In the AD-AS model and the Keynesian Cross model, the market interest rate is an external (exogenous) variable and is set in the money market relatively independently of the equilibrium of the commodity market. The main purpose of analyzing the economy using the IS-LM model is to combine the commodity and money markets into a single system. As a result, the market interest rate turns into an internal (endogenous) variable, and its equilibrium value reflects the dynamics of economic processes occurring not only in the money, but also in the commodity markets.
The IS-LM model (investments – savings, liquidity preference – money) is a model of commodity-money equilibrium that allows you to identify economic factors that determine the function of aggregate demand. The model allows you to find such combinations of the market rate of interest R and income Y, at which equilibrium is simultaneously achieved in the commodity and money markets.
The basic equations of the IS-LM model are:
1) Y = C + I + G + Xn is the main macroeconomic identity.
2) C = a + b(Y-T) is the consumption function, where T = Ta + tY.
3) I = e – dR is the investment function.
4) Xn = g – m’Y – n’R is the net export function.
5) 
is a function of the demand for money.
Internal variable models: Y (income), C (consumption), I (investment), Хn (net exports), R (interest rate).
External model variables: G (government spending), MS (money supply), t (tax rate).
The empirical coefficients (a, b, e, d, g, m’, n, k, h) are positive and relatively stable.
In the short term, when the economy is underemployed (Y≠Y*), the price level P is fixed (predetermined), and the values of the interest rate R and total income Y are mobile. Since P = const, the nominal and real values of all variables coincide.
In the long run, when the economy is in a state of full resource employment (Y=Y*), the price level P is mobile. In this case, the variable MS (money supply) is the nominal value, and all other variables of the model are real.
The IS curve is the equilibrium curve in the commodity market. It is a geometric place of points characterizing all combinations of Y and R, which simultaneously satisfy the identity of income, the functions of consumption, investment and net exports. At all points of the IS curve, the equality of investment and savings is observed. The term IS reflects this equality (Investment = Savings).
The simplest graphical output of the IS curve is related to the use of savings and investment functions (see Figure 9.1). In Fig. Fig. 9.1, and the savings function is depicted: with an increase in income from Y1 to Y2, savings increase from S1 to S2. In Fig. Figure 9.1,b shows the investment function: rising savings reduce the interest rate from R1 to R2 and increase investments from I1 to I2. In this case, I1 = S1, and I2 = S2. In Fig. Figure 9.1,c shows the IS curve: the lower the interest rate, the higher the income level.
Similar conclusions can be drawn using the Keynesian Cross model (Figure 9.2). In Fig. Figure 9.2 shows the investment function: an increase in the interest rate from R1 to R2 reduces the planned investment from I(R1) to I(R2). In Fig. Figure 9.2,b depicts a Keynes cross: reducing planned investments reduces income from Y1 to Y2. Figure 9.2,c shows the IS curve: the higher the interest rate, the lower the income level.
The is curve equation can be obtained by substituting equations 2, 3, and 4 into a basic macroeconomic identity and its solution with respect to R and Y.
The equation of the CURVE IS with respect to R is:
,
where 
.
The equation of the CURVE IS with respect to Y is:
,
where 
.
The coefficient 
characterizes the angle of inclination of the IS curve relative to the y-axis, which is one of the parameters of the comparative effectiveness of fiscal and monetary policy.
The IS curve is more gentle provided that:
the sensitivity of investment (d) and net exports (n) to interest rate dynamics is high; the marginal propensity to consume (b) is high; the marginal tax rate (t) is small; the marginal propensity to import (m’) is small.
Under the influence of increased government spending G or tax cuts T, the IS curve shifts to the right. A change in tax rates also changes the angle of its inclination. In the long term, the angle of inclination of IS can also be changed through income policy, since high-income families have a marginal propensity to consume relatively lower than that of low-income families. The remaining parameters (d, n, m’) are practically not affected by macroeconomic policy and are mainly external factors determining its effectiveness.
