Hello and welcome to the session. I am Asha and I am going to help you with the following question which says if A and B are distinct integers, prove that A minus B is a factor of A raise to the power n minus B raise to the power n whenever n is a positive integer. We have given a hint, write A raise to the power n is equal to A minus B plus B raise to the power n and expand. Let us now begin with the solution and with the help of this end we will solve the problem which will make it very easy. Now as we know A can be written as A plus B minus B or on rearranging it can be written as B plus A minus B therefore A raise to the power n will be equal to B plus of A minus B raise to the power n. On expanding the right hand side with the help of binomial theorem we have on the left hand side A raise to the power n and on the left hand side nC0 B raise to the power n into A minus B raise to the power 0 plus nC1 B raise to the power n minus 1 into A minus B raise to the power 1 plus nC2 B raise to the power n minus 2 into A minus B raise to the power 2 plus nCn B raise to the power 0 into A minus B raise to the power n or it can further be written as A raise to the power n is equal to B raise to the power n plus nC1 B raise to the power n minus 1 into A minus B plus nC2 B raise to the power n minus 2 into A minus B whole square plus goes on up to plus nCn is 1 and B raise to the power 0 is again 1 then we have A minus B raise to the power n. Now taking B raise to the power n on the left hand side we have A raise to the power n minus B raise to the power n is equal to and now taking A minus B common from all the terms we get A minus B and inside the bracket we have nC1 B raise to the power n minus 1 plus nC2 B raise to the power n minus 2 plus goes on up to nCn is 1 and here we have A minus B raise to the power n minus 1 and this shows A minus B is a factor of A minus B raise to the power n this completes the session hope you enjoyed it take care and have a good day.


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The title of this lecture is Austrian theory of the business cycle, but as I think you'll see it's actually a part of a much broader aspect of Austrian economics something I call capital-based macroeconomics and The business cycle theory is just something of a corollary to the more general Capital-based macroeconomics, I think you'll see how that works So what we're going to do is is Sketch out the whole business of this sustainable and unsustainable growth and I don't mind giving the punchline away It depends on what the central bank is doing and what they're doing with interest rates and that sort of thing if interest rates are market Determined it turns out we get sustainable growth if they're manipulated by the central bank you get something else and it comes in the form of the Austrian business cycle To put this in perspective I just list the elements of capital-based macroeconomics and you'll see that most of the elements are things that you already know Production possibilities frontier you've seen that In standard textbooks and even in my earlier lectures loanable funds market very broadly conceived Market for funds. It's a the markets plural that Channeled savings into the business community for investment purposes The structure of production we've got to work out of that yesterday. So we'll be able to Almost skip across that because you've already seen it and then labor markets. Well, of course, that's a That's treated heavily in Keynesian economics and monitors economics and so on the only difference here is that we have stage Specific labor markets. They don't all go up and down at once depends on what stage of production we're talking about and Our applications here are sustainable growth as supported by saving on one hand an unsustainable growth triggered by credit expansion Now what you'll see is I'll take up the bulk of the time with the first part the sustainable growth and this comes from a Methodological point that I got from Hayek. This is a paraphrasing but he says before we could even ask how things might go wrong We must first explain how they could ever go right. Okay? And so that first part explaining how they can go right is the bigger job And once we get through that then it becomes obvious what happens When things go wrong when the Federal Reserve is Manipulating the interest rate That's a methodological point by the way that virtually all other schools ignore Keynes for instance assume that the market was Unstable and it doesn't go right period So let's start out with the production possibilities from here And To with the macroeconomic Application we're looking at the trade-off between consumption and investment So under favorable conditions a fully employed economy Is what we're talking about features consumption and investment as alternative uses of the economy's resources So it allocates resources in both uses Making the most of the trade-off and of course that's the market at work for you and for me So we may be at that point on the frontier and by the way the frontier is Is the locus of sustainable combinations of consumption and investment we'll see later That is possible to go beyond the frontier but not sustainably in other words you get into a region with it's unsustainable namely a boom that will end in a bust Okay, the PPF Here I've just