Proving the Pythagorean Theorem The Pythagorean Theorem is one of the most well known and important theorems in all of mathematics. It was named after a Greek mathematician and philosopher. However, there is evidence to suggest that the Babylonians knew about this relationship some 1000 years earlier. The Chou-pei, an ancient Chinese text containing a table of Pythagorean triples also shows that the Chinese were aware of the relationship between the sides of a right triangle long before Pythagoras. It was, however Pythagoras, or one of his colleagues in the Pythagorean society, that proved the theorem and so it was named after him. Since the time of Pythagoras, there have been many proofs for this theorem. Here we will examine several. Pythagorean Theorem: A triangle is a right triangle if and only if the square of the hypotenuse is equal to the sum of the squares of the shorter two sides. 1) Stockbroker’s Proof In 1830, Henry Perigal, a London stockbroker, discovered a way to indicate the truth of the Pythagorean Theorem using paper and scissors. • • • • • • Using a straight edge, draw a right triangle. Label the longest side of your right triangle c and the other two a and b. Draw a square off of side a. Draw a square off of side b. Using your protractor (to ensure right angles), draw a square off of side c. What is the area of each of these squares? _________ _________ _________ (Label the areas) • • Find the center of 𝑏 2 , mark this point w. Using your protractor, draw a line segment through point w that is perpendicular to side c of your triangle. Draw a line segment that passes through point w that is perpendicular to the line segment you just drew. (Square 𝑏 2 should now be broken into 4 pieces) • • • • Cut out all three of your squares (𝑎2 , 𝑏 2 , 𝑎𝑛𝑑 𝑐 2 ) Cut square 𝑏 2 into 4 pieces using the line segments that you have drawn. Using squares 𝑎2 and the 4 pieces from square 𝑏 2 , construct a square congruent to square 𝑐 2 . (Hint: you may want to lay the pieces on top of square 𝑐 2 .) 2) Pythagoras’s Proof Pythagoras lived in the fifth of sixth century B.C. and founded the Pythagorean School in Crotona where students studied philosophy, mathematics, and natural science. Pythagoras and his colleagues are credited with many contributions to mathematics. • Draw two squares of the same size. • On Square 1, draw a vertical line segment so that you split the top and bottom sides of the square into two unequal pieces. Label the shorter piece a and the longer piece b. Next, draw a horizontal line segment so that you split the right and left sides of the square into two pieces of size a and b. • • ➢ What is the length of one side of the square? _______________________________________________________________ ➢ What is the area of the square (express using an exponent)? _______________________________________________________________ ➢ What are the areas of each of the four pieces that your square has been cut up into? ______________ ______________ ________________ _______________ ➢ Find the sum of these four areas. ________________________________________________________________ ➢ Write an equation representing the relationship between the area of the large square and the area of its parts. ________________________________________________________________ • • • On Square 2, split each of the four sides into segments of lengths a and b and mark this point. Be sure to alternate between a and b so that no two a’s will be consecutive and no two b’s will be consecutive as you move around the square. Connect the top point with the point on the left side of the square. Label this segment c. Connect the left point with the bottom point. Connect the bottom point with the point on the right side of the square. Connect the point on the right side of the square to the top point. ➢ What is the length of one side of the square? _______________________________________________________________ ➢ What is the area of the square (express using an exponent)? _______________________________________________________________ ➢ What are the areas of each of the five pieces that your square has been cut up into? ___________ ___________ ____________ ___________ ____________ ➢ Find the sum of these five areas. ________________________________________________________________ ➢ Write an equation representing the relationship between the area of the large square and the area of its parts. ________________________________________________________ • Using substitution, set the equation found for Square 1 equal to the equation found for Square 2 and simplify. 3) President Garfield’s Proof President Garfield discovered this proof 5 years before becoming president of the United States. He came up with this proof while having a mathematics discussion with some of the members of congress. • • Using a straight edge, draw a right triangle. Label the longest side of your right triangle c and the other two a and b. • Extend side a by b units (in the direction that is not by the right angle). You should now have a segment that is (a+b) units long. Construct a new right triangle, with the same dimensions as the first, using your newly drawn segment b. (Be sure that the segments labeled a do not touch.) Connect the two vertices of the two triangles that are currently not connected (not the right angles). Note that you have now created a third right triangle. Use your protractor to verify. • • • What shape have you created? ________________________________ 1 ➢ So, knowing that the area of a trapezoid is 2 (𝑏𝑎𝑠𝑒1 + 𝑏𝑎𝑠𝑒2 )𝑥 ℎ𝑒𝑖𝑔ℎ𝑡, find the area of the trapezoid you have drawn. (Remember the height of the trapezoid is the distance between the two parallel bases.) _______________________________________________________________ ➢ What are the areas of each of the three triangles that your trapezoid has been cut up into? __________________ __________________ ___________________ ➢ Find the sum of these three areas. ________________________________________________________________ ➢ Write an equation representing the relationship between the area of the trapezoid and the area of its parts. ➢ ________________________________________________________________ 4) Proof of the Pythagorean Theorem using Similar Triangles • Start with circle O having diameter GH and radius of length c. Point F lies of the circle so FO is of length c. Drop a perpendicular from F to point K on the diameter GH. Denote the length of FK as a and the length of KO as b. Joining points F, K, and O forms a ____________ triangle. For ∆FKO, the length of the hypotenuse is ___________ and the lengths of the legs are _________ and __________. F c a G • c b O H K Construct segments FG and FH. Since FK is perpendicular to GH (by construction), we note that both ∆FKH and ∆GKF are _______________ triangles. F c a G • c O b K H Note that angle HFG is inscribed in a semicircle, and as such, is a right angle. Therefore, the measure of angle HFG = ___________. Angles HFG and KFG share a common side FK and lie in a right angle; therefore, angle HFK is complementary to angle KFG. But, angle KFG is also complementary to angle FGK since the two non-right angles in a right traignle must have measure that sum to 90 degrees. Therefore, angle HFK must be congruent to angle FGK. It follows then that right triangles FKH and GKF are similar triangles. That is ∆FKG ~∆GKF. • • In ∆FKG, why is the length of leg HK equal to c-b? __________________________________________ __________________________________________________________________________________ In ∆GKF, why is the length of leg GK equal to c+b? __________________________________________ __________________________________________________________________________________ F a c-b G • c+b K H Since ∆FKG ~∆GKF, corresponding parts are proportional to one another. The triangles are shown below with the same orientation so that it is easier to see the proportional relationships. H F c-b F a K a G c+b • • K Write the proportional relationship between the two legs of the triangles above. Simplify algebraically. Does the result look familiar? p Principles and Standards for School Mathematics advocates an experimentation approach to middle-grades geometry study (NCTM 2000). Students are asked to explore and examine a variety of geometric shapes and discover their characteristics and properties using hands-on materials. They also create inductive arguments about the Pythagorean relationship. This empirical approach to the Pythagorean theorem, for example, will lay the foundation for analytical proofs. Incorporating experimentation as part of middle school geometry is also consistent with research on how students learn best. According to van Hiele, students need opportunities to develop their geometric thinking through five levels: (1) visualization, (2) analysis, (3) informal deduction, (4) deduction, and (5) rigor (Fuys, Geddes, and Tischler 1988; van Hiele 1986). Levels 2 and 3, analysis and informal deduction, are especially relevant in the middle grades. At the analysis level, students can go beyond their perception of shapes and analyze the components, parts, and properties of shapes. Students use descriptions instead of definitions. They also discover and prove properties or rules by folding, measuring, or using a grid or a diagram. At the informal deduction level, students learn the role of definitions, analyze the relationships between figures, order figures hierarchically according to their characteristics, and deduce facts logically from previously accepted facts using informal arguments. The activities presented here will help students make the transition from level 2, analysis, to level 3, informal deduction. In activities 1 and 2, middle-grades students explore the Pythagorean theorem by using jelly beans to estimate the areas of squares and semicircles. In activity 3, students use the result they obtained for squares and apply deductive reasoning to establish the result for semicircles on the Jelly Beans Jeong Oak Yun and Alfinio Flores 202 Mathematics Teaching in the Middle School ● Vol. 14, No. 4, November 2008 Copyright © 2008 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM. iStockphoto.com The Pythagorean Theorem with sides of the right triangle. In activity 4, students use their discovery about the areas of semicircles and deductive reasoning to establish further relations among figure areas. THE JELLY BEAN STRUCTURE Activity 1 uses a cardboard mat that is constructed with sides to hold jelly beans. Figure 1 shows a right triangle and its corresponding squares. Figure 2 illustrates the mat with cardboard fencing to hold one layer of jelly beans. To construct the mat, draw the shapes by hand or with computer software and paste the colored figures on cardboard that is about the size of a shoe box. To make the fences, cut strips of corrugated cardboard that are about 2 cm wide. (We recommend cutting perpendicularly to the ridges Jeong Oak Yun, joyun@asu.edu, is a graduate student at Arizona State University and a teacher of public secondary school in Seoul, Korea. She likes to use dynamic models to help students understand mathematical concepts. Alfinio Flores, alfinio@udel.edu, is a professor of mathematics education at the University of Delaware in Newark, where he teaches mathematics and mathematics education courses. He likes to make mathematical concepts more tangible for students and help teachers develop a profound understanding of school mathematics. Fig. 2 The jelly bean structure (a) The mat’s outside fence is glued in place. (b) The triangular fence is added but not glued. Photographs by Jeong Oak Yun; all rights reserved Fig. 1 The mat (c) A layer of jelly beans covers the legs. Vol. 14, No. 4, November 2008 ● Mathematics Teaching in the Middle School 203 given different mats based on different shapes, such as isosceles right triangles (see fig. 2c) and scalene right triangles (fig. 3), and then compare results. Students could move from one station to the next, where the layout of each mat is different, and conduct a similar experiment. Photographs by Jeong Oak Yun; all rights reserved Fig. 3 A scalene right triangle ➞ Photographs by Jeong Oak Yun; all rights reserved Fig. 4 Jelly beans on the move (a) Slide the jelly beans from the squares on the two legs to the square on the hypotenuse. (b) Jelly beans cover the square on the hypotenuse. so that the strips can be bent easily.) Paste the strips on the mat along the outside of the figures with instant glue. Make the fence for the inner right triangle separately and do not glue it down, since it will be removed during the experiment. Estimating the area with jelly beans works well as long as students abstain from eating the candy because (1) jelly beans are easy to layer; (2) empty space, or gaps, around the jelly beans will be kept to a minimum; and (3) students like the bright colors. Although jelly beans are threedimensional objects, we are using one layer, which is two dimensional. In this activity, students will not have to count the number of jelly beans to compare areas. The activities are designed to allow students to move jelly beans from one section of the mat to another to see whether or not all the jelly beans fit into a new shape. There are several ways in which the activities could be conducted. The teacher can demonstrate the first activity and discuss the relationship between the areas of the squares. Students can also work with the mat in groups. Different groups could be 204 Mathematics Teaching in the Middle School ● Vol. 14, No. 4, November 2008 Activity 1: The Pythagorean relationship Students will use a mat that consists of a right triangle with squares on each of the three sides. With the cardboard fence glued around the three squares, students insert the triangular fence around the triangle (see fig. 2b). They should notice that a square is formed on each side of the right triangle. Students fill the two squares on the legs of the right triangle with jelly beans, ensuring that jelly beans completely fill the two squares without overlapping, forming just one layer (see fig. 2c). Next, students remove the cardboard triangle that is inside the frame and push all the jelly beans into the square on the hypotenuse (see fig. 4a). Finally, they re-insert the triangular fence and flatten the layer of jelly beans. Students verify whether the jelly beans completely cover the square on the hypotenuse (see fig. 4b). The teacher guides students to explicitly express in their own words what they found about the sum of the areas of the squares on the legs of the right triangle compared with the area of the square on the hypotenuse. Then students label the two legs of the right triangle as a and b, and the hypotenuse as c, and express the relationship using an algebraic equation: a 2 + b 2 = c 2. Activity 2: EXTENDING the Pythagorean theorem Students can also use jelly beans to explore the relationships between areas when other similar shapes are constructed on the sides of a right triangle. Students use a mat with Fig. 6 Extending