The Keynesian cross is a simple explanation. Equilibrium in the commodity market, simple Keynesian model or "Keynesian cross" model
Investment – Savings Model (I – S)
Let's compare the Keynesian approach with the neoclassical one. In terms of approach:
- Keynesian - savings are a function of income (see Fig. 3.8), and investments are a function of the interest rate (see Fig. 3.10, b);
- neoclassical - both savings and investments are functions of the interest rate (Fig. 3.14).
Rice. 3.14.
Is it possible to combine both approaches?
Yes, they are combined in the J. Hicks model investments - savings (IS ), which describes the equilibrium conditions on real market (Fig. 3.15).
Rice. 3.15.
The beginning of the study is quadrant IV. Assume that the interest rate in the money market is set at r 1 It determines the amount of investment dependent on it at the level I 1 This position is shared by both neoclassicists and Keynesians.
Move counterclockwise to quadrant III. It contains information that is also shared by both economic schools, namely: the condition for macroeconomic equilibrium is the equality of investment and savings. On the graph, this is shown by the bisector I = S. Since the amount of investment has already been determined, it means that for the equilibrium state, the amount of savings must be the same, i.e. S 1.
The next quadrant II contains the Keynesian information that saving is a function of income. Therefore the projection S 1 to the right horizontal axis of the national N.I. income allows you to reach its equilibrium level Y 1.
We approached quadrant I with ready coordinates of the point: Y 1, r 1. In fact, this point contains information about the equilibrium volume of national income N.I. at a certain amount of the interest rate, which is ensured by maintaining a balance between investment and savings, so it can be denoted as IS 1.
Through successive iterations, using different interest rates, one can obtain an infinite number of such equilibrium points IS. They all come together in a curve IS.
economic sense crooked IS lies in the fact that it shows the relationship between the rate of interest r and national income N.I. in equilibrium between savings S and investments I.
Any point on the curve IS reflects both the level of investment and the level of savings.
"Keynesian Cross"
The time has come to combine knowledge about both components of aggregate demand (in the interpretation of J. M. Keynes): consumer spending and investment spending. We will use the graph of consumer spending as a basis (see Figure 3.8) and include investment spending in it. In the "income - expenses" coordinate system, investments look like a horizontal line (see Figure 3.10, A ), so the total C + I will repeat the shape of the graph WITH, but will move parallel to it higher by the amount of investment I (Fig. 3.16).
Rice. 3.16. "Keynesian Cross"
In this case, the macroequilibrium point will shift from to , and the value of the equilibrium national income N.I. will increase by (). We have the right to ask ourselves how these two increments relate: investment and income. Let's say the increase in investment was 100 den. units Will this increase the national income also by 100 den. units?
To answer this question, it is necessary to consider the multiplier effect.
Multiplicative effects
Autonomous spending multiplier is a coefficient showing the dependence of changes in national income on changes in any element of autonomous spending, i.e. independent of the dynamics of national income.
Recall that in the Keynesian model we are talking about consumer and investment spending:
where is the autonomous spending multiplier; - change in national income; – change in autonomous expenses.
This dependence is multiplier (increasing): with the growth of autonomous spending, the growth of national income will occur in a much larger amount than the initial additional spending.
This is because the latter create a chain reaction in the economy, stimulating economic activity and additional employment in related industries. As a result, the "national pie" is growing by leaps and bounds, increasing several times compared to the initial additional autonomous costs.
Let's look at this using the investment multiplier as an example.
Investment multiplier is a coefficient showing the dependence of changes in national income on changes in investment.
In a formalized form, the investment multiplier looks like this:
where is the investment multiplier; - change in national income; - change in investment.
We transform the above formula as
The economic meaning of this formula is that, by advancing investments of a certain size, it is possible to achieve an increase in national income by a predetermined amount.
But what determines the value of the multiplier?
Obviously, from the share of disposable income that is used for current consumption. Let the worker hired as a result of additional investment divide the received wages into two parts: 80% for consumption and 20% for savings. He uses these 80% to buy bread, meat, milk, shoes, etc., thereby allowing the baker, butcher, milkman, shoemaker to place orders with their suppliers, and they, in turn, with theirs, etc. This is how the animation process works. If the ratio between the consumed and saved parts of the employee's income is different (for example, 40% for consumption and 60% for savings), then the volume of current purchases will be half as much, and, consequently, the multiplier process will be more restrained. Thus, the investment multiplier is inversely proportional to marginal propensity to savings, i.e.
