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## The IS-LM model

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**The model**• The IS-LM model was developed in 1937 by John R. Hicksin an attempt to authentically interpret the “General Theory of Employment, Interest and Money”, the famous book published by John Maynard Keynesin 1936.**The model**• The model tries to explain the movement of output and interest rate in the short run. • To this end, it uses two curves: the IS (short for Investment and Saving) and the LM (short for Liquidity and Money). • The IS curve represents equilibrium in the goods market. • The LM curve represents equilibrium in the financial markets.**The IS curve**• We will try to use the goods market to establish a relationship between the interest rate and output. • We already know [from introductory macro…] that output in a closed economy is the sum of consumption (C), investment (I) and government expenditures (G). • Y = C + I + G**The IS curve (the Keynesian cross)**• We also know that output (Y) is by definition equal to income and that it represents the amount of spending undertaken by households, firms and the government. • But, how much do we want to spend? In other words, what is our demand for goods and services? • If we denote demand with Z, then: Z = C + I + G • So, demand (like output) is simply equal to the sum of consumption, investment and government expenditures.**The IS curve (the Keynesian cross)**• If we try to elaborate a bit more on the form of consumption, we can say that consumption must depend on our disposable income. • Our disposable income must be equal to total income (Y) minus the taxes that we pay to the government (T). • So, consumption is a function of our disposable income C(Y-T).**The IS curve (the Keynesian cross)**• What if we want to be more specific about the functional form of the consumption function. • Let’s assume that we must cosume something anyway in order to survive, like food. We call this amount autonomous consumption and let’s denote it by c0. • The rest of our consumption depends on our disposable income (Y-T). It is reasonable to assume that we consume a percentage of our disposable income and that we save the rest of it. We call this percentage marginal propensity to consume (MPC) and let’s denote it by c1. Since we consume less than our disposable income, c1 must be a number between 0 and 1. So, the non autonomous part of consumption must look like that: c1(Y-T). • Therefore, consumption in general must be equal to: C = c0+c1(Y-T)**The IS curve (the Keynesian cross)**• If this is the form of consumption, then demand in total must be equal to: Z = C + I + G => Z = c0+c1(Y-T) + I + G => Z = (c0- c1T + I + G) +c1Y • This last equation tells us that demand is equal to a sum of some variables that are exogenously given, namely: c0: the amount of autonomous consumption, c1T: the amount of taxes times MPC, I: investment, which for now we can assume that it is constant, and G: government expenditures. We will call this whole expression (c0- c1T + I + G), autonomous spending. • It also tells us that demand is a positive function of income (Y) and, moreover, that the slope of this positive function is c1, which is less than one. So the slope of the demand is flatter than the 45o line (the slope of which is 1).**The IS curve (the Keynesian cross)**• Now we have the first building block of the Keynesian cross. We are going to graph the demand as a function of income. We already proved earlier that the demand is a positive function of income and this is what we are going to graph now.**The IS curve (the Keynesian cross)**Z This is a picture of the demand as a function of income. The vertical intercept of the line ZZ which represents the demand, is the autonomous spending and its slope is the marginal propensity to consume. ZZ Slope: MPC 1$ Vertical intercept: autonomous spending Y**The IS curve (the Keynesian cross)**• Our economy is in equilibrium when actual production is equal to the demand, i.e. Y = Z. • The only place that generally satisfies this equilibrium condition is the 45o line in our previous graph. So, our equilibrium must be on that line.**The IS curve (the Keynesian cross)**• If we assume further, that there is no inventory investment, then output (= income) must always be equal to the demand. • So, in that case, not only are we always on the 45o line, but also always on the intersection of the demand function with the 45o line, which is our equilibrium point.**The IS curve (the Keynesian cross)**Actual Production Y=Z Z This is a picture of the Keynesian cross. We observe that in equilibrium, demand is equal to income and production along the 45o line. In our model, since there are no inventories, we are always in equilibrium. ZZ Y* 45o Y* Y**The IS curve (the multiplier)**• If we combine the equilibrium condition Y = Z, with the expression for the demand that we derived earlier, Z = c0+c1(Y-T) + I + G, we get: Y = c0+c1(Y-T) + I + G => Y = c0+c1Y - c1T + I + G => Y- c1Y = c0- c1T + I + G => Y(1- c1) = c0- c1T + I + G => Y = [1/(1- c1)] (c0- c1T + I + G)**The IS curve (the multiplier)**• We have already called the (c0- c1T + I + G) part of the above equation, autonomous spending. • Now, we will give a name to the 1/(1- c1) part and we will call it the multiplier. The reason for that name is that this fraction is greater than one (remember that 0<c1<1).