the Pythagorean theorem for semicircles Photographs by Jeong Oak Yun; all rights reserved Fig. 5 Precise semicircles ➞ Photographs by Jeong Oak Yun; all rights reserved (a) Three semicircles surround a right triangle. (b) Three fences form three semicircles. three semicircles adjacent to each side of a right triangle (see fig. 5a). The diameters of the semicircles are the sides of the right triangle. With the cardboard fence around the three semicircles glued to the mat, students insert the triangular fence around the right triangle. They need to notice that the two fences together form three semicircles on the sides of the right triangle (see fig. 5b). When making the structure for students, be sure that the arcs are precise semicircles. In the same way as in the first activity, students pour one layer of jelly beans in the two semicircles on the legs of the right triangle, completely covering the two semicircles (see fig. 6). Then they remove the cardboard triangle that is inside the frame and slide the jelly beans into the semicircle on the hypotenuse. Next, they re-insert the triangular fence, flatten the layer of jelly beans, and verify whether the jelly beans completely cover the semicircle on the hypotenuse. Students can describe in their own words the relationship they see between the sum of the areas of the semicircles on the legs of the right triangle and the area of the semicircle on the hypotenuse. The teacher will also guide students to see that in each of the activities, the three shapes on the sides of the right triangle are similar to each other, that is, squares are similar to other squares, and semicircles are similar to other semicircles. These two activities correspond to van Hiele’s level 2, because the verifications are done empirically. Other Extensions Students can also experiment with different shapes constructed on the sides of the right triangle, as long as all three shapes are similar and their corresponding sides are placed on the sides of the right triangle (see figs. 7a and 7b). Of course, these extensions to the Pythagorean theorem in terms of similar shapes on the sides of a right triangle are not Vol. 14, No. 4, November 2008 ● Fig. 7 Other extensions of the Pythagorean theorem (a) An equilateral triangle is along each side. (b) A quarter circle is along each side. new (Flores 1992; Heath 1956; Pólya 1948), but students are usually surprised that the Pythagorean cc2 2 relationship holds true for shapes C C other than squares. aa2 2 Mathematics Teaching in the Middle School 205 a1 c1 c1 a1 t t Activity 3: Formulas for the areas of the semicircles and connections with algebra Students can derive the extension of the Pythagorean theorem with semicircles by using some algebraic skills, such as factoring and simplifying algebraic expressions. They can do this activity working in pairs or small groups. Students can denote the two legs of the right triangle as a and b, and the hypotenuse as c (see fig. 5a) and express the relationship among the areas of the squares on the sides of the right triangle as a 2 + b 2 = c2. They need also remember that the diameter of each semicircle is congruent to the corresponding side of the triangle and that the sum of the areas of the semicircles on the legs of the right triangle is equal to the area of the semicircle on the hypotenuse. The teacher can guide students to write algebraic expressions for the areas of the semicircles and for their relationships. Students can write an algebraic expression for each radius of the semicircles in terms of the length of each side of the triangle and write an expression for the area of each semicircle. They can use algebraic notation to state that the sum of the areas of the semicircles on legs a and b of the right triangle is equal to the area of the semicircle on xxxxxx the hypotenuse, c: 2 2 2 1  a 1  b 1 c × π+ ×  π = ×  π 2  2  2  2 2  2 Students then factor and simplify both sides of the equation to show that it is equivalent to the equation a 2 + b 2 = c 2. To verify, they can reverse all the steps; starting with a 2 + b 2 = c 2, they can show how to obtain from this equation the relation among semicircles. This activity corresponds to van Hiele’s level 3 because students make deductive arguments based on previously accepted facts. 206 Fig. 8 Crescents formed by semicircles c2 C a2 a1 c1 t A B Activity 4: Area of the crescents of Hippocrates Hippocrates of Chios, a Greek mathematician from the fifth century BC, discovered that two figures bounded by arcs of circles have the same area as a figure bounded by straight lines. In figure 8, semicircles are cons

ENTREPRENEURSHIP 1 Entrepreneurship Institution Affiliation Date: ENTREPRENEURSHIP 2 Opportunity for exploration In the technology market, a lot of opportunities and market gaps exist; making room for improvements and new market entries. Our team will be looking at the available entrepreneurship opportunities and investment opportunities. We need to put into consideration the market gap, the risks involved and also the profitability on the investment opportunity. We will look at different companies, the services they provide, the importance of those services and the need to improve or not to deliberate in the company. With the importance information on the financial statements of the companies and the revenues derived from them, we will choose to work on analyzing only one business and evaluate whether it is advisable for us to invest or recommend someone to invest in the business. Our deliberation will be based on examination of its business model, market strategy, completion and the capital structures. There is also need to do a Strengths, weaknesses, opportunities and Threats (SWOT) analysis on the company in order to make sure that we find a viable investment solution. Investments should be for business expansion and improved service delivery to the business clients. It ought to bring a solution to an existing problem in a company or business. ENTREPRENEURSHIP 3 SUNNING LIMITED EXECUTIVE SUMMARY Sunning Limited is a private limited company that was …

Questions and Data Analyses Assignment and Peer Review Part One 1) Think of a topic or issue that you are passionate about. Regarding your topic, what questions do you think are most important? (For example, you may love dogs and care about health. You could ask “Can owning a dog be good for one’s health?”) 2) Search internet for articles describing data analyses that address your questions. (For example, the article https://www.ahajournals.org/doi/10.1161/CIRCOUTCO MES.119.005554 addresses the example question given in 1) 3) Critique one of these articles using the concepts that are covered in the till now. Compose one-two page paper (500-750 words) describing your critique. Please include: a. Summary of your question and why you picked the question b. Summary of the article and how it addressed your question. c. Review the Elements of Data Style. Categorize the analysis (or analyses) (descriptive, exploratory, inferential, predictive, causal, or mechanistic). Explain the reasoning behind your choices. d. Address context, resources, and audience of the analysis in the article that you chose. e. Include limitations, perspective, alternatives, assumptions, boundaries and weakness f. Make at least five direct references to our readings in your paper.