It should be understood that the multiplier effect works both ways, but only in a part-time economy.
Let's imagine what happens in a full-employment economy when investment increases. To do this, we use the first macroequilibrium model AD-AS, because it fixes the position of full employment, which is absent in the "Keynesian cross" model (Fig. 3.17).
Investments are an integral part of aggregate demand, so their increase will cause an increase in aggregate demand. Since there are no free resources in the economy, it will respond to the increased aggregate demand not by expanding real production, but only by increasing prices.
Rice. 3.17.
Thus, in a full-employment economy, there are no reserves for production growth, and therefore the investment multiplier does not work.
(keynesian cross) is a macroeconomic model in economic theory showing a positive relationship between total spending and general level prices in the country.
The theory of aggregate demand is often referred to as Keynesian economics. The Keynesian model proceeds from the identity of total expenditures and total income (Say's model): \mathrm V\;=\;\mathrm E , where \mathrm V - income, output; \mathrm E - expenses.
Distinguish between real and planned costs. Planned expenses represent the amount of spending that all economic agents plan to spend on goods and services. Real costs occurs when firms are forced to make unplanned investments in inventory in the face of an unexpected change in sales.
If the economy is closed, then planned spending can be defined as the sum of consumption, planned investment, and government spending:
\mathrm E\;=\;\mathrm C\;+\;\mathrm I\;+\;\mathrm G.
We express the consumption function by the identity: \mathrm C\;=\;\mathrm C\;\cdot\;(\mathrm V\;-\;\mathrm T), the investment function - \mathrm I\;=\;\mathrm I" (investments are fixed), the amount of government spending and the amount of taxes are stable, i.e. \mathrm G\;=\;\mathrm G" and \mathrm T\;= \;\mathrm T" , in this case in a closed economy:
\mathrm E\;=\;\mathrm C\;\cdot\;(\mathrm V\;-\;\mathrm T)\;+\;\mathrm I"\;+\;\mathrm G".
This equality means that the amount of planned expenditures is a function of income, planned investments and planned government purchases.
On the graph, point \mathrm A is the point of equality between actual and planned costs. At the same time, the volume of output is equal to the potential. This model was named Keynesian Cross. If aggregate demand (\mathrm(AD)) rises to (\mathrm E)_1 and aggregate supply grows faster than aggregate demand (\mathrm(AS)\;>\;\mathrm(AD)) , that is, firms increase output in more As aggregate demand grows, unplanned stockpiling occurs. If aggregate demand falls to (\mathrm E)_2 and firms reduce supply to (\mathrm V)_2 , there will be an excess of aggregate demand over aggregate supply: (\mathrm(AD)\;>\;\mathrm(AS) ), it will be satisfied by reducing stocks. The reduction in inventories will stimulate the growth of production and the economy will begin to shift towards natural output.
The equilibrium output (\mathrm V)_0 may fluctuate depending on the change in the value of any component of total expenditure. The growth of any of the components shifts the planned expenditure curve upward, which affects the growth of the equilibrium level of output. The decline in any of the components of aggregate demand entails a decrease in the level of employment and the equilibrium volume of output.
If actual output is less than potential ((\mathrm V)_0\;<\;\mathrm V") , then this suggests that aggregate demand is inefficient, i.e. total spending in the economy is insufficient to ensure full employment of resources. The effect of insufficient aggregate demand has a depressing effect on the economy - there is recessionary gap(even though \mathrm(AD)\;=\;\mathrm(AS) ). In order to overcome this recessionary gap, as well as to ensure full employment, it is necessary to ensure an increase in aggregate demand to a level that ensures that the actual volume of output is equal to the potential one: (\mathrm V)_0\;=\;\mathrm V" .
If the actual output is greater than the potential ((\mathrm V)_0\;>\;\mathrm V"), then this suggests that the total spending in the country is excessive. Due to the excess of aggregate demand, there is inflation boom: the price level therefore rises. Firms do not have the opportunity to expand production in proportion to the increasing aggregate demand, because. all available resources are already occupied in production - there is an inflationary gap. The inflationary gap is overcome by containing aggregate demand.
The Keynes cross can only be used for the purposes of macroeconomic analysis in the short term. implies fixed prices and cannot be used to analyze the consequences of macroeconomic policy in the long run, which are associated with an increase or decrease in inflation.
The Keynesian cross only shows how equilibrium output is established at a given level of planned investment, government spending, and taxes.