**The IS curve (the multiplier)**• Therefore, whatever the change is in any of the parts of autonomous spending, the change in output is a multiple of that change. • So, if the government, e.g., decides to increase G by an amount x, this will result in an increase of Y by x times the multiplier. • Graphically, the demand will shift up by as much as the change in autonomous spending (the vertical intercept) but output will increase by more than that.**The IS curve (the multiplier)**• The size of the multiplier obviously depends on c1, the marginal propensity to consume. The larger the MPC, the smaller the denominator and the larger the multiplier. • Graphically, a large MPC corresponds to a steeply sloped demand curve. Shifts of a steep demand curve have large effects on income. • A small MPC corresponds to a relatively flatter demand curve. Shifts of a flatter demand curve have relatively milder effects on income. • We will now graph those two cases.**The IS curve (the multiplier)**The Keynesian cross with a flat demand (small MPC). The shift in demand has a milder effect on output. The Keynesian cross with a steep demand (large MPC). The shift in demand has a large effect on output. Actual Production Y=Z Actual Production Y=Z Z Z ZZ2 Y2 ZZ1 ZZ2 Y1 Y2 ZZ1 Y1 45o 45o Y Y1 Y2 Y1 Y2 Y**Deriving the IS curve**• The Keynesian cross is an important building block toward the IS curve but our mission is not accomplished yet. • However, from this point the derivation of the IS curve is straightforward. It relies on the relaxation of an assumption that we made earlier, namely that the level of investment is constant.**Deriving the IS curve**• Constant investment is a clear simplification of the Keynesian cross model. We already know that investment is not constant but rather a negative function of the interest rate. • At this point we also add that investment is also positively related with output. The line of reasoning is that as firms see their volume of sales going up, they will undertake more investment to accommodate this increase. But the level of sales is just proportional to output, since if output is increasing, more goods are going to be sold and if output is decreasing less goods are going to be sold. So, the bottom line is that we observe a positive relationship between the level of investment and the level of output. • Therefore, investment is a negative function of the interest rate and a positive function of output. In symbols we write: I = I(Y,i)**Deriving the IS curve**• Focusing on the interest rate, we can say that if the interest rate increases, this reduces the level of investment, shifts down the demand and consequently, through the multiplier, reduces the level of income. • On the other hand, if the interest rate decreases, the level of investment increases, the demand shifts up and the level of income increases.**Deriving the IS curve**• We have therefore shown that there exists a negative relationship between the interest rate and income. • This negative relationship is what is known as the IS curve. • The mathematical form of the IS curve is called the IS relation and it is simply: Y = C(Y-T) + I(Y,i) + G • A more specific form of this equation is the already familiar to us equation: Y = [1/(1- c1)] (c0- c1T + I + G) • This is the graphical derivation of the IS curve.**Deriving the IS curve**Actual Production Y=Z Z A decrease in the interest rate increases the level of investment (Panel A), which shifts up the demand and increases income (Panel B). The IS curve sums up these movements in the goods market (Panel C). ZZ2 Y2 Panel B ZZ1 Y1 45o Y1 Y2 Y i i i1 i1 Panel A Panel C i2 i2 IS I I1 I2 I Y1 Y2 Y**Shifts of the IS curve**• As always in economics, here too we are interested in curve shifts. • So, we are going to mix things up a bit, shift the curves around and see what happens.**Shifts of the IS curve**• So, what could possibly move the IS curve? • First, let’s recall the IS relation, Y = C(Y-T) + I(Y,i) + G, the general equation that describes the IS curve. • Which part of this equation could move the IS curve? • Maybe, it’s better if we start by what could by no meansmove the IS curve: income (Y) and the interest rate (i). • Why? Because, these are the endogenous variables of our model. These are the variable that we are trying to explain. They are the variables on the two axes of our graph (like price and quantity in a supply and demand diagram). So, if these two variables move, we move along the curve. We don’t shift it.**Shifts of the IS curve**• So, what could move the IS curve is any of the other variables that are exogenous, i.e. they are taken as given outside the model, namely: • G, government expenditures (variable controlled by the government), • T, taxes (variable controlled by the government), • C, consumption patterns that are independent of disposable income, if for example, the households decide to consume more because an asteroid is going to hit the earth (variable controlled by household preferences), and • I, investment patterns that are independent of the interest rate and income, if for example firms go into an unexplained investing spree (variable controlled by the animal spirits of the investors).