Problem Set #6 Economics 2180 Due on December 11th. Please put your name, your section and your TA’s name on the problem set. You are allowed to work together, but each student must hand in their own problem set in their own words. Please make sure to do all problems. Skipping a problem is much worse than getting one wrong. Late problem sets will face a substantial penalty. #1 Suppose that American firms become more optimistic and decide to increase investment expenditure today in new factories and office space. How will this increase in investment affect output, interest rates, the exchange rate, and the current account? (show on ISLM with FX diagram & explain) [assume shock is temporary] #2 a. Use the ISLM to illustrate the impact on Y, i, E, and the trade balance if there was a sudden temporary increase in consumer confidence (they feel better about their job prospects and debt levels). Draw the shock, and then explain what happened to each variable. b. do the same as in a), but assume that the home country has a fixed exchange rate maintained by the central bank. #3 a. Use the ISLM to illustrate the impact on Y, i, E, and the trade balance if there was a temporary fall in foreign government spending. b. Explain what policy response you would take if you were allowed to adjust either fiscal or monetary policy and had as your goal maintaining full employment. (assuming NO fixed exchange rate in this scenario) #4 use the symmetry integration diagram to place the following countries on the diagram and say if they should peg or not (just use your general knowledge about them, no outside research is required). That is, draw up the diagram and place four points (one for each below) and then just write a sentence for each as to whether it seems they should peg or not. a. b. c. d. Should El Salvador peg to the U.S. Should Brazil peg to the U.S. Should Lithuania peg to the euro Should El Salvador peg to the euro OPTIONAL PRACTICE PROBLEM #5 Lithuania was pegged to the euro. Using the IS-LM-FX model for home (Lithuania) and foreign (euro) show how each of the following impact Lithuania: a. The Eurozone reduces its money supply b. Lithuania cuts government spending Lectures for Chapter 18 Summary of intervention • Central banks frequently intervene • Often sterilized and often ineffectual • If they want to move E, they can, by doing unsterilized intervention (or just moving the interest rate). • They MIGHT have an impact if they sterilize, but usually only if there are special circumstances (capital controls) or if they can somehow move expectations – Joint operations are much more successful 2 So far… • Thought about trade flows and impacts on firms and factors of production • Considered how to think about the current account • Developed a model for the exchange rate • Thought about price levels and money • In the end though, what matters to firms and people? – Output • Jobs, Standard of living • Business conditions, profit opportunities 3 Output in the short run • Again, two countries (or home and RoW) • Short run so prices are… – Sticky • If prices are sticky, and we are in a long run equilibrium, we’ll say expected inflation is zero • We assume fiscal policy (G & T) are set by policy (they are exogenous) – Recall, G is just spending, not transfers • Other parts of Y=C+I+G+(X-M) will fluctuate with different variables. – Holding transfers and foreign income at zero so – CA = X-M – Assume all in rest of world (RoW, or “foreign”) are constant unless we state otherwise 4 What determines consumption? • Consumption = C(Y-T) – That is, consumption is a function of income – People consume income at some rate (the marginal propensity to consume or MPC) 5 Reasonable? • YES! People due tend to consume a particular amount of their income • It can change: – – – – – – We’ll often “shock” the MPC Fear of a recession or losing job >>> MPC falls Fear of having too much debt >>> MPC falls Optimism >>> MPC rises Sudden ability to borrow more >>> MPC rises More wealth means MPC may rise • Key point is MPC is constant unless we give you a story saying how/why it changed 6 What determines investment • Firms investment is a decreasing function of the real interest rate. That is, investment falls when the real interest rate rises. 7 Reasonable? • YES: firms often borrow to make their marginal investment. They’ll have an expectation about “rate of return”, as borrowing costs rise, they’ll undertake fewer projects. • This is just a repeat from our S, I, CA model • It can change: – Again, many things can change it • “animal spirits” • Optimism / pessimism • State of the financial sector (ability to borrow at a given interest rate) 8 What determines the trade balance? • What should determine the trade balance? – The real exchange rate (q) recall q = EP*/P (up is a depreciation) • Increasing function: as our goods get cheaper, we export more and import less • Referred to as expenditure switching – Home income (how much can we spend) • Decreasing function: as our income rises, we consume more imports – Foreign income (how much can they spend) • Increasing function: as their income rises, they can buy more of our stuff. 9 Trade balance * * * TB = TB(  E P / P , Y − T , Y − T    ) Increasing function Decreasing function Increasing function • Upward sloping against q, changes in Y or Y* are seen in a shift in the curve 10 A bit more careful • If home depreciates, what happens to the volume and price of home exports? – P: Nothing – Volume: up – Goods stayed same price at home, but got cheaper to world, so volume up. • If home depreciates, what happens to the volume and price of home imports? – P: up (same price in foreign currency, now more expensive here) – Volume: down (due to increase in price) • Overall, volume effect says TB improves, BUT, price effect is negative: we’re paying more for our imports. • KEY ASSUMPTION: we will assume the volume effect wins out at least over time. • We’ll return to these issues 11 Still… you do see effects • Over time, a change in E should change prices people face, and you do see a broad correlation between q and TB. – Operates with a lag – Lots of omitted variables • Expenditure switching does happen and is important, but is not as simple as it may 12 sound at first. Exogenous shocks • In addition to changes in G or T (or Y*), we can also let the curves shift • E.G., could be optimism about economy >>> more consumption, or more investment, or a taste shock towards home • Means that the curves shift. 13 Equilibrium in economy • We know that Y=C+I+G+(X-M) • We also know that the amount we demand or spend must be equal to our production / income (having taken into account (X-M)). – In a closed economy, it is obvious. – In an open economy, the CA tracks the gap • So, we can think about all of these things we’ve determined make up demand and know they must = income (Y). 14 Putting it together • Equilibrium is where Demand equals supply in the overall economy ( ) Y = C (Y − T ) + I (i ) + G + TB EP * / P , Y − T , Y * − T *  D 15 intuition • Slope of D line is more flat b/c as income rises, demand for home production does not rise 1 for 1 • Always need to be where D = Y • At point 3. we’re in excess supply – Inventories will build up, firms produce less, shift over to 1 – Opposite if at point 2 16 What happens if I goes up? (surge in optimism) • Moves I function out, which shifts demand up which leads to higher output 17 Factors that shift the demand curve • What would shift out the demand curve? Fall in taxes T Rise in government spending G Fall in the home interest rate i Rise in the nominal exchange rate E Rise in foreign prices P* Fall in home prices P Any shift up in the consumptio n function C Any shift up in the investment function I Any shift up in the trade balance function TB        Demand curve D  up  shifts  Increase in demand D  at a given level of outputY      • Anything that would make demand at home rise 18 How are i and Y related i IS Y • We will sketch out the relationship that when i is high, Y is lower (and vice verse) • Changing “i“ is a slide along the curve • Other things can shift the curve (things that will change Y at any given i) 19 Putting everything together • We will use the cross diagram to motivate what is happening with demand (IS curve) • THEN use the interest rate we get to figure out what is going on in the FX market • Just like using our MS/MD diagram next to the FX market. • After 5 minutes from now, won’t draw the cross, it is just in the background. 