Lecture Note No
Advanced Macroeconomics1 Eco320 Note #1
Mathematical Approach of the IS-LM Curve Model
We still maintain the assumption about the fixed price level: The price level P is assumed to be fixed in the IS-LM model unless it is specified otherwise.
1. What Are We Trying to Do?
The IS-LM is the most broadly used frame of reference in macroeconomics theory.
In the IS-LM Curve Model, the interactions between the goods (output) market and the financial market gives the equilibrium Y* and the equilibrium interest rate i*.
As this new Y*and i* satisfy the equilibrium conditionsin the goods market as well as the money market.
We would like to proceed in the following manner:
First, we will establish an inverse relationship between interest rates and investment in the goods market. This eventually leads us to the IS curve or the various combinations of i and Y, which satisfy the equilibrium condition in the goods market or make the demand equal to the supply in the goods market.
Second, we will establish an inverse relationship between interest rates and (real) money demand in the money market. This leads us to the LM curve or the various combinations of i and Y, which satisfy the money market equilibrium or make the demand equal to the supply in the money market.
Third, by equating the Ys or the i’s that we have obtained from the IS and the LM curves, we will solve for the national income Y* or the interest rates i*, which satisfy the goods and money market equilibrium conditions at the same time. Graphically, they are obtained from the intersection of the IS and LM curves.
Fourth, we will examine various “comparative statics”, which is represented by the multiplier. In this section, we examine the impact of exogenous changes in government expenditures, money supply, monetary market conditions, etc.(these are only a few out of many possible exogenous changes in the economy), on the equilibrium national income Y* or the equilibrium interest rate i*.
2. IS curve
1) Components
(1) Consumption Expenditure
Funtional Form of Consumption
C = C0 + c1 (Y-T)
For now, we assume that T = T0. However, later we should introduce a more realistic assumption that T = T0 + t1 Y.
(2) Investment Expenditure
How does a high real interest rate dampen economic activities? Actually it works in two ways; first it reduces the investment, and thus the AE, eventually reducing Y*. It also reduces the real money demand. At this moment ignore the second impact.
The (real) interest rate constitutes a cost of obtaining (financial) capital for additions to capital stock or investment. A higher interest rate means a higher cost, a lower profitability and the lower rate of return on investment projects. Note well that the investment is a decreasing function of real, not nominal, interest rates.
Some investment projects, which used to be marginally profitable or managed to make both ends meet, are no longer profitable. So the desired investment will decrease as the interest rate increases.
Functional Form of Investment
I = I0 – b i,
I0 is the autonomous investment and bis the elasticity of investment with respect to interest rates. b>0. Here b measures the responsiveness to changes in investment to changes in the interest rate.
The larger the value of b, the more responsive the investment with respect to changes in interest rates. In other words, the larger the value of b, the more interest-rate elastic the investment.
A numerical example would be I = 100 – 5 i:One percentage increases in investment will bring about a 5% decrease in investment.
(3) Government Expenditure
(4) Net Exports
X – M
As long as the prices are set to be constant (no changes in the relative price level or competitiveness), and there is no change in Foreign exchange rates, they are
X = X0 or exogenous (as the domestic country has no control over its exports)
M = M0 - m1 Y
*Question: Compare the consumption function and the import function. What is the major difference? Why is M not a function of Y-T, but Y itself? – Why is T not deducted from the Import Expenditure Function?
2) IS Curve: Goods Market Equilibrium
For simple illustration, for now, we assume that X-M =0 or that the economy has no exports or imports. Later this should be relaxed.
The IS curve shows various combinations of national income and interest rates which bring out the equilibrium, or the equality between demand and supply, in the goods market.
Let’s plug the aforementioned modified investment function into the aggregate expenditure function, and solve for Y* and i*.
(1) Algebraic Derivation
Recall there are three different cases of AE.
Case 1. All taxes are lump-sum or autonomous.
Under 3 unrealistic but simplying assumptions that T = T0, X-M =0 and all prices are fixed.
Suppose that we are dealing with the aggregate expenditures with only lump-sum taxes and no exports or imports (This is Case 1 in the last chapter of the Keynesian Cross Diamgram).
The AE will be
AE = C0 + c1 (Y – T0) + I0 – bi + G0
= c1 Y + (C0 - c1 T0 + I0 + G0 – b I0 )
At equilibrium,
Y= AE
Y = c1 Y + (C0 – c1 T0 + I0 + G0 – b i )
Y – c1 Y = C0 – c1 T0 + I0 + G0 – b i
Solve for Y* and i*: We can rewrite this equation as a functional relationship between Y* and the interest rate i. That is, solving for Y* or i*, depending on what we need to know about:
This is the algebraic expression of the relation between (i, Y) which represents equilibrium in the final goods market.