The LM curve is the equilibrium curve in the money market. It fixes all combinations of Y and R that satisfy the functions of demand for money at the value of the money supply specified by the Central (National) Bank MS. At all points of the LM curve, the demand for money is equal to their supply. The term LM reflects this equality (Liquidity Preference = Money Supply) (see Figure 9.3).
Graphical output of the LM curve.
Rice. Fig. 9.3 shows that in the money market, an increase in income from Y1 to Y2 increases the demand for money, and therefore raises the interest rate from R1 to R2. Rice. Figure 9.3,b shows the LM curve: the higher the income level, the higher the interest rate.
The equation of the curve LM can be obtained by solving equation 5 of the model with respect to R and Y.
The equation of the curve LM is:
(relative to R);
(relative to Y).
The coefficient 
characterizes the angle of inclination of the LM curve relative to the y-axis, which, similar to the angle of inclination of the IS curve, determines the comparative effectiveness of fiscal and monetary policy.
The LM curve is relatively gentle provided that:
the sensitivity of the demand for money to the dynamics of the market interest rate (h) is high; the sensitivity of demand for money to the dynamics of GNP (k) is small.
An increase in the MS money supply or a decrease in the price level P shifts the LM curve to the right.
Equilibrium in the model is achieved at the point of intersection of the IS and LM curves (see Fig. 9.4.).
Algebraically, the equilibrium volume of production can be found by substituting the value of R from the IS equation into the equation LM and solving the latter with respect to Y.
(provided that 
.
At a fixed price level P, the equilibrium value of Y will be the only one.
The equilibrium value of the interest rate R can be found by substituting the equilibrium value of Y into the IS or LM equation and solving it with respect to R.
Relative effectiveness of fiscal and monetary policy
Fiscal expansion. Rising government spending and tax cuts lead to a crowding-out effect that significantly reduces the impact of stimulus fiscal policies (see Figure 9.5).
If government spending G increases, then total spending and income increases, which leads to an increase in consumer spending C. An increase in consumption, in turn, increases total spending and income Y, and with a multiplier effect. The increase in Y contributes to the growth of demand for MD money, as more transactions are made in the economy. An increase in demand for money with a fixed supply causes an increase in the interest rate R. An increase in interest rates reduces the level of investment I and net exports Xn. The fall in net exports is also associated with an increase in total income Y, which is accompanied by an increase in imports. As a result, the growth of employment and output caused by stimulating fiscal policy is partially eliminated by crowding out private investment and net exports.
If there were no crowding out of investment and net exports, the increase in Y due to the increase in government spending (or tax cuts) would be equal to (Y0Y2). However, due to the displacement effect, the actual increase in Y is only (Y0Y).
Monetary expansion. An increase in the money supply allows for short-term economic growth without a crowding-out effect, but has a contradictory effect on the dynamics of net exports.
The increase in the Ms money supply is accompanied by a decrease in R interest rates (see Figure 9.6), as the resources for lending expand and the price of credit decreases. As a result, total expenditures and income Y increase, causing an increase in consumption C. The dynamics of net exports of Xn is influenced by two counteracting factors: the growth of total income Y, which is accompanied by a decrease in net exports, and a decrease in the interest rate, which is accompanied by its growth. The specific change in the value of Xn depends on the magnitudes of the changes in Y and R, as well as on the values of the marginal propensity to import m’ and the coefficient n.
The relative effectiveness of fiscal and monetary policy is determined depending on:
(a) The sensitivity of the investment and net export functions to market interest rate dynamics (coefficients d and n);
b) the degree of sensitivity of the demand for money to the dynamics of the market interest rate (coefficient h).
The relative effectiveness of stimulus fiscal policy is determined by the magnitude of the displacement effect. If the displacement effect is less than the output growth effect, then, all else being equal, fiscal policy is effective.