20 Cross will operate in the background • • We’ll think about the relationship between i and Y, and use the goods market to derive it first. [the IS curve] AND will keep track of what happens to E. When i falls, Y goes up BOTH due to the investment impact AND due to the trade balance (via the interest rate impact on E) 21 Anything else that moves demand, moves the IS curve • Nothing happens to the interest rate or FX yet until we put another curve on the IS diagram • Note: change in i is how we derive IS (slides along curve) the rest shift it 22 Factors that move IS out • What shifts the IS out? Fall in taxes T Rise in government spending G Rise in foreign interest rate i* Rise in future expected exchange rate E e Rise in foreign prices P* Fall in home prices P Any shift up in the consumptio n function C Any shift up in the investment function I Any shift up in the trade balance function TB • •        Demand curve D IS curve   shifts right up shifts    Increase in Increase in demand D  equilibrium outputY at any level of outputY  at a given and at a given home interest rate i home interest rate i     i* and Ee are on here because it shifts FR out and thus depreciates home currency, increasing the TB Prices move q and therefore move TB too. 23 How else are Y and i related? • Money demand. • An increase in Y leads to an increase in money demand and an increase in i. That is our “LM” curve 24 LM will move if i changes for a given Y Rise in (nominal) money supply M Any shift left in the money demand function L LM curve   down or right  shifts   Decrease in equilibrium home interest rate i at given level of outputY 25 Reality check • There are some other ways of conceptualizing the LM – Think of it as the Fed’s “reaction function” • If Y goes up more than they wanted, the would raise interest rates (hence upward slope) • If they decided to be more aggressive with monetary policy (i.e. cut rates) that is shifting the LM curve down • Some rename it the “MP” curve (monetary policy), but really winds up working fairly similarly – To avoid confusion, we’ll use the “LM” terminology, but you should know it can be more broadly applied with similar results 26 Equilibrium • Wind up where both goods and money markets are in equilibrium. • The interest rate coming out of that will give us the exchange rate. • Now, we can have a variety of shocks and variety of policies and can keep track of what happens to Y, i, and E. 27 Using the framework • Goal is to let us have a range of shocks that will give us a new Y, i, and E 28 Temp Increase in the MS • What moves? – LM moves out (lower interest rate for a give level of output in our MS/MD model) • What happens? – Lower i, higher Y, higher exchange rate (more depreciated home currency) 29 Temp Increase in the MS • Money supply rises, LM goes out, i falls, exchange rate depreciates, and Y rises • NOTE: not moving IS curve, sliding along. The increase in investment and exports is built into the IS slope 30 Same as last week (but now we see Y too) • This is what we did last time. Temp MS up got us a lower interest rate and a more depreciated exchange rate • Note this picture is holding Y constant. Innovation this week is endogenous Y 31 Temp increase in G • What moves? – IS curve shifts out (greater demand) • What happens? – i goes up, Y goes up, E goes down (home appreciates) 32 Increase in G • Note the crowding out as both the interest rate rises and exchange rate appreciates – If interest rate did not go up, Y would have gone out more – This is what the upward slope of the LM gets us 33 What about holding E fixed? • As we’ll discuss next class, sometimes the central bank may want to hold the exchange rate fixed. • That will greatly change how things work. 34 What if CB didn’t want to let E change • They CAN’T increase the MS – Doing so would change E, and they don’t want to – So you can’t use monetary policy 35 Temp increase in G (E fixed) • A soon as i goes up, E starts to move, CB must respond. How? • Increase M so LM shifts out and i stays same so E stays same • Y has gone up a lot (no crowding out) 36 Fiscal policy if E is constant • Here, if G goes up, there’s pressure for i to rise and hence E to fall (home appreciate) (shift from IS1 to IS2) • CB must shift out LM to get to point 2 so E stays fixed. • We never get to that equilibrium at IS2 and LM1. CB responds right away 37 Other shocks • Note: increase in G or M are just two of many shocks • BUT, in a lot of ways, they all work the same. • Any of our shocks that shift up demand will look like those fiscal shocks – And don’t forget things can expand OR contract demand 38 The Long Run • Note: everything in this model is short run • Book largely is stopping there when it comes to stabilization policy. • In the long run, what happens? – PRICES ADJUST! • What does that do? – Will eventually get us back to equilibrium 39 Prices rise in response to permanent MS increase Long run Short run • • Short run is a big depreciation with Y up and i down (FR moves due to expectations) In Long Run, prices should rise, bringing M/P back to balance, so LM comes back. – ISLM equilibrium is back at 1 • Note: FR moves out permanently (just as in the overshoot), so we wind up at a more depreciated exchange rate in the long run despite going back to the same Y and i in the ISLM figure 40 A Drop in Investment with no response • • • If IS falls back and nothing is done, there is some cushion right away due to more depreciated E and lower interest rate (so IS does not go back as far as it would) Also, though, once people realize nothing will be done, they expect P to fall, FR shifts out, over time IS shifts back in some and LM out (as M/P goes up with falling prices) Gets messy: model is really about the short run and assumes you WON’T just sit in a recession indefinitely 41 In the Long Run • What is going to matter is productive capacity – Population growth – Productivity • We’re abstracting from these for the short run, but in the long run, they create the “Speed Limit” • Think of Yf as the “full employment Y” effectively the maximum potential output of the economy without overheating • If Y < Yf, eventually prices fall to accommodate that. If Y>Yf eventually prices rise to accommodate that. 42 So…. • Use ISLM to think about shocks to the economy or policy shocks and responses • Remember that there is a long run anchor based on Yf. • Will turn next to thinking about policy a bit more carefully and thinking a bit about the special challenges of policy in the current crisis. 