Basics of economic theory. Lecture course. Edited by Baskin A.S., Botkin O.I., Ishmanova M.S. Izhevsk: Publishing House "Udmurt University", 2000.
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Add comments4. Actual and planned expenditures, the Keynesian cross
All actual investment collectively, they are subdivided into planned And unplanned, such as investment in inventory. Due to this, the latter are a kind of leveling mechanism for investments in the equality of investment = savings and make it possible to restore macroeconomic balance in the market.
The costs of all economic entities are also classified according to the time of decision-making: planned, i.e. those that were planned for implementation, the acquisition of a certain set of goods or resources at specific prices, and real actually produced. Real, accordingly, differ from the planned in the case when firms make sudden investments in inventories, or in the dynamics of market pricing.
The planned expenditure function is defined as the total expenditure of all economic entities at a certain level of employment, output, and prices. In other words, this function has the same mathematical expression as the function for determining national income or GDP:
E = C + I + G + Xn,
where C is household expenditures on current consumption;
I - expenses of firms for investment;
G - government spending on maintaining public goods and budgetary institutions;
i.e., the costs of foreigners to purchase our products minus our country's costs to consume imported products.
Based on this, autonomous values are of great practical importance: they do not depend on income, or on the interest rate, or on the price level, etc. Accordingly, the autonomous costs of all subjects market system are defined as (a + I + G + g), where C = a + b ? Y d ; respectively a \u003d C - b? Y d ; b is the marginal propensity to consume; Y d is the amount of disposable income, which is obtained by deducting all taxes from the total income.
Net export function:
X n \u003d g - m'Y,
where g is autonomous net exports independent of income;
m' is the marginal propensity to import, i.e. m' = ?M / ?Y; ?M - change in the cost of purchasing imported goods; ?Y is the change in income.
If the income of subjects increases, they begin to purchase more both domestic and imported products. The share of exports itself does not at all depend on the income structure of our population, but is determined by the incomes of those entities that will buy our domestic products abroad. That is why in the function of net exports, the “-” sign precedes income, which means that exports are negatively dependent or independent of the income of the exporting country.
Below is the Keynes Cross, or graph (Figure 1), showing the relationship between total output and planned spending:
Rice. 1. Keynes Cross
The bisector of the angle of the plane is the line of equality of planned expenditures to gross output, i.e., everything that is produced will be consumed, at the same time, this indicates the equality of investment and savings. In fact, this equality (Y = E) occurs by chance and cannot be constant. Accordingly, the planned expenditure curve (E = C + I + G + X n) only crosses the bisector. Therefore, it turns out that the planned costs either exceed the volume of output, or, on the contrary, are much lower than it. Point A reflects the equilibrium value of output and consumption, it is in it that the economy is in a state of macroeconomic equilibrium, when the interests of all subjects are taken into account. This is the benchmark of the market, to which all its actors are striving.
Thus, the segment Y 2 - Y 0 shows that the costs that economic entities plan to carry out significantly exceed the volume of output that was actually produced. Demand exceeds supply, and firms are forced to supply the market with once created stocks to meet the needs of buyers. Thus, an equilibrium value of the level of output and consumption is reached.
The case on the interval Y0 – Y1 is characterized by overproduction. The entire volume of produced goods and services is not in high demand, the number of entities wishing to purchase them is small. In this case, the optimal solution for firms is to transfer unsold products to inventory for use in case of need in future periods. When surplus production is removed from the market, equilibrium is restored.
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Equilibrium model "national income - total expenditure" or "Keynes's cross":
- used in the analysis of the impact of macroeconomic conditions on national income and expenditure flows;
- clearly shows what impact on national income can have a change in each of their components of total expenditure.
Aggregate demand in the Keynesian model depends on such important categories as the consumption function and the savings function. Both consumption and saving are functions of income. For a better understanding, the following quantities are used:
- marginal propensity to consume (MPC) calculated by the formula: , (1)
– marginal propensity to save (MPS): ,
Example. If additional income household is 100 rubles, of which 75 rubles. used for consumption, and the remaining 25 rubles. - for additional savings, then the MRS will be 75/100 = 0.75, and LZS - 25/100 = 0.25.
The marginal propensity to consume is between zero and one: 0< МРС < 1. Их сумма всегда равна единице.
Average propensity to consume (APC) calculated by the formula: 
Average propensity to save (APS)
Let's construct a simplified model, assuming that all total expenditures (AE) are only consumption expenditures.
