Aggregate demand and Keynesian multiplier
In the model AD-AS the interest rate is an exogenous variable. The main purpose of the IS-LM model is to integrate money market and goods market into the single system. Thus, interest rate turns into endogenous variable. IS-LM – a model of money-goods equilibrium which helps to define factors influencing AD.
Тема 11. Совокупное предложение. Макроэкономическая сбалансированность и экономический рост
We can use the Keynesian Cross to examine the impact of a change in any of the exogenous
variables C, I, G or NX on the value of Y. We can express the expenditure of households, firms, foreigners and the government on domestic goods and services as E = C + I + G + NX (C-consumption, I-investment, G-government expenditures, NX net export). We specify the following consumption function that relates consumption (C) to disposable income (after-tax income): C = C + b[Y − T] where b is MPC=deltaC/deltaYd.
In equilibrium, Expenditure = Output, E = Y . We can now express the model as
Y = C + I + G + NX
Y = C + b[Y − T] + I + G + NX (C is autonomous consumption)
Solving for Y from the above, the equilibrium level of output in this model can be algebraically expressed as Y =C + I + G + NX − bT/1 − b.
From the above expression, we see that a 1 unit increase in any of these variables increases Y by 1/1−b units. This increase is > 1 because the marginal propensity to consume, b, is < 1. The translation of a 1 unit change in any of the exogenous variables into a more than 1 unit increase in Y is known as the multiplier effect, with 1/1−b being called the spending multiplier (из того, что нашла в Интернете следует, что он же и является мультипликатором Кейнса). When income has gone up by 1 unit because of the direct effect, consumption will increase by b units since the consumer spends b of every dollar that he/she receives. This is called a feedback effect. This feedback effect does not end here, the increase in consumption causes income to go up by b units which in turn causes consumption to rise by b2 units and so on. If we denote the change in government purchases by deltaG and the corresponding impact on output by deltaY then we can express the net impact of an increase in G on Y as:
deltaY = deltaG[1 + b + b^2 + · · · ] = deltaG*(1/1 − b)
2. The interconnection of AD-AS and IS-LM. The equilibrium in IS-LM.
In the model AD-AS the interest rate is an exogenous variable. The main purpose of the IS-LM model is to integrate money market and goods market into the single system. Thus, interest rate turns into endogenous variable. IS-LM a model of money-goods equilibrium which helps to define factors influencing AD. It helps to find R and Y which bring an equilibrium both in money and goods markets. Thus, IS-LM concretizes AD-AS.
IS curve shows all the combinations of Y and R which satisfy equations 1, 2, 3, 4. At any point I=S.
LM curve combines Y and R which satisfy the equiation 5. At all the points Md=Ms
An increase in Gov expenditures and a decrease in Taxes lead to a crowding out effect which decreases the efficiency of stimulating fiscal policy.
Money supply increase can provide a short-term economic growth without a crowding-out effect, but it can affect contrarily net export dynamics.
NX is influenced both by Y increase (which lowers NX) and R decrease (which increases NX). The real changing NX depends on the value of delta Y and delta R and on the value of m and n coefficient of NX sensitivity to R.
Relative efficiency of budgetary and credit-monetary policy depends on:
Relative efficiency of stimulating fiscal policy depends on the crowding out effect.
It is small if: 1) I and NX are not sensitive to R increase (IS is more steep) 2) money demand is sensitive to R rise and increase in R shall be small enough to bring money market in equilibrium. (LM is more sloping)
So, fiscal policy is the most efficient if IS is steep and LM is sloping. (ну, и наоборот, соответственно)
Relative efficiency of monetary policy depends on the stimulating effect of money supply increase and R decrease on the dynamics of I and NX.
This effect is big if: 1) I and NX are sensitive to interest rates dynamics. (IS is sloping) 2) Demand for money is not sensitive to R dynamics. (LM is steep).
Thus, monetary policy is relatively efficient if LM is steep and IS is sloping.
Both of these policies are connected with a move of AD curve upwards. (это к вопросу влияния макроэкономической политики на совокупный спрос. Более светлых идей по этому поводу у меня нет).
Врезка по теме из Мэнкью (Научный креатив, чтоб его): The ISLM model shows how monetary and fiscal policy influence the equilibrium level of income.The predictions of the model, however, are qualitative, not quantitative. The ISLM model shows that increases in government purchases raise GDP and that increases in taxes lower GDP. But when economists analyze specific policy proposals, they need to know not only the direction of the effect but also the size. For example, if Congress increases taxes by $100 billion and if monetary policy is not altered, how much will GDP fall? To answer this question, economists need to go beyond the graphical representation of the ISLM model. Macroeconometric models of the economy provide one way to evaluate policy proposals. A macroeconometric model is a model that describes the economy quantitatively, rather than only qualitatively. Many of these models are essentially more complicated and more realistic versions of our ISLM model. The economists who build macroeconometric models use historical data to estimate parameters such as the marginal propensity to consume, the sensitivity of investment to the interest rate, and the sensitivity of money demand to the interest rate. Once a model is built, economists can simulate the effects of alternative policies with the help of a computer.
If prices increase from P1 to P2, real money supply goes down, LM moves left. It provokes R increase and I fall and relatively decreases XN. Thus the production volume decreases from Y1 to Y2.
Stimulating budgetary-tax policy (price level changing)
Economy starts t the point C. G increases (or T decreases), IS moves right to IS reflecting AD increase up to AD. It provokes inflation of demand P0 goes up to P1. It provokes real money supply decrease and moves LM curve to the left to LM. At the point A we have a short-term macroeconomic equilibrium. (Ура!) But under inflation conditions, nominal salary grows. Thus, firms expenses grow and profits decrease. They cut down their production volume and AS moves to the left to AS. It provokes again prices increase (expenses inflation) from P1 to P2. LM goes to the left to LM. R increases all