43 Stabilization policy • In general, we will often assume the economy is at some Yf (full employment level of output) to begin with. • A shock hits taking us below Yf. Now what? • What can the CB do? – Can try to counteract the shock by increasing M • Will everything go back to where it was? – Depends on the shock. If an IS shock, no, monetary policy leaves a different end result 44 What does it look like? • What would a negative shock to consumption look like? • What would the CB response look like ? 45 Shock and response • Note: Monetary Policy does not just put things back to where they started if shock is to IS (fiscal would) 46 Example: Latvia vs. Poland • Poland aggressively used Monetary and Fiscal Policy to cushion the crisis. Latvia did neither 47 So how should CB behave? • Different choices: – – – – Target P (or inflation) Target GDP (or unemployment) Target E Some combination • Many CB simply target inflation and leave the rest. • Others also look at GDP or employment (FED has a dual mandate officially) – (and many inflation targeters do this de facto) • One way to look at this is something called the “taylor rule” or a “monetary policy reaction function” – Basically assumes the CB raises rates when inflation goes up and lowers them when unemployment goes up, and that you get some sort of function that averages these. 48 For example • fed funds = 8.8 + 1.65*π – 1.65*Unemployment – Using core inflation and full time worker unemployment • Wide variety of different rules depending on the weight on inflation or employment 49 CB tools • We are saying CB just adds / subtracts money • Again, more realistically: – Can change interest rates (that’s what taylor rule is emphasizing) • In some countries both borrowing and lending, various Repo rates, etc. In general, changes the overnight borrowing rate to change banks cost of funds. – Can change behavior by banks which alters how they generate money 50 Running out of room • How do we think about policy at the zero lower bound? • What should it look like on ISLM ? 51 Back to ISLM, but LM is flat • • The idea is that LM is bounded by zero, and given where we are, shifts in IS don’t affect interest rates: calls for fiscal policy This suggests shifting LM can’t do any good. There are other ways to think about it once we acknowledge not ALL interest rates are at zero (just short risk free) 52 How do we get there? i=0 • Shift back IS far enough AND start with i1 not too far above zero • Desired money market outcome is below zero, but you can’t get there. • Again, means shifting LM out does nothing and shifting IS out only affects Y, not i or E. 53 Basic Fed Balance Sheet • “Normal” Fed balance sheet • Could change this to try to do something else – Change expectations of E or inflation – Change the long run interest rate by buying those assets – The latter one doesn’t really go on the ISLM framework 54 Fiscal Policy • Thus far we’ve tried to respond to shocks with monetary policy • Could also try fiscal (changes in G or T) 55 Most simple case • In a simple world, consumption could shift in, but government spending increases or taxes drop and we’re right back to where we were. 56 If we just increase G, though, many things change • What is different? – i goes up and E goes down – Crowding out (I falls, TB weakens) • So, if C fell and G goes up in response, you may wind up with lower I than before. – Same Y, but different composition 57 Not all cases are simple • If shock is to LM, don’t get back to initial E – LM shock could be surge in money demand OR MS contraction • Could also have a shock to FR (say foreign interest rate goes down) 58 Foreign interest rate shock (down) (no CB response, but G up in response) • Shock moves in FR and moves in IS, government increases G in response to get IS back to its starting place • Note: if no response, E goes down (home appreciates) and Y falls a bunch. With response, E goes down MORE, but no change in Y. >>> Distributional impacts 59 Quick note • Can get confusing. • Earlier, we had FR move after a MS shock, but we did not go back and move the IS with it. • Key point is if the shock originates in domestic – (in which case the exchange rate impact is already built into the slopes of the lines) • or in the foreign – (in which case we move FR and the impact on E feeds back into a move in IS) 60 Raises an issue of coordination • In theory, who should respond to what shock? – Fiscal to IS shocks, Monetary to LM or foreign money market shocks • What are some problems? – Hard to tell which shocks sometimes (lots of things happening at once) – Different preferences from CB and fiscal authority – CB may want to “make” the government do something • Issue right now in Europe 61 What if CB CAN’T respond • Recall liquidity trap – Here all the pressure is on the fiscal to respond – BUT, no crowding out, so fiscal policy should be more effective 62 What other policy challenges • What’s the magnitude of the shock • Lags in policy implementation – M policy operates with a lag – Fiscal policy may take a while to pass/spend • Other country responses – If you expect a depreciation to help, they may expand M too! – Can’t all export out of a shock at the same time • Private sector offsets – Ricardian equivalence issues • should only be an issue with temporary tax cuts, spending impacts go through as do permanent tax cuts or temp tax cuts to liquidity constrained households – Market responses • If market fears inflation too much, short term rate cut may not lower long rates (either expecting rate reversal or inflation) • If market fears long run solvency, rates could rise a lot when fiscal expands • Lower level government responses: – State and local may be liquidity constrained and may cut while Federal expands AND if you spend through them, they may not spend all of it. 63 Fiscal policy catch 22 • If close to a solvency border, there’s a challenge: – Stimulate economy, could lead to rising interest rates which constrain economy – DON’T stimulate economy hurts solvency by constraining growth 64 Change in E and change in TB • There are two other issues that ISLM skips • The first is the “J-curve” – The idea is that when the exchange rate first changes, we likely don’t see quantities move right away • Contracts already signed • Takes a while to find new customers – That means buy same number of imports, but they cost more (they got more expensive in local currency terms when exchange rate depreciated). >>> total value of imports rises – Value of exports unchanged in local currency at first – So at first X-M goes DOWN from a depreciation – After the brief period (6 months?) the quantities adjust, and Trade Balance improves • Looks like a “J” 65 The J-curve • We are skipping the transition and just assuming depreciation for home leads to CA improves 66 Pass-through We see evidence for this in how much trade is invoiced in foreign currencies • Another issue is that prices don’t always change 1 for 1 with exchange rates (pass-through = 1 if change in E is passed through into change in P) – Foreign firms might price in local currency so as not to lose market share due to volatility – If so, the price consumers see won’t move – No expenditure switching from exchange rate changes • NYT article gets at this: – Crate and barrel buyer “we certainly won’t import less” – If firms and consumers don’t change, then impact from a change in E is limited – Very little impact on IS when E changes 67 Evaluating fiscal policy • What is the multiplier – The extent to which Y goes up when G goes up • In a normal (non liquidity trap) economy, what is the multiplier ? – Should be less than 1 if there is any crowding out – Multiplier > faster bounce, but hard to disentangle from other things going on – G20 tried to get coordination: Why? 72 Impact of G* going up • • • • G* up means Y* goes up means i* goes up This shifts out FR, and depreciates home currency. Shifts out the IS curve (i* up moves out IS is one of our shocks) Implication: great if the other country spends. You get some benefit, but none of the debt. To avoid free-riders, major economies all agreed to act 73 When to do stimulus • In a recession – Especially a long and severe one • When monetary policy is constrained – Especially at the zero bound • When market responses are not going to wipe it out – When you have “fiscal space” • When there are things you “need to do” anyway – Can simply accelerate spending from future on things like infrastructure. In that case, have not changed the long run budget picture. • In a larger country – Smaller the country, more of the spending goes to spillover into foreign economy. 