The above two are identical. However, in economics, it is customary to put the price or interest rate on the vertical axis, and the quantity on the horizontal axis when drawing a graph or a diagram. So the second one is in line with the illustrative tradition. We can draw the IS curve by putting i on the vertical axis and Y on the horizontal axis.
Note that the slope has a negative sign and thus the IS curve is downward sloping. This means that in equilibrium, of the goods market, the interest rate and income move in the opposite direction; if interest rate increases for some reason, in order to stay at the same equilibrium in the goods market, national income should decrease.
(2) Intuitive Explanation of the IS curve.
We can also give the following intuitive explanation about the negative slope of the IS curve;
Let us start from one equilibrium: Y = YS = AE = C + I + G. Here let us change the interest rate and examine the responsive changes in Y. If i and Yturn out to be moving in the same direction, the slope of the IS curve will be positive, and vice versa.
Let us suppose that the interest rate decreases from i0 to i1. If investment is inversely related to the interest rate, there will be an increase in investment and thus an increase in the AE. This means that there will be an excess aggregate demand (now Y < AE’). How can we re-establish the equality between AE and Y? The answer is by increasing Y. In the equality of AE and YS, or the equilibrium of the goods market, when interest rates goes down and national income goes up. The interest rate and national income should move in the opposite direction.
We can express the above relationship with a curve in a graph with Y* on the horizontal axis, and the interest rate i* on the vertical axis. This curve is called the IS curve because, at equilibrium, AE = Y, which means C + I + G + G + X – M = C + S + T. As C in both side cancels out, the equilibrium condition of the goods market can be expressed as I + G +X = S + T + M. The first letter of each side of the equality read ‘I’ and ‘S’. So along the IS curve, I + G + X = S + T + M. So comes the word ‘IS curve’.
3. LM Curve for Money Market Equilibrium
LM cure shows the combinations of the interest rate and the income (i, Y) which satisfies the equilibrium in the money market.
1) Components
(1) ‘Nominal’versus ‘Real’ Money Supply/Demand
We should make distinction between Nominal Supply or Demand and Real Supply or Demand. The first one is in monetary terms, and the second in quantity terms.
In microeconomic analysis of equilibrium, we define the demand and supply in real terms, not in monetary or nominal terms. For instance, if we say that $20,000 worth of hamburgers are demanded (or supplied), the statement is not clear enough. This $20,000 is nominal demand in monetary terms. What about the real demand or quantity? If the price is $1 per hamburger, in real terms, 20,000 units of hamburgers are demanded. If the price is $10, in real terms 2,000 hamburgers are demanded.
In the same vein, for the analysis of the money market equilibrium, the quantity of money should be also defined in real terms, not in nominal or monetary terms. The nominal quantity of money is the face value of the money, and the real quantity of money is the face value divide by the price level;
Real quantity of money = Nominal quantity of money/Price level.
m = M/ P
The real quantity of money supply m is the nominal money supply divided by the price level. For instance, nominal money supply is 2,000,000,00 dollars or $ 2 billion. The price level is measured by a price index. Suppose that the price index is 100 (or 1.00) right now. The real money supply m = 20,000,000,000/100 or 20/1.0 (units do not matter as long as there is a consistency).
(2) Money Supply
The nominal quantity of the money supply is determined by the monetary authority, which usually is the central bank.
MS = M0
Money supply varies depending on the scopes of money: it may include only cashes (in circulation) in a narrow scope, and may include cashes and all deposits in a broad scope.such as M2. The different scopes of money supply will be discussed in full in the separate chapter.
For instance, M0 = $20,000,000,000 or $20 billion.
The monetary authority does not have to determine the nominal money supply on the basis of any variables in any given manner over time. Thus, we regard the nominal money supply as an exogenous variable, and regard it as arbitrarily determined by the monetary authority.
Mathematically, this means that the nominal money supply curve is vertical, being independent of interest rates. As the money supply is independent of the interest rate, when drawn in the interest rate and real quantity dimension, the money supply curve is vertical, being the same regardless of the level of the interest rate.
At one point of time it is fixed. However, of course, over time it can be changed by the monetary authority. In fact, the monetary authority sets the nominal money supply in each period.
(3) Real Money Demand
(a) Uniqueness of Real Money Demand
A few important things to remember about real money demand:
First, note that the money market equilibrium should be defined in terms of real money supply and demand;
Nominal money supply is equal to nominal money demand at all times, i.e., at and out of equilibrium. The nominal quantity of money demanded by the society as a whole is always equal to the nominal quantity of money supplied by the government; MS = MD at all times.Suppose the government is handing out newly printed paper monies or notes on the street. IS there anyone who would refuse them? Every dollar of money supply will be gladly demanded.