The y-axis plots planned or desired consumption expenditures (C), which represent all total expenditures. The amount of output, income Y is plotted on the abscissa. If total expenditure exactly corresponded to total income, then this would reflect any point lying on a straight line drawn at an angle of 45º expressed by the equation C \u003d Y. But in reality such a coincidence does not occur, MPC is less than 1, and only part of the income is spent on consumption . The slope of the curve C at all its points is determined by the marginal propensity to consume. Therefore, the graph of the consumption function should deviate from the 45º line down. Hence the consumption function:
Consumption function without autonomous consumption
Consumption exists even at zero income.
Autonomous consumption consumption is independent of the level of current income.
Graphically, autonomous consumption is a line running parallel to the x-axis.
Autonomous consumption
Factors determining autonomous consumption:
- expectations.
- wealth Example. A fall or a sharp rise in the performance of stock exchanges will affect the behavior of those who store their wealth in the form of securities.
- credit.
Let's assume that autonomous consumption will be 100 billion rubles. Taking it into account, the formula for determining the consumption function takes the form:
Graphically, this means that the consumer spending curve is plotted as a vertical summation of the curves. The consumption graph does not proceed from the origin of the coordinate axes, but from a point lying on the ordinate axis corresponding to the level of autonomous consumption.
Consumption function taking into account autonomous consumption.
To build a model in the simplest version, it is necessary to solve a system of two equations:
Y = C (graphically represented by a 45º line
(plot of consumption function)
Example. If MPC = 0.75, and autonomous consumption is 100 billion rubles, then we get: C = 100 + 0.75 Y. Since Y = C, then, substituting the symbol Y for C, we can write: Y = 100 + 0 .75Y. Consequently, the equilibrium level of income will be 400 billion rubles. The intersection of the 45º line and the consumption curve at point E means the level of zero savings. To the left of this point, one can observe a shaded area representing negative savings (i.e. "living in debt"), and to the right, positive savings.
Equilibrium is observed at point E, since only here income and expenses are equal. With an income level equal, for example, to 700 billion rubles, the value of consumption C will be: 100 + (0.75 x 700) = 625 billion rubles. The vertical distance between the 45º line and the consumption curve, denoted by the letter S, is the savings value of $75 billion. rub. (700 - 625).
"Keynesian Cross": simplest model
The savings function is defined
The most important component of the planned total expenditures is investment, I. The level of investment has a significant impact on the volume of the national income of the society.
Investments (capital investments) across the country affect economic growth. The source of investment is savings. Savings is disposable income minus personal consumption expenditures: (Y - T) - C. The problem is that savings are carried out by some economic agents, while investments can be made by completely different groups of economic entities.
Factors affecting investments:
1. Expected rate of return
2. Interest rate level
3. Changes in technology and innovation
4. Level of taxation
5. The rate of inflationary depreciation of money
Investments, the value of which does not depend on current income, are standalone investment.
With the revival of business activity, the growth of employment, the desire of various groups of entrepreneurs to invest will increase. These investments are called derivative, or induced, they depend on the dynamics of national income. The higher the level of income in society, the more investments are made, and vice versa.
Investment is a function of the interest rate: I = I (r)
This function is decreasing: the higher the interest rate, the lower the level of investment. The balance between investment and saving is determined by the flexible interest rate.
According to Keynes, savings is a function of income, not of the interest rate: S = S(Y).
In a closed economy without taking into account government spending, the most important macroeconomic identity: Y = C + I
On the one hand, the national income when it is used is equal to the sum of expenditures on consumption (C) and investment (I).
On the other hand, the produced national income can be represented as:
Y = C + S , where S is a function of current income.
The Keynesian cross model is interpreted on the basis of the injection approach.
Keynesian cross model
The third component of spending is government spending (G). Let us assume that their value is 200 billion rubles.
The fourth component of autonomous total expenditures is net exports (Xn). It also does not depend directly on current income. Suppose the value of net exports is 50 billion rubles.
The y-axis shows the value of all components of autonomous costs:
C a = 100 billion rubles, I = 150 billion rubles, G = 200 billion rubles, Xn = 50 billion rubles. In total, the value of autonomous expenses is designated A and is equal to 500 billion rubles. The highest line (C a + I + G + Xn) reflects all components of autonomous spending. At any equilibrium point located on the 45º line, the total expenses AE are equal to the income Y and can be represented by the formula: .
Macroeconomic equilibrium is shown by points E 1, E 2, E 3, E 4, reflecting more high level income as the autonomous components of total expenditure increase. Line F shows the level of full employment, at which income is equal to 2000 billion rubles.