74 Using ISLM • The key lessons from this material: • ISLM can let us look at all sorts of shocks – Changes in government spending or money supply – Changes in optimism, consumption, investment, preferences etc. • We can make different assumptions about a shock and then the policy response – Could ask you to think about a fiscal (G,T) or monetary (M) policy response to a shock – Could ask you to think about a fixed E response or float • Should know the model and how to manipulate it AND some of the aspects that are simplifications – Should be able to tell me what happens to E, i, Y, and trade balance 75 Chapter 15 slides Where we are • We’ve introduced the concept of exchange rates as asset prices where investors decisions are based on expected returns • We’ve introduced the notion of PPP as a guide for long run behavior of the exchange rate. • We’ve introduced money and output (largely via the quantity equation) as the driving forces behind prices in the long run. • Also, though, money and output can drive interest rates via influence on money supply and money demand. • NOW: put everything together to get a model for what is happening to exchange rates in the short run and long run. 2 Recall • Fundamental equation of UIP. • We’ll say Ee is anchored (by PPP) and i and i* are anchored by CB policy and macro shocks. • Now, we’ll let shocks to these things alter the exchange rate 3 Poll Question • Home i is 5%, foreign is 3% Ee is 1.224 and E is 1.16. Would you rather hold – A: Home – B: Foreign – C: makes no difference 4 Simple calculations • Note: i, i* and Ee don’t change. • By changing E, the expected depreciation of home changes, so the expected return on the foreign bond changes • Equilibrium is where the expected return is the same – That is, where column 1 = column 6 5 Example • • Take bottom row: If Ee is 1.224 and E is at 1.24, you’d be expecting APPRECIATION of home. – Hence -1.3% in column 5 • If Home has higher interest rate too, you’d win twice – Foreign earns 3% + expecting home to APPRECIATE 1.3% >> return on foreign = 1.7% much worse than return on home (5%) • Calculate expected change in E, add to i*, compare to home i. Should be the same. 6 We can graph it • The more appreciated home is to begin with, the better deal foreign is. – If we were at point 2, everyone wants to hold euros, so they sell dollars and buy euros such that dollar depreciates (E goes up) • Or, the cheaper home is today, the better deal home is today – If we were at 3, dollar is too cheap, everyone wants to buy it, so dollar appreciates 7 Key concept: we know the future • The idea is we know where the exchange rate is going, so if the dollar depreciates today, that doesn’t signal a run on the dollar, but signals an appreciation in the future. • Think of it like a stock. You have a long run price target (say 50$). If Stock is below 50, you buy. If the price rises, it makes you less likely to buy. 8 Three shocks: what happens • Home interest rate rises • Foreign interest rate falls • Expected exchange rate goes down (we now expect a more appreciated exchange rate in the future) • In all three cases, we’ll want to think how the graph works, what the intuition is, and how it works in the equation 9 Home interest rate rises: what happens • 1. DR line shifts up (home rate is higher) • 2. we find new equilibrium where home and foreign returns are equal (up where E is lower or home more appreciated) 10 3 shocks: 1. home interest rate • Home increases interest rate: – E goes down – Sets up larger expected depreciation of the dollar, making foreign deposits more attractive (makes up for the higher home interest rate) 11 Is this right? • i = i* + %ΔeE – At first: .05 = .03 + .02 – Now: .07 = .03 + ??? Has to be .04 – If Ee didn’t change, E must have APPRECIATED, setting up a bigger depreciation expectation – E goes down roughly 2% • INTUITION: – At first I was happy with money in either place – Home rate goes up, so I (and everyone else) move money to home, making it appreciate – In equilibrium, homes expected depreciation offsets the higher interest rate. 12 Foreign interest rate falls: what happens • 1. FR line shifts down (lower return on foreign) • 2. we find new equilibrium where home and foreign returns are equal (up where E is lower or home more appreciated) 13 3 shocks: 2. foreign interest rate falls • Foreign cuts interest rate: – E goes down – Sets up larger expected depreciation of the dollar, making foreign deposits more attractive, which makes up for their lower interest rate 14 Is this right? • i = i* + %ΔeE – At first: .05 = .03 + .02 – Now: .05 = .01 + ??? Has to be .04 – If Ee didn’t change, E must have APPRECIATED, setting up a bigger depreciation expectation – E goes down roughly 2% • INTUITION: – At first I was happy with money in either place – Foreign rate goes down, so I (and everyone else) move money to home, making it appreciate – In equilibrium, homes expected depreciation offsets the higher interest rate. 15 3 shocks: 3. Ee falls • Expected exchange rate goes down (more appreciated for home): – E goes down – Since home is expected to depreciate less than before, it needs to appreciate some right now to get us back to the same expected change in E that we had before 16 Is this right? • i = i* + %ΔeE – At first: .05 = .03 + .02 – Now: .05 = .03 + ??? Has to still be .02 – If Ee changed (appreciated), E must have APPRECIATED by the same amount, keeping the same depreciation expectation – E goes down roughly 2% (same as expected E) • INTUITION: – At first I was happy with money in either place – I think home will be more appreciated in the future, so I (and everyone else) move money to home, making it appreciate – In equilibrium, homes expected depreciation offsets the higher interest rate. 17 Expectations matter • You can trace many exchange rate movements to shocks to expectations. • Famous example is the value of the confederate dollar during the civil war. 18 Fleshing out the model • Right now, only E can change unless we change it. – In economics jargon, we say everything else is exogenous • Now, we’d like to expand the model so we have fundamental shocks that move both the interest rate, the exchange rate, and expectations • More moving parts. – First up: interest rates 19 A key assumption • Prices are sticky (nominal rigidity) • We’ve mentioned this a few times: Why are they sticky? – “menu prices” • Literally changing prices • Making decisions – Wage contracts – Supply contracts • We will tend to talk about the “short run” as a time period when prices are sticky. – You can think of it as about a year or so. • Asset prices (exchange rates, interest rates, etc.) are flexible 20 Interest rates • There are all sorts of interest rates in the economy • We are going to focus on the short term interest rate, think of it as a money market rate set overnight. • A longer term rate might be more expensive (or perhaps less), and riskier borrowers would be charged more, but we’ll think of these gaps as consistent across the world and focus on 1 rate per economy. 