indicated that it's used for a lot of things ever everybody's seen it in the textbook no matter which one They use and it's used for emphasizing the concept of scarcity illustrating the implied trade-off and Expositing theories of capital and interest economic growth international trade But it rarely appears in a macroeconomic construction Certainly not in Keynesian economics not in Monitorism because the monitor is simply add C plus I and call it output But in the Austrian theory It's it's critical so investment here represents gross investment, which is typical of the way this Diagram is used so it includes replacement capital What Frank Knight called maintenance In my lecture yesterday and so we have that whole That whole distance of the Horizontal axis is gross investment a good part of that in a developed economy is replacement capital but some of it is new investment Okay, net investment investment over and above what it takes to replace So with positive net investment the economy grows you get more Capital and produce more stuff produce more investment goods more consumption goods Get more stuff the PPF shifts outward Premier to year permitting increasing levels of both consumption and investment and we can show that here Watch the economy grow you hear the economy grow okay Okay Okay, so four periods of growth are shown with consumption as well as in saving and investment increasing in each period the actual rate of expansion of the PPF depends on many factors We won't rehearse them here would take too long But importantly a change in saving preferences see what I've done so far haven't changed anything Although people are saving they've been saving to keep saving But a change in saving preferences which provokes movement along the initial PPF Effects the rate at which the PPF expands outward So suppose people become more thrifty more future oriented they reduce their current consumption and save instead So watch the movement along the PPF and this is just ordinary market mechanisms Making that work as we'll see moves along And so already We've departed from any kind of a Keynesian notion because in Keynes consumption and investment Always move in the same direction Who couldn't possibly move along the PPF which slopes downward? Okay, with the increased saving and investment the economy grows at a faster rate So this watch that Grows at bigger intervals. All right It's the increase in saving that makes the difference we can compare The high-income growth economy with the original low-income growth economy put them both together and show With no initial increase in saving the economy grows at a modest rate look on the left You see it growing at a modest rate With the initial increase in saving investment increases at the expense of consumption after which both consumption and investment increased Dramatically from period to period so watch that first decrease then increase and then you draw a line across there Where you can see that after four periods people are consuming more than they did Before and that's the whole business of saving. Okay, you save now in order to be able to purchase more in the future and while you're saving of course you're getting accrued interests not in this country we're down to about half a percent but That's that's the way it works now. Don't get me wrong. I'm not your father trying to Admonish you to say Okay, I'm just showing you what the consequences are of saving that you are able to Consume more in the future than you otherwise would But you probably should say more Okay, the market for loanable funds That's the interest rate Plotted against saving and investment. We have a supply of it and a demand for it So saving constitutes the supply of loanable funds Upward sloping as you would expect demand reflects the business community's willingness to borrow and undertake investment projects and that's the demand for loanable funds and of course The market mechanism here will then settle on that intersection So a straight application straightforward application of marshaling and supply and demand analysis And then you can see that vertical or a horizontal distance is the investable resources In other words, those are funds that are handed over to the investment community with which they can buy equipment and Materials and so on for beefing up the economy and the savings that comes out of income the income Was earned by people working and making stuff and if they didn't consume that much that meant they left some stuff over Okay And that's equivalent to their saving and it lets the business community take possession of that stuff that was left over and Increase the productive capacity of the economy. That's the market that worked for you and for me So here just points out loanable funds was closely identified with David Robinson. Yeah, there he is That was before Keynes And then on the suggestion of Roy Herrod Roy Herrod couldn't believe That Keynes was taking loanable funds theory out out of his macro theory and and Herrod read the draft of the general theory and so He advised Keynes that if he if for sure he was taking that theory out he should make that clear to the readers and so Keynes put that diagram in To emphasize that he was throwing it out. Okay, if you thumb through the general theory You'll see that diagram is the one and only diagram in