**Shifts of the IS curve**• Out of those four parameters, we are mostly interested in the first two (G and T), because it is only those that policy makers can control. The other two cannot be affected directly by government policies. • So, our analysis will be primarily focused on government expenditures and taxes. • However, just bear in mind that changes in consumption and investment patterns affect the IS curve in exactly the same way as changes in government expenditures.**What happens if the government decides to increase**government expenditures (G↑)? • We will use the Keynesian cross to explore the effects of such a move. • First, let’s recall the equation for the demand that we derived earlier: Z = (c0- c1T + I + G) +c1Y • If, ceteris paribus, the government decides to increase G (by ΔG), then it is obvious that the demand curve would shift up by an amount equal to ΔG. The vertical intercept would move up by ΔG but the slope would remain the same.**What happens if the government decides to increase**government expenditures (G↑)? • But would happen to income after this shift of the demand curve? • Now, we have to recall the IS relation in the specific form that we also mentioned earlier: Y = [1/(1- c1)] (c0- c1T + I + G) • If G↑, then Y would go up by as much as ΔG times the multiplier 1/(1- c1), so by more than ΔG. • So, it looks like it’s a good deal for the government to increase G, since with an initial amount of increase, it can get income to increase more through the multiplier.**What happens if the government decides to increase**government expenditures (G↑)? • But, what does this mean for the IS curve? • It means that the increase in government expenditures caused an increase in income for a given level of interest rate. Remember that the interest rate did not move at all. • This corresponds to a shift of the IS curve to the right. For a given level of interest rate now we have more income. • Let’s look at this effect graphically.**The effects of G↑**Z Actual Production Y=Z ZZ2 Y2 Panel A ΔG An initial increase in G shifts up the demand by ΔG, which increases income by ΔG/(1 - c1) in Panel A. This means that for a given level of interest rate, the IS curve in Panel B must shift to the right by ΔG/(1 - c1). ZZ1 Y1 ΔY=ΔG[1/(1 - c1)] 45o Y1 Y2 Y i ΔY=ΔG[1/(1- c1)] Panel B i* IS1 IS2 Y1 Y2 Y**What happens if the government decides to decrease**government expenditures (G↓)? • The process that we follow must be clear by now. • Again we use the demand equation: Z = (c0- c1T + I + G) +c1Y • If, ceteris paribus, the government decides to decrease G (by ΔG), then it is obvious that the demand curve would shift down by an amount equal to ΔG. The vertical intercept would move down by ΔG but the slope would remain the same.**What happens if the government decides to decrease**government expenditures (G↓)? • Then we use the IS relation to see what happens to income: Y = [1/(1- c1)] (c0- c1T + I + G) • If G↓, then Y would go down by as much as ΔG times the multiplier 1/(1- c1), so by more than ΔG. • This means that the decrease in government expenditures caused a decrease in income for a given level of interest rate. • This corresponds to a shift of the IS curve to the left. For a given level of interest rate now we have less income.**The effects of G↓**Z Actual Production Y=Z ZZ1 Y1 ΔG Panel A An initial decrease in G shifts down the demand by ΔG, which decreases income by ΔG/(1 - c1) in Panel A. This means that for a given level of interest rate, the IS curve in Panel B must shift to the left by ΔG/(1 - c1). ZZ2 Y2 ΔY=ΔG[1/(1 - c1)] 45o Y2 Y1 Y i ΔY=ΔG[1/(1- c1)] Panel B i* IS2 IS1 Y2 Y1 Y**What happens if the government decides to decrease taxes**(T↓)? • Again we use the demand equation: Z = (c0- c1T + I + G) +c1Y • If, ceteris paribus, the government decides to decrease T by ΔT (so ΔT is negative), then it is obvious that the demand curve would shift up by an amount equal to -c1ΔT. The vertical intercept would move up by -c1ΔT but the slope would remain the same.**What happens if the government decides to decrease taxes**(T↓)? • Then we use the IS relation to see what happens to income: Y = [1/(1- c1)] (c0- c1T + I + G) • If T↓, then Y would go up by as much as ΔT times [- c1/(1- c1)]. • The expression [- c1/(1- c1)] is the version of the multiplier when taxes are changed by the government. • We observe that in the numerator of this expression there is c1, which corresponds to a number that is less than one. Therefore, if we compare this version of the multiplier with the general version [1/(1- c1)], we conclude that the general version is larger. This means that expansionary fiscal policy is normally more effective if conducted through increases in government expenditures rather than decreases in taxation.**What happens if the government decides to decrease taxes**(T↓)? • So, in effect the decrease in taxes caused an increase in income for a given level of interest rate. • This corresponds to a shift of the IS curve to the right. For a given level of interest rate now we have more income.