21 Interest rates • Last time, we talked about money supply having to be equal to money demand. – – – – – We’re going to focus on real money supply and real money demand: Recall our money demand equation We just replace MD with MS and… d M Have an equation where = L   Y P real MS equals real MD A constant Real income  Demand for real money • What would make that happen? • Interest rates can change so make supply and demand equal. • Think of the interest rate as the “price of money” • So, “L” is going to be a function of the interest rate 22 Remember our money demand • Downward sloping money demand – Less money demand if interest rate is higher • Money demand can move with output 23 Interest rates MS i i1 MD M/P • How will we determine home interest rate? – Start with our money demand and add…. • Money supply (vertical line) – Interaction of home money supply (set by CB) and home money demand (based on L function derived last time) 24 Money Market Equilibrium • Note that at 2. and 3. there is no equilibrium • Why not? – MS does not equal MD 25 Interest rates: getting to equilibrium • At point 3: what is wrong? – Too much money demand – People sell bonds (price of bonds fall, interest rate goes up) – Eventually we’re at 1. 26 Changes in M in the short run MS1 MS2 i i1 i2 MD M1/P1 M2/P1 M/P • We are going to largely define the short run as the period when prices are sticky. • So, what happens to M/P when the MS goes up? – In the short run, M changes, P is constant so M/P goes up (shifts out) – i goes down. 27 Intuition • CB increases supply of M • In the short run, P is sticky, so it is a REAL increase (ie real MS goes up) • If supply of money goes up, price of money should fall, so interest rate goes down. • Note: this is different than case with an increase in the growth rate where you are primarily affecting expectations of inflation and hence and increase in M growth >> R going up. – Our diagram is “static” we’re looking at just Short Run, not thinking about ongoing inflation 28 Increase in Y in the short run MS i i2 i1 MD M/P • So, what happens to M/P when the Y goes up? – NOTHING, MS is set by CB • What happens on the figure then? – MD shifts out (because of an increase in Y which increases transactions demand for cash) – In the short run, i goes up 29 Some implications • CB in this example can directly control the interest rate and does so by changing MS. • In practice, CB USUALLY sets a target interest rate for very short end of yield curve and lets MS adjust to get the rate it wants • Complications – Long rates may not obey the CB – If short rate hits zero, now what? 30 For Example • • • Here we see the failure of long rates to respond to the Fed raising rates and subsequently fail to respond when it cut rates. This makes monetary policy appear ineffective Need to consider the counterfactual (where would rates be without the Fed actions) 31 A bit more traction • On a number of other rates we see a response to Fed Funds 32 rate (the rate the Fed sets) A bit more traction • • Enough that we will assume that the CB can control rates to some extent Exception will be the zero lower bound at which point it may need to do more 33 From the book • You do see some things (treasury and mortgages) moving with fed funds rate • Others, though, you likely see a re-pricing of risk at the point of the crisis. Makes it harder to tell what’s going on 34 Changes in M in the short run MS1 MS2 i i1 i2 MD M1/P1 M2/P1 M/P • We are going to largely define the short run as the period when prices are sticky. • So, what happens to M/P when the MS goes up? – In the short run, M changes, P is constant so M/P goes up (shifts out) – i goes down. 35 Increase in Y in the short run MS i i2 i1 MD M/P • So, what happens to M/P when the Y goes up? – NOTHING, MS is set by CB • What happens on the figure then? – MD shifts out (because of an increase in Y which increases transactions demand for cash) – In the short run, i goes up 36 Putting it together • We now have a way to set the local and foreign interest rate in the money market • Shocks to the money supply, output, expectations, etc. can all generate changes in the exchange rate, interest rates, or both. • CRUCIAL ASSUMPTION: capital (or financial) mobility. If no one can move money across borders, all the talk of “moving money to take advantage of better returns” is moot. 37 Recall our ForEx market • Now add our money market as the thing driving DR 38 Putting i and E together • Now we just let the interest rate be determined in the money market and then trace things to the ForEx market • FOREIGN money market is not shown, changes there just show up in a change of the FR curve 39 What happens if??? • Temporary increase home money supply ? • Temporary increase foreign money supply ? • Note: temporary for now because we don’t want to have to think about what happens if prices move yet. 40 Home money supply up • 1. MS line shifts out • 2. interest rate falls • 3. exchange rate goes up (home depreciates) 41 Home MS up • Two steps: – Start with impact on money market. – Take new interest rate to the FX market to get new E 42 May look complicated, but… • All we’ve done is taken two figures that are (hopefully) fairly intuitive • Money supply going up pushes interest rates down • Lower interest rate means the home currency depreciates immediately • What happens in the long run? – Everything just goes back. Temporary shock 43 Foreign MS up • Change in foreign has no impact on domestic Money market • Now everything is going on in the FX market • Now it is just a change in foreign interest rate (already done) 44 What happens if M up permanently • We know the short run impact on MS • What else moves? – Expectations of prices – Which means expectations of the exchange rate move – So the FR curve moves • Crucial thing is that expectations move on NEWS. Once people know prices WILL move in the long run, expectations for the future change NOW (even before prices start moving) 45 M up permanently • • • • 1. MS line out 2. interest rate falls 3. AND !!!! FR line moves out (Ee goes up) So E goes up A LOT 46 Short run after permanent shock • Still short run, but fact that it will be permanent means expectations move right away – Prices have not moved yet, but we know they will 47 Short run after permanent shock • So, we have both the MM response lowering interest rate AND an expectations shift – So a permanent shock has an even bigger impact that temporary EVEN IN THE SHORT RUN 48 What happens in the Long Run? • Prices adjust – (that’s always the answer to “what happens in the long run) • So what moves ? – M/P goes back – E adjusts 49 Splitting out the variables • Note: prices are adjusting slowly – M/P and hence i moves with it 50 Overshooting • Very famous result: overshooting of the exchange rate (can help explain some of the volatility) 51 Is this “OK” • Is it sensible to have this expected appreciation after the overshoot? – Yes, UIP still holds. Need the expected appreciation to balance out the lower interest rate. – If we increase the home money supply, home interest rate falls. If we started at i=i*, we now have i