Second, while an individual cannot control real money demand, the general public as opposed to the monetary authority can control real money demand;
When an individual receives some new paper monies, her/his nominal (and real) balances increase. S/he may succeed in decreasing the nominal money demanded or the real money balanced back to the initial level by spending the excess money holdings. However, because her/his expenditures will become someone else’s receipts, some other members are getting the increased money supply. So from an individual’s view point the nominal money demanded may be controllable, while it is not controllable from the entire society’s viewpoint. What is true for individuals is not necessarily true for the society as a whole. This is the ‘fallacy of composition’ commonly founded in macroeconomics.
As individuals are busy getting rid of the excess of money holding over the desired level of demand (“I would like to have $200 in my pocket, but as government gives me a new $100 bill, now I have the excess of money holding by $100. I would like to go back to the desired level of money demanded, that is $200 by spending $100 away.”) The increased money becomes a kind of ‘hot potato’. What does this mean in terms of the real money demand? The real money demand, which is the nominal money demand ( = the nominal money supply) divided by the price level, is going back to the initial level. The increased speed o spending and expenditure will eventually push up the price level. The general public are collectively changing the price level and thus controlling the real money demand.
Suppose MS = M = MD = $200 billion and P = 1.00 initially in the equilibrium; the real money demand is MD/P = 200/1 = 200 and should be equal to the real money supply at the equilibrium. This real money demand is at the desired level at the equilibrium in light of all the determinants of the demand including the income level and the interest rate.
Now the monetary authority increases the nominal money supply MS to $400 billion.
First, all the increased nominal money supply will be demanded. So the nominal money demanded is equal to the new nominal money supply; MD’ = MS’ = M’ = $400 billion.
In the short-run, the price does not change, and thus the actual amount of the real money holding will be m’ = m’’ = M’/P = $400/1.00 = 400. This is much larger than the desired real money demand, that is, 200. As there are no change in the determinants of the real money demand, there should not be any change in the level of real money balances the general public wish to hold. There is an excess of real cash balances over the desired real money demand; ‘actual’ real money balances > ‘desired’ real money balances.
As individuals with excessive money balances try to recover the desired real money balances by spending the excess money receipt, the price level is going up to P’. At this new price level, the new ‘actual’ real money balances (M’/P’) become equal to the desired level of real money balances.
Specifically, the price level will go up to the level of 2 (or the index number 200). The actual real money demand will be 400/2 = 200, the same level as before any changes.
However, if P is fixed, there have to be permanent changes in real money demand, which is only possible when there is a change in its determinants such as interest rates and real income. The idea is that if there is an excess of liquidity, in the money market interest rate should fall and this should in turn boost investment and national income. This is the case in hand.
(b) Functional Form of Real Money Demand
The real money demand is given a functional form such as
md = L ( i, Y ).
The above equation defines the real money demand as a decreasing function of interest rates and a increasing function of national income. What determines the desired level (quantity) of real money demand? Just as the desired quantity of hamburgers is determined by the consumers’ income and the price of hamburger, the real demand for money is determined by the income level of the economy, that is the national income, and the price of the money, that is, the interest rate.
Let us examine the second point in the above statement: the price of money is the interest rate. In other words, the opportunity cost of holding money balances is the interest rate.
Money is one of many assets, which include bonds, stock, equities and real assets. Money and other assets are substitutes. The major difference between money and other assets is that money does not bring in any positive pecuniary returns. Actually it is very often subject to the erosion of real value due to inflation, and other assets do have pecuniary returns. However money, or cash balances in a precise term, renders a unique non-pecuniary service, which is known as ‘liquidity’. Money is the most generally accepted medium of exchange and most ‘liquid’. So when you decide to hold assets in the form of cash balances instead of any other, you are showing your preference for liquidity over pecuniary returns. This is the reason why the money demand is called’ liquidity preference’, and the money demand function ‘liquidity preference function.’
The interest rate represents the foregone pecuniary return or the economic sacrifice you have to take when you are choosing cash balances over other assets, as your mode of holding assets; in other words, the interest rate is the opportunity cost of holding cash balances. When the interest rate goes up, the cost of holding cash balances increases and naturally you would like to hold less assets in the form of cash balances and more interest bearing assets. This means that the demand for money is inversely related to the interest rate.
Now we have another major factor to be considered, which affect the real money demand; the income level. When real income increases, in most cases, the demand for money increases in real terms, too. To name one reason, when real income increases, there occur more transactions, and then more cash balances should be held to back up the increased transactions.
We can give the liquidity preference function the following specific functional form;
md = kY – h i + u,
where K is the elasticity of real money demand with respect to the national income; h is the elasticity of real money demand with respect to interest rates; and u is the random component of real money demand.
The liquidity preference curve is negatively sloped when drawn with the interest rate on the vertical axis and the amount of real money on the horizontal axis. The variables Y and u are the shift parameters of the real money demand curve.
Also, we can draw a set of liquid preference curves for different levels of income; the higher the level of national income, the larger the demand for real money balances. You may remember, from the class of introductory economics, that an increase in income shifts the demand curve to the right.