the whole general theory no diagrams in the general theory except this one Kind of a screwy version as you'll see later of that one but And so on page 180 I won't make you look for it Market for loanable funds So if people have become more future oriented they increase their saving Causing the interest rate to fall and thereby encouraging business community to undertake more investment projects. All right So watch the savings curve shift right? There it goes. Okay lower interest rate Which inspires investors to undertake more investment With a given technology saving and investment are pre-requisite prerequisite to genuine sustainable economic growth now I've put it up two diagrams here and it turns out we can use them together Loanable funds market shows how interest rates bring saving and investment in the line with one another The PPF shows the trade-off is Struck between consumption and investment So market adjustments and output prices wages and Other input prices keep the economy functioning on its PPF as under normal conditions normal normal market conditions Okay, these two capital based macroeconomics Show these two elements Show the pattern of movements of consumption saving and investment and the interest rate that are consistent with the change in Intertemporal preferences now you have to watch these both at once. They sort of move in tandem with one another Before people become more as before people become more future oriented they save more which transmits a single of Lower interest rates to the business community So watch that process here it goes Okay, so it's the lowering of the rate of interest that causes That point on the frontier to move along it in that direction Okay, the lower that The lower rate of interest establishes a new equilibrium in the loanable funds market and the economy moves along the PPF in the direction of More investment and less current consumption It'll be more future consumption because they're undertaking investment and that investment will materialize in the future Businessmen know business people know that when people save They save up for something Okay, they don't save for fun, you know saving isn't fun okay, they save up for something and How in the world of business people figure out what that something is well that that job is the Entrepreneurial job. Okay. Now if you don't have any entrepreneurs in your markets in trouble But fortunately in this country at least we do have entrepreneurs Even the possibility of a market economy could work this way is at odds with Keynesian theory because it shows consumption and Investment moving in opposite directions that never happens in Keynes That's what that says Okay, so according to Keynes any reduction in consumer spending will result in excess inventories which in turn would cause Production cutbacks worker layoffs spiraling downwards of the economy and expenditures the economy would go into a recession The business community would commit itself to less not more investment and otherwise. He's saying it would fall inside the PPF All right. Well in his theory it would because he's thrown out the market mechanisms that would keep it from doing that Keynes call that the paradox of thrift don't don't save you wish you hadn't now if retail inventories Were representative investment then Keynes would be right here the derived demand effect dominates reduced consumer spending Means reduced inventory replacement. Well, duh. Yeah In general late-stage investments move with Consumer spending But here's the other part of the story that Keynes leaves out however, the interest rate effect dominates in the long term or Early-stage investments a lower interest rate can stimulate industrial construction for instance or product development the longer the production process takes place then The more critical the interest cost is Those projects even if it's multi-structured all along the way The interest rate costs are pretty heavy if the interest rate is high If the interest rate falls then that that dramatically lowers the cost of Engaging in long-term investment To keep track of changes in the general pattern of investment We need to consider the structure of production and the stage specific labor markets So we can do that and here I can zip forward because you've seen this yesterday So we've got a tame hyakian triangle here With time on the horizontal axis and final output on the vertical axis and the sequence of stages that lead from the early stages to ultimate consumption and We've got our friendly long-term investor a researcher and our retail manager, which is late-stage So late-stage investment activities is simplified by inventory management. Yeah, that's right for pedagogical convention Convenience the initial capital structure is shown as having five stages. That's arbitrary. How many stages does the economy actually have? You can't answer that in more than you can answer how many good how many different goods are produced in the economy No one has an answer to that question either, but there are stages of production. It does take time and so on but Pedagogically we represented as five stages With growth though the number will increase That's that's a developed economy the more developed economy is typically the more stages of production you have so although five of these stages are in operation