The graph of consumer spending, built by us earlier, rises by an amount equal to the sum of all components of autonomous spending.
Macroeconomic equilibrium at full employment can be graphically achieved by shifting the aggregate expenditure curve upward.
At point E 4, the equilibrium level of income at full employment is 2,000 billion rubles. The equilibrium value Y E is determined by the formula:
A is the value of any autonomous expenses.
An upward shift in the graph of total expenditures is caused by an increase in any of the components of autonomous total expenditures. This is explained by changes in the determinants of autonomous spending on consumption, investment, government spending and net exports.
Multiplier
"Multiplier" means "multiplier". The essence of the multiplier effect is as follows: with an increase in any of the components of autonomous spending, the national income of society increases, moreover, by a value greater than the initial increase in spending.
M value 
is the autonomous spending multiplier.
The multiplier formula can also be expressed in terms of the marginal propensity to save: M
The higher the propensity to consume and, accordingly, the lower the propensity to save, the greater Mk and the greater the increase in national income will accompany the initial increase in investment. Thus, the multiplier can be defined as the ratio of the change in income to the change in any of the components of autonomous expenses M
The multiplier effect does not only work in the direction of increasing the level of income, or output. Reducing any of the components of autonomous spending will cause a multiple reduction in income and employment. Then, in our case, the schedule of total expenditures will shift down, and macroeconomic equilibrium will be established at an ever lower level of income.
Lecture 7. EQUILIBRIUM IN THE COMMODITY MARKET. SIMPLE KEYNESIAN MODEL or "KEYNESIAN CROSS" MODEL.
7.1. Commodity market and its equilibrium
In order to determine the value of equilibrium output (equilibrium national
income) should be equated to the amount of planned expenses: where
What happens if costs increase? Keynes showed that an increase in spending leads to an increase in income, but income increases to a greater extent than the increase in spending that caused it, that is, with a multiplier effect. The multiplier is a coefficient that shows how many times the total income (output) increases (decreases) with an increase (decrease) in spending per unit. The action of the multiplier is based on the fact that the expenses made by one economic agent necessarily turn into income of another economic agent, which spends part of this income, creating income for the third agent, etc. As a result, the total amount of income will be greater than the initial amount of expenses.
Assume that a household increases its autonomous spending by $100, i.e., purchases goods and services with this amount. This means that the producer of these goods and services receives an income of $100, which he spends on consumption and savings. Let us assume that the marginal propensity to consume mpc = 0.8, which means that for each additional $1 of income, an economic agent spends 80 cents (i.e. 80%) on consumption and saves 20 cents (i.e. 20%) ( i.e. marginal propensity to save mps = In this case, having received $100 additional income, the producer will spend $80 on consumption (Y x mps = 100 x 0.8 = 80) and $20 will go to saving (Y x mps = 100 x 0.2 = 20 $80 he spent on consumption (to buy goods and services) will create additional income for another seller, who in turn will spend $64 on consumption (Y x trc = 80 x 0.8 = 64) and save $16 (respectively 80 x 0.2 = 16 ), etc. The process will continue until the increase in costs reaches 0.
Let's sum all the income received to find out how much the total income increased as a result:
We got an infinitely decreasing geometric progression (and this is the mathematical meaning of the multiplier) with a base (trc) less than one. Therefore, its sum is
It is a multiplier of (autonomous) consumer spending. In our example, the multiplier is 5 (1/=5). Therefore, with an increase in autonomous consumer spending by $100, the increase in total income was $ x 5 = 500).
Similar reasoning applies to changes in (autonomous) investment spending. By increasing investment, the firm purchases investment goods, creating income for their producer, who in turn spends part of this income on consumption, providing additional
move to the manufacturer of these consumer goods etc. As a result, the growth in total income will be several times greater than the initial increase in investment, i.e., there will be a multiplier effect, and the multiplier (but in this case, investment costs) will also
will be equal to
The formula for the autonomous spending multiplier can also be derived algebraically. Because: 
A graphical depiction of the effect of an expense multiplier (for example, the investment multiplier) is shown in Figure 7.5.
The figure shows that each subsequent increase in income is less than the previous one. The multiplication process continues until the increase in income becomes equal to zero.
|
|
The higher the marginal propensity to consume (trc), the larger the autonomous spending multiplier. So, for example, if trc = 0.9, multiplier =/, and if trc = 0.75, multiplier = 4 (1/= 4). And since trc determines the slope of the planned expenditure curve, the larger the trc, the steeper the curve.