**The effects of T↓**Z Actual Production Y=Z ZZ2 Y2 An initial decrease in T shifts up the demand by -c1ΔT, which increases income by ΔT[- c1/(1 - c1)]in Panel A. This means that for a given level of interest rate, the IS curve in Panel B must shift to the right by ΔT[- c1/(1 - c1)]. Panel A -c1ΔT ZZ1 Y1 ΔY=ΔT[- c1/(1 - c1)] 45o Y1 Y2 Y i ΔY=ΔT[- c1/(1 - c1)] Panel B i* IS1 IS2 Y1 Y2 Y**What happens if the government decides to increase taxes**(T↑)? • Again we use the demand equation: Z = (c0- c1T + I + G) +c1Y • If, ceteris paribus, the government decides to increase Tby ΔT (now ΔT is positive), then it is obvious that the demand curve would shift down by an amount equal to -c1ΔT. The vertical intercept would move down by -c1ΔT but the slope would remain the same.**What happens if the government decides to increase taxes**(T↑)? • Then we use the IS relation to see what happens to income: Y = [1/(1- c1)] (c0- c1T + I + G) • If T↑, then Y would go down by as much as ΔT times [- c1/(1- c1)]. • So, in effect the increase in taxes caused a decrease in income for a given level of interest rate. • This corresponds to a shift of the IS curve to the left. For a given level of interest rate now we have less income.**The effects of T↑**Z Actual Production Y=Z ZZ1 Y1 -c1ΔT An initial increase in T shifts down the demand by -c1ΔT, which decreases income by ΔT[- c1/(1 - c1)]in Panel A. This means that for a given level of interest rate, the IS curve in Panel B must shift to the left by ΔT[- c1/(1 - c1)]. Panel A ZZ2 Y2 ΔY=ΔT[- c1/(1 - c1)] 45o Y2 Y1 Y i ΔY=ΔT[- c1/(1 - c1)] Panel B i* IS2 IS1 Y2 Y1 Y**The LM curve**• To get to the LM curve, we have to use financial markets and go through the theory of liquidity preference. We have to understand why people decide to hold money in their pockets or in non- interest bearing bank accounts (checking accounts). In other words why we choose to forgo the interest rate that the banks offer us when we hold illiquid bank products (e.g. CDs, etc.).**The LM curve**• The answer is very simple: convenience and security. • It is true that having highly liquid assets, such as cash or immediately available, through an ATM, checking accounts makes our life easier. • Imagine if we had to go to the bank to liquidate part of our investments every time we needed to go to the grocery store. Also having liquid assets provide us with a sense of security, that we will, no matter what, have some money immediately available in case an emergency (or a new financial opportunity) occurs.**The LM curve**• Since we have answered why, now we have to answer how much money we hold. • To answer this question, first we have to define what is money. • Generally, for our purposes money is cash and checking (non interest bearing) bank accounts. This is known as M1. • There are also other measures of money but we are not really interested in them.**The LM curve**• Then we have to come up with a measure of money. We call the measure of money with the interesting name: real money balances or real money stock (M/P). • To determine how much money we hold, as always in economics, we will look for an equilibrium. • The equilibrium between the supply of real money balances and the demand for real money balances.**The LM curve (Money supply)**• The supply of real money balances is easy because it is exogenously given. It is controlled by the central bank through the ways that we learned in introductory macro (open market operations, discount rate, required reserves ratio). So the supply is just a number decided by the central bank and we do not need to worry about it.**The LM curve (Money supply)**i Since money supply (Ms) is independent of the interest rate, it can be represented by a vertical line. The amount of money supplied only depends on the decision of the central bank and nothing else. Ms M/P**The LM curve (Money demand)**• The demand for real money balances is more complicated. The amount of real money balances that we demand, depends on what? • Well, first it depends on income (Y). The more income in an economy, the more transactions will occur and the more money we will demand to effect these transactions. So, there is a positive relationship between demand for real money balances and income. • But also, it depends on the interest rate. The higher the interest rate on illiquid financial products (e.g. CDs), the less money we will demand, since money pays no interest whereas these illiquid products do. Because we do not want to lose a lot of interest, as interest rates go up, we will hold less and less real money balances. So, there is a negative relationship between demand for real money balances and interest rate.**The LM curve (Money demand)**• If we wanted to write down in symbols what we just said in words, we would write this expression for money demand: (M/P)d = L(i,Y) • Demand for real money balances is a function L of the interest rate and income. • Or, if we want to assume that money demand is exactly proportional to the level of income in an economy, we can even more simply write: (M/P)d = YL(i)

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