement 60 / 60