during any time period, okay, we could take a field trip and see things going on at each and every stage of production but if we step back and Look at the whole process. We see Goods moving from stage to stage to stage to stage. It's just two different interpretations two different perspectives on The structure of production so watch the resources or goods in process Move through the stages. We've seen this too But we won't bother with it bother with the model a fords and so on To get on the sequence of stages form a hyacinth triangle a Summary description of the economy's intertemporal structure of production. That's right in a growing economy See how I've eliminated the stages. They're still there. I'm just not showing pictures of them In a growing economy the triangle increases in size Along with the outward expansion of the production possibilities frontier Now this is with a given rate of saving not a changing rate a given rate of changing when the PPF is Expanding well, so does this So does this triangle so watch that process like so when people choose to save more they send to seemingly conflicting with emphasis on seemingly signals to the market and Canes would be baffled by This seemingly conflicting signals decrease in consumption Decrease consumption dampens the demand for investment goods that are close to the temporal proximity With consumable output. This is the derived demand effect Reduced interest rates, which means lower borrowing costs stimulates demand for investment goods that are temporarily remote Temporarily remote from the consumable output. This is the time discount or interest rate effect Okay, so derived demand and time discount Are in conflict only if investment is conceived of the simple aggregate such as Keynes C plus I plus G so Keynes consider well, here's I does it go up or does it go down? There's no in between on that one But in capital-based macroeconomics capital and hence Investment is conceived as a structure changes in the demand for investment then can add Differentially to and or subtract differentially from the several stages of production that make up the structure and Here it just says well This is this is a significant Keynes theorizing in terms of aggregates Rather than in terms of structures underlies high-ex claim Quoted here. Mr. Keynes aggregates conceal the most fundamental mechanisms of change Okay, so increased saving results in a reallocation of resources among the stages of production The two effects drive demand and time discount have their separate and complementary effects on the capital structure The drive demand effect here decreased demand for consumption goods dampens the investment activities in the late stages of production time discount it Stimulates investment in the more remote stages of production. So watch the structure of production respond to an increase in saving There it goes. Okay, you even get a sixth stage. Okay, and may get a seventh and eighth with further Saving and now you notice consumption is down, but that's only an initial result because As the economy develops it will grow at a faster rate because of that increased saving and consumption will be up within four or five Periods as shown in earlier graph, okay increased saving then has the effect of both of the magnitude of the investment aggregate and the temporal pattern of The capital creation so look at it. What's the economy respond to an increase in saving? It's all coordinated here looks like that The PPF shows that more saving permits more investment There's a little arrow I can try and go shows that capital creation in the late stages such as retail Inventories is decreased while capital creation in the early stages such as product development is increased See so you get a two-way thing you get a skewing, okay, you never find that in Canes So the structure of production is given more of a future orientation Which is consistent with the savings that made the restructuring possible people are saving because they're Planning on being able to buy more in the future So we hope the increased growth rate here. See so we're doing same thing as before only Showing how it changes once there's an increase in saving Alright now this this I think is Illuminating what we can track here As tracked by both the PPF and the Hayekian Triangle consumption is Seen to fall as The economy is adapting to a higher growth rate Okay, after which consumption rises more rapidly than before and eventually surpass surpasses The old projected growth path look at it on the right side Down look at it on the triangle side Down Two different views are the same thing. Let's plot it against time consumption against time You have a reduction in consumption followed by an increase in consumption activity and if you compare that to What the growth path would have been without the saving? You can see that saving implies giving up some consumption in the near future and that would be That area In order to enjoy more consumption in the immediate future and possibly far future Okay, and that would be that area now stage specific labor markets Here I'm misreplacing our Chemical researcher and our retail manager with supply and demand curves and there's supply and demand for their services And so we can see that that shift then brings about Changes in demand for labor but in opposite directions