And the steeper the curve of planned expenditures (ie, the greater the trs and, consequently, the multiplier, the greater the increase in income will give the same increase in expenditures). This is illustrated in Figure 7.7. In Fig. 7.6 (b), the marginal propensity to consume (trc) is greater, and therefore the planned expenditure curve is steeper and the multiplier effect of income growth with the same amount of expenditure growth is greater (®Y2 > ®Yi) than in Fig. 7.6 (a) .
By adding government sector In our analysis, we obtain a three-sector model in which three macroeconomic agents operate: households, firms, and the state. Government spending is an important component of total spending (aggregate demand). Unlike C and I, government spending is an exogenous value or a so-called control parameter. Government spending does not depend on the level of income and is entirely determined by the macroeconomic (primarily fiscal) policy of the government.
Public expenditures arise in connection with the need for the state to perform its many functions, the main of which in the modern economy are:
1) determination of the rules for conducting economic activity, i.e., the “rules of the game” (antimonopoly law, supporting the development of the private sector of the economy, protecting property rights, protecting freedom of competition, protecting consumer rights, etc.);
2) maintaining economic stability (fighting inflation and unemployment and ensuring economic growth);
3) production of public goods (ensuring security, law and order, education, healthcare, development of fundamental science);
4) social policy (social security of the poor through redistribution of income, payment of pensions, scholarships, unemployment benefits, etc.);
Table 1. Taxation systems
|
Proportional tax |
progressive tax |
regressive tax |
||||
|
tax rate |
Tax Amount |
tax rate |
Tax Amount |
tax rate |
Tax Amount |
|
With proportional tax tax rate does not depend on the amount of income. Therefore, the amount of tax is proportional to the amount of income.
Direct taxes (with the exception of income tax and in some countries income tax) and almost all indirect taxes are proportional.
With a progressive tax, the tax rate increases as income increases and decreases as income decreases.
An example of a progressive tax is the income tax. Such a system of taxation contributes to the maximum extent to the redistribution of income.
In a regressive tax, the tax rate increases as income decreases and decreases as income increases.
The explicitly regressive system of taxation in modern conditions not observed, i.e. there are no direct regressive taxes. However, all indirect taxes are regressive, and the higher the tax rate, the more regressive it is. The most regressive are excise taxes. Because the indirect tax- this is part of the price of the goods, then, depending on the amount of income of the buyer, the share of this amount in his income will be the greater, the less income, and the less, the greater the income. For example, if the excise tax on a pack of cigarettes is 10 rubles, then the share of this amount in the budget of a buyer with an income of 1000 tons. , is equal to 0.1%, and in the budget of the buyer, who has an income of 5000 tons. - only 0.05%.
In macroeconomics, taxes are also divided into: autonomous (or lump-sum), which do not depend on the level of income and are denoted by T and income, which depend on the level of income and the value of which is determined by the formula: tY, where t is the tax rate, Y is the total income ( national income or gross national product)
The amount of tax revenues (tax function) is equal to: Т= Т + tY Distinguish between the average and marginal tax rates. The average tax rate is the ratio of the tax amount to the amount of income: tav = T/Y. The marginal tax rate is the amount of the increase in the tax amount for each additional unit of income increase, (it shows how much the tax amount increases with an increase in income per unit): 
Let's assume that the economy has a progressive taxation system, and income up to $50,000 is taxed at a rate of 20%, and more than $50,000 at a rate of 50%. If a person receives 60 thousand dollars of income, then he pays an amount of tax equal to 15 thousand dollars (50 x 0.2 + 10 x 0.5 = 10 + 5 = 15), i.e. 10 thousand dollars from the amount of $50,000 and $5,000 over $50,000, i.e. $10,000. The average tax rate would be 15:60 = 0.25 or 25%, and the marginal tax rate would be 5 :10 = 0.5 or 50%. Under a proportional taxation system, the average and marginal tax rates are equal.
Taxes affect both aggregate demand and aggregate supply. However, our cost-income model, because it is a Keynesian model, only considers the impact of taxes on aggregate demand.
Within the framework of the “expenditure-income” model, taxes, as well as government purchases, act on national income (total output) Y with a multiplier effect.