The effect of displacement is relatively insignificant in two cases:
if investment and net exports are insensitive to rising interest rates in the money market, that is, if the sensitivity coefficients d and n are relatively small. In this case, even a significant increase in R will cause only a slight displacement of I and Xn, and therefore the total increase in Y will be substantial. Graphically, this situation is illustrated by the steeper IS curve (see Figure 9.7). The slope of the LM curve is of secondary importance in this case. If the demand for money is highly sensitive to an increase in interest rates and a slight increase in R is enough to balance the money market. Since the increase in R is insignificant, the effect of displacement will be relatively small (even with relatively high coefficients of sensitivity I and Xn to the dynamics of R). Graphically, this situation is illustrated by a more gentle curve LM (see Figure 9.8). The slope of the IS curve is of secondary importance in this case.
Stimulus fiscal policy is most effective when combined with a relatively steep IS and a relatively gentle LM (see Figure 9.9). In this case, the displacement effect is very small, since both the increase in interest rates is very insignificant, and the coefficients d and n are very small. The total increase in Y is (Y0Y1).
Stimulative fiscal policy is relatively ineffective if the crowding-out effect exceeds the output growth effect.
The displacement effect is significant if:
investment and net exports are highly sensitive to the dynamics of interest rates, that is, the coefficients d and n are very large. In this case, even a slight increase in R will cause a large decrease in I and Xn and therefore the total growth of Y will be small. Graphically, this situation is illustrated with respect to the gentle IS curve (see Figure 9.10) The slope of the LM curve is of secondary importance in this case. Demand for money is insensitive to the dynamics of R. In this case, in order to balance the money market, a very significant increase in R is needed. This causes a very strong displacement effect even at relatively small coefficients d and n. Graphically, this situation is illustrated by a steeper LM curve (see Figure 9.11). The slope of the IS curve in this case is of secondary importance.
Stimulative fiscal policy is the least effective when combining a relatively flat IS and a steep LM (see Figure 9.12). In this case, the increase in Y equal to (Y0Y1) is very small, since both the increase in interest rates is very large, and the coefficients d and n are significant.
The relative effectiveness of the stimulating monetary policy is determined by the magnitude of the stimulating effect of increasing the money supply and lowering interest rates on the dynamics of investment and net exports. This stimulating effect is the opposite of the displacement effect.
The stimulating effect on I and Xn is relatively large in 2 cases:
if I and Xn are highly sensitive to the dynamics of interest rates. Graphically, this corresponds to a relatively gentle IS (see Figure 9.13). In this case, even a slight decrease in R in response to the growth of the money supply leads to a significant increase in I and Xn, which significantly increases Y. The angle of inclination of the LM curve in this case is of secondary importance. if the demand for money is insensitive to the dynamics of R. Graphically, this corresponds to a relatively steep LM. In this case, the increase in the money supply is accompanied by a very large decrease in interest rates, which greatly increases I and Xn even with relatively insignificant coefficients d and n. The angle of inclination of IS in this case is of secondary importance.
Accommodative monetary policy is most effective when combined with a relatively steep LM and a gentle IS (see Figure 9.14). In this case, the reduction in interest rates is very significant and the coefficients d and n are significant. Therefore, an increase in Y equal to (Y0Y1) is relatively large.
Stimulative monetary policy is relatively ineffective with a high sensitivity of demand for money to the dynamics of R, as well as low sensitivity of investment and net exports to the dynamics of the interest rate.
A flat LM means that the money market comes into equilibrium with a very small decrease in R in response to an increase in the money supply. Even if I and Xn are very sensitive to the dynamics of R, such a small reduction in interest rates is not enough to significantly increase investment and net exports. Therefore, the total increase in output (ΔY) is very small (see Figure 9.15).
A steep IS means that even with a significant decrease in R, investment and net exports will increase very slightly, since the coefficients d and n are very small. Therefore, the overall increase in output will be insignificant even with a large increase in the money supply and a significant decrease in interest rates (see Figure 9.16).
Monetary policy is the least effective when combining a steep IS and a gentle LM (Figure 9.17). In this case, both R decreases slightly, and the reaction to this from I and Xn is very weak. Therefore, the total increase is very small and equal to (Y0Y1).