as far as late stage and early stage So we can watch that. Okay, watch the demand curves in those Lower diagrams so you have a demand for Late stage going down don't need to stock so much retail stuff, but a demand in the research Wing they're going up because At lower interest rates. It's more profitable to increase in those activities The difference the differential shifting of labor demand gives rise to a wage rate gradient And I discovered this when I when I made this diagram I put in this gradient. It just shows that that gradient will persist until the economy is Fully adjusted to the new saving rate And that very term wage rate gradient plus a graph is included in Footnote in the second edition of high X prices in production. It's I think it's a little cryptic But once you once you see what it is and you can you can pick it out. So look for a footnote in Edition two of prices in production That has a graph in it and you'll see What that gradient looks like Okay, we're getting to the end of our First round here before we get into business cycles. So look what we've got going for us. We've got a loanable funds market Production possibilities frontier. We've got the triangle and we've got the different labor markets Now so we can watch the whole thing at once okay, I'm gonna do it twice because Can't see everything So we're gonna show an increase in saving And it's like poetry and motion in motion here everything is coordinated me back up and do it again Okay, now I'm gonna I'm gonna skip past this because this is just Steve hanky Saying that Hayek was right and I'm gonna skip this I love this I actually took that picture And it's only Hayek saying the hanky is right And I put both of those in there to make the case that I'm right all right We've got there okay Now credit expansion see hey, I've got 15 minutes left. I can do it. I think I can do it credit expansion This is a whole new ball game Increases in the money supply how'd that happen? Increases the money supply enter the economy through credit markets and they do the central bank litter literally lends money into existence The new money masquerades as saving it's not saving it doesn't represent some work that somebody's done to produce goods That can be used for investment purposes Okay, it's not saving it's new money That is the supply of loanable funds shifts rightward, but without there being any increase in saving okay, so The watch these opposing movements, this is a whole different ball game here Okay now it kind of looks at you squint your eyes it just kind of looks like okay I've been increased in saving no no that many increase in savings Savings have been padded by newly created money. So it's it's not s prime. It's s plus delta m And now you can see what looks like to equilibrium it really can't be too equilibrary. So it's really a It's really a disequilibrium All right because you're seeing that on the one hand saving is reduced It's probably says that responding to a lower interest rate people actually save less and Consume more I mean why save If all you guess 1% go ahead and consume at the same time Well says the result is not new sustainable equilibrium, but rather a disequilibrium That for a time is masked by the infusion of the loanable funds so Pumping new money through credit markets drives a wedge between saving and investment Of course it does Investors move down along their demand curves taking advantage of the lower borrowing cost. We've got it Shows the arrow down there there go the Investors are borrowing that new money Savors move down along their unshifted supply curves in response to the weakened incentive to save All right The discrepancy between saving and investment is papered over literally With newly created money which itself represents no investable resources Now I think you can start to say and probably can see the whole thing That this really is just a corollary to our Capital-based macroeconomics capital-based macroeconomics Starts out showing how things can work out right how markets can Work all right and then this shows that a Key instance very relevant instance of how things can go wrong if policy makers Disturb the market system Okay, and so you have that You have that division or that difference between Quantity supply and quantity demanded and of course that difference is precisely the amount of new money that was pumped into the economy because that's that's the right word shift of s to s plus delta m So much of high-ex-riding on money is Aimed at shifting the focus away from the bedrock Relationship between money and the general level of prices see that's the monetary stuff mv equal pq and the whole focus is on Relationship between m and p and Freedmen pretty much nailed it here Inflation causes prices to rise. Yeah, that's that's correct. Okay, but mv equals pq watch q q is a combination Added up combination of consumption and investment they're lumped together And so, you know, he doesn't look inside of q and see what's going on. So high-ex shifted the Ships of the focus toward the intertemporal Discoordination that is caused by credit expansion. Okay, now we can trace this around Favorable credit conditions spur investment activity spur. Yeah Which suggests a clockwise movement along the PPF in the direction of investment? All right, we can trace that up and see that