There are two types of tax multiplier: 1) multiplier of autonomous (chord) taxes and 2) income tax multiplier 7.7. Autonomous tax multiplier
Let us first consider the effect of the multiplier of autonomous taxes, i.e., those that do not depend on the level of income. Since the simple Keynesian model assumes that taxes are levied only on households, i.e., they affect the amount of consumer spending, the consumption function changes with the inclusion of taxes in our analysis, taking the form: С = С+ trs (Y - Т).
A change in taxes leads to a change in the amount of disposable income. (RD = LD - T). Tax increases reduce disposable income, while tax cuts increase disposable income. If, for example, taxes are reduced by $100, then disposable income increases by $100. But disposable income is divided into consumption (C) and saving (S). If mpc = 0.8, then for a $100 increase in disposable income, consumption increases by $80 (100 x 0.8 = 80), and since the spending multiplier in this case is 5 (1/(= 1/0.2 = 5), then the increase in total income as a result of a change in taxes by $100 will be $400 and not $500, as in the case of a change in government purchases by the same $100, i.e., the multiplier effect is less. the member of the geometric progression will not be 100, but 80).
Let us now determine the value of the tax multiplier. Taxes act on aggregate demand through changes in consumer spending.
The value is the tax multiplier. And since (1 - trs) is nothing but
mps (marginal propensity to save), the tax multiplier can also be written as (-mpc / mps). In our example, it is equal to / (= - 0.8 / 0.2 = - 4). The tax multiplier is a coefficient that shows how many times the total income will increase (reduce) when taxes are reduced (increased) per unit.
We derive the multiplier of autonomous taxes algebraically. We substitute the consumption function ^ = C + mpc (Y-T) into the national income function Y = C + I + G, we get: Y = C +
mpc (Y - T) +1+ G, whence 
. If we denote the multiplier of autonomous
taxes 
and therefore
You should pay attention to 2 points:
1) the tax multiplier is always negative. This means that its effect on
total income is the opposite. An increase in taxes leads to a decrease in total income, and
cutting taxes - to the growth of total income. In our example, tax cuts on
Led to $400 increase in total revenue
2) in its absolute value, the tax multiplier is always less than the multiplier
autonomous spending. So, the multiplier effect of taxes is less than the multiplier
catalytic effect of public procurement (obviously,
how much a change in government purchases affects aggregate demand directly
directly (they are included in the aggregate demand formula), and a change in taxes affects
indirectly through changes in consumer spending. For example, if at trs = 0.8 and go-
government purchases, and taxes are increased by $100, then an increase in government purchases
increases total income by a growth
taxes reduces total income by 400). That is, as a result, total income (output) increased by $100.
Based on this circumstance, it is possible to derive a balanced budget multiplier for autonomous (chord) taxes.
7.10. Balanced budget multiplier
The budget is called balanced if government purchases and taxes increase by the same amount (G = T). As our example shows, a $100 increase in both government purchases and autonomous taxes resulted in a $100 increase in national income Y, which means that the balanced budget multiplier is 1 (100:100 = 1).
Let's derive the balanced budget multiplier algebraically. Let's compare the multiplier effect that a change in the autonomous spending of the state and taxes gives. Change
the value of government purchases leads to a change in income: 
and a change in autonomous taxes leads to a change in income: 
The overall change in Y will occur under the combined influence of these two effects, i.e. 
Hence 
Let's add the foreign sector to the analysis. As a result, we obtain a four-sector model of the economy. Foreign sector spending is an important component of total spending and is known as net export spending. Net export is one of the types of relations of a given country with other countries ( international trade). Net exports equal the difference between exports and imports. Exports are autonomous, that is, they do not depend on the income level of a given country, but are determined by the income level in other countries (trading partner countries) (direct relationship) and the level of the exchange rate (inverse relationship). Exports represent the demand of the foreign sector for the goods and services of a given country. Therefore, the higher the level of income in other countries, the more willingly they will buy goods produced in this country, i.e., exports will increase. And the higher the exchange rate of the national currency, the more expensive and therefore less attractive they become for foreigners, so exports fall. The export function can therefore be expressed by the formula: 
where is income in other countries, e is the exchange rate of the monetary unit of this country.
As for imports, one part of it may not depend on the level of the aggregate income of a given country and represent autonomous imports, but the other part necessarily depends on the level of income, since the growth of the national income of this country leads to an increase in demand for goods and services, including imports, that is, with an increase in income, imports increase. Thus, imports are divided into autonomous and non-autonomous (induced) and therefore the import formula can be presented: where Гт - autonomous imports, and mpm - marginal propensity to import. (Note that imports depend on national income, not disposable income.) The marginal propensity to import is a value that shows how much imports will increase (decrease) with an increase (decrease) in income per unit: 
0 < mpm < 1
In addition, imports also depend on the exchange rate of the national currency. Moreover, the dependence is direct, that is, the higher the exchange rate of the national currency, the cheaper and more attractive imported goods become for domestic buyers).
Since net exports are the difference between exports and imports, the net export function is:
where (Ex - Im) is autonomous net export and (mpm Y) is induced import.
The slope of the curve of planned total expenditures in the four-sector model of the economy is less (it is flatter) than in the three-sector one, since it is determined by the value (mpc (1-t) - mpl), and in the presence of induced investments by the value (mpc (1 - t) + mpl - trt) (Fig. 7.11). Therefore, the multiplier effect in an open economy is smaller than in a closed one.
A change in the value of autonomous net exports shifts the curve of planned
buying expenses. An increase in autonomous net exports leads to a parallel shift in the aggregate expenditure curve, while a decrease leads to a downward shift.
An increase in the marginal propensity to import changes the slope of the planned expenditure curve and the value of the multiplier. The larger the TPT, the flatter the curve, so the multiplier effect is smaller.
Let us include the function of net exports in the equation of equality of total income (output) Y to the total expenses of all macroeconomic agents:
Value 
is the cost multiplier. Let's call it KA
Note that the expression in brackets is the sum of all autonomous, i.e., income-independent expenses. A change in any of the components of autonomous total expenditures leads to a multiplicative change in the value of equilibrium income Y. Therefore, an increase in autonomous net exports leads to a multiplicative increase in income: 
So, the autonomous spending supermultiplier is:
where A_ is the amount of autonomous expenses (not depending on the level of income). Super tax multiplier:
Transfer Super Multiplier: 
The denominator of the supermultiplier (the reciprocal of the multiplier) is called the marginal leakage rate (MLR):
7.17. Savings Paradox.
From simple
Keynesian model
it followed that for the growth
economy needs
increase the total
expenses that are injections and cause the growth of total income, moreover, with a multiplier effect. And all withdrawals from the flow of spending multiplierly reduce total income, leading the economy to recession and even depression. A paradoxical conclusion followed from this: the more the economy saves (accumulates), the poorer it becomes. (The paradox is that if a person increases his savings, then he becomes richer, and the economy becomes poorer with an increase in savings). The graphical interpretation of the savings paradox is presented in Fig. 7.12 in two different versions: 1) on the graph of investments and savings (Fig. 7.12. (a)) and 2) on the graph of injections and withdrawals (Fig. 7.12. (6))
Since in the Keynesian model, savings positively depend on the level of income, and investment is an autonomous value, the savings curve has a positive slope, and the investment curve is horizontal (Fig. 7.12 (a)). An increase in savings leads to a left-up shift in the savings curve from Si to S2. If the amount of investment does not change, then the growth of savings leads to a reduction in total income (output) from Y| to Y2 Thus, as a result of the growth of savings, the economic situation worsens.
Figure 7.12.(6) shows the curve of autonomous costs (injections), which do not depend on
from the level of income and therefore are represented by a horizontal line and a withdrawal curve, the value of which is a certain share of total income, equal to. The slope of the seizure curve is determined by the MLR. The graph allows you to study the impact on the economy of any type of withdrawals (eg taxes, imports), not just savings. As seizures increase, the MLR rises and the slope of the seizure curve becomes steeper. As a result, with a constant value of autonomous expenditures, the total output is reduced from "
However, the gloomy picture of the savings paradox exists only in the Keynesian model. In the classical model, saving is always equal to investment. Therefore, in accordance with classical ideas, if savings increase, then investments increase by the same amount. Graphically, investment growth looks like an upward shift in the investment curve from As a result, no reduction in income (output) occurs (Fig. 7.12. (a)). Similarly, if the marginal seizure rate increases as a result of an increase in any of the types of seizures, then this is offset by a corresponding increase in injections, and the value of aggregate output does not change (Fig. 7.12.(6)).
A simple Keynesian model allows you to show a way out of a recession. Such a measure should be the active intervention of the state in the economy. It is no coincidence that the measures proposed by the Keynesians were called the policy of state activism. Keynes and his followers proposed using fiscal policy to stabilize the economy, and first of all, such an instrument as changing the amount of government spending, since this allows you to directly, and, therefore, to the maximum extent influence aggregate demand and with a multiplier effect on aggregate output and income.
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