Derivation of the aggregate demand curve from the IS-LM model and economic policy under changes in the price level
Fig.9.18. a graphical method for deriving the aggregate demand curve from the IS-LM model is presented.
The aggregate demand equation can be derived from the algebraic expression for equilibrium Y (see 9.1. of this lecture), provided that flexible prices are introduced. In generalized form, it can be represented as:
where α, β, γ, θ are generalizing coefficients.
An increase in the price level from P1 to P2 reduces the real money supply, which graphically corresponds to a shift in the LM curve to the left (see Figure 9.18-A). A decrease in the money supply raises the interest rate R, which leads to a decrease in investment I and relatively reduces the net exports of Xn. As a result, the volume of production of Y decreases from Y1 to Y2 (see Figure 9.18-B).
Increased government spending, tax cuts, or increases in the money supply shift the aggregate demand curve to the right. The same type of shift in the AD curve accompanies completely different shifts of the IS and LM curves, corresponding to the measures of stimulating fiscal and monetary policy (see Figure 9.19 and Figure 9.20).
Let us consider in more detail the stimulating fiscal policy when the price level changes. Suppose that the economy starts at point C (see Figure 9.21). With an increase in government spending G (or a decrease in taxes T), the IS curve shifts to the IS’ position, which reflects the growth of aggregate demand to AD’. The AD shift causes demand inflation – the price level rises from P0 to P1. Under the influence of rising prices, the real money supply is relatively reduced and the LM curve goes to the left (to the LM’ position). At point A, a short-term macroeconomic equilibrium is established.
In the context of demand inflation, economic agents gradually adjust their behavior: nominal wage rates are raised. This increases average unit costs and reduces firms’ profits. Gradually, firms begin to reduce their output and the AS curve slowly shifts to the left.
The reduction of AS causes a further increase in prices (cost inflation) from P1 to P2. This price increase shifts the LM’ curve even further to the left, to the LM position.” Since the money supply is relatively reduced all the time, the interest rate increases all the time (from R0 to R2). At point B, a long-term macroeconomic equilibrium is established at a higher level of prices and interest rates than at point C, and a change in the structure of the economy in favor of the public sector. In this case, the level of production is equal to the potential.
The essence of the stimulating monetary policy when the price level changes is as follows. Suppose that the economy starts at point B (see Figure 9.22). The increase in the money supply shifts the LM curve to the right to the position of LM’, which reflects the growth of aggregate demand AD to AD’. The AD shift is accompanied by demand inflation – prices increase from P0 to P1. This relatively reduces the real money supply and the LM’ curve shifts back to the left to the LM position.” At point A, a short-term macroeconomic equilibrium is established.
In the future, cost inflation leads to a decline in aggregate supply: the AS curve shifts to the left to AS’. The price increase from P1 to P2 returns the LM curve to its original LM position, as the real money supply is constantly declining. At point B’ a long-term macroeconomic equilibrium is established with the potential level of production, the initial level of interest rates R0 and the price level increased from P0 to P2. Obviously, in the long run, the growth of the money supply causes only an increase in prices with the invariability of real variables. This phenomenon is called the neutrality of money.
With a single and unexpected increase in the money supply in a short period, both real and nominal interest rates decrease (although changes in the nominal rate may be insignificant). In the long run, both interest rates remain unchanged.
If there are long-term changes in the growth rate of the money supply, then in the long term the nominal interest rate increases, “pushed” by the growth of the expected inflation rate. This does not exclude, however, short-term reductions in the nominal rate due to current monetary policy measures. The expected growth rate of the money supply, according to Fisher’s equation, does not have a noticeable effect on the real interest rate.
Both fiscal and monetary expansion cause only a short-term effect of increasing employment and output, without contributing to the growth of economic potential. The task of ensuring long-term economic growth cannot be solved with the help of a policy of regulating aggregate demand. The incentives for economic growth are linked to aggregate supply policies.