investors are trying to pull the economy that way All right But income earners Are actually saving less and hence consuming more which suggests the counterclockwise movement along the PPF in The direction of consumption and there that is Okay, so now if you look at the axis consumers are pushing upward Parallel to the axis Particle axis and investors are pushing outward rightward parallel to the investment So the wedge between saving an investment translates into a tug of war Between consumers and investors. It's not a regular tug of war. They're not pulling 180 degrees or pulling nice They pull 90 degrees They simply get the economy off of its production possibilities frontier Consumers pulling up and investors pulling right. Okay, and the economy can go out in that direction But it's not sustainable growth. The PPF is sustainable growth If you go out in that direction Then you're in unsustainable growth and let's let's do the vector that resolves those two in other words and that effect is to Is to push economy outward beyond the frontier to what I call a virtual equilibrium is kind of Wavy up there. They can't you can't get there.  Structure of production well it gets pulled in both directions against the middle And this is something that the Austrians haven't paid enough attention to and yet it's hardcore in Mises It's not at all hardcore in Hayek right and and let me show you the lower interest rate Consistent with a future orientation stimulates investment activities in the early stages Yeah, but without sufficient resources being freed up elsewhere many of these investment projects will never be completed and so we see the initiation of Projects, but turns out they can't be completed because the resources Aren't there to complete them now that part is recognized by Mises by Hayek by Rothbard by all the Austrians but some of them either ignore or even deny this other effect which is Critical I think to the theory Compounding the inter temporal discord nation Increased consumer demands and who could doubt there's increased consumer demands because can't get much interest for the savings that might as well spend Okay, that draws some resources toward the late stages further reducing the prospects for completing a new capital structure So we can We can show that too So the dynamics of boom and bust entail both over investment as shown in the PPF diagram and malinvestment and Then an unsustainable lengthening of the Hayekian triangle which shows up in the Hayekian triangle You could call it over investment in the early stages But the general term is malinvestment Meaning that over investment in the early stages. It doesn't mean over investment overall But what gets left out is the metal Okay, so you get over investment which is shown in Both of those diagrams these distortions are Compounded by over consumption as shown in both the PPF and the Hayekian triangle like so and It's critical and I dug this out of Mises. It wasn't hard to dig out. I was sort of there But people somehow overlook this Mises repeatedly used the phrase malinvestment and over consumption Hayek soft-pedal even more than soft-pedal just ignored the notion of over consumption And even has some passages that imply there isn't over consumption But there is I think Mises was right Hayek was wrong on that point Okay So the tug-of-war that fits consumers against investors pushes the economy beyond the PPF The low interest rate favors investment and increasingly binding resource constraint keeps the economy from reaching that extra PPF point so Here I show look at the PPF point you get the you get the economy rising up above it unsustainably and being Skewed around one way or the other skewed around historically normally in that direction and eventually caving in okay the temporarily conflicted structure production dueling Triangles that term was given to me by John Cochran of Metropolitan State in Colorado Eventually turns boom into bust and the economy goes into recession Possibly into deep depression. So what's that? orange Arrow some people have told me the orange and blue is not really the two colors. I should be using But it's Auburn University, you know, that's Where the orange and blue comes from Okay Now look what we've got here. I mean this This is the vocabulary of booms and busts You know a wedge between saving and investment tug of war between consumers and investors dueling triangles This is not going to turn out pretty, right? That's the idea okay, and I Hesitate to do it this way that I'm going to talk about the triple P's that sounds like a management course, you know Uh Patting the supply of loanable funds with new money drives a wedge between saving and investment Yeah, it does Papering over the difference between saving and investment gives play to the tug of war between consumers and investors and pitting early stage Stages against late stages distorts the high-hacking triangle in both directions the temporarily Discoordination eventually turning boom into bust Okay, so watch the economy respond to credit Can any of you identify that guy that's Joe the plumber in Election ray several years ago, okay, and that's it. I mean the Joe the plumbers the one that's getting screwed in the and the politicians that do the business then Get off the hook Okay, it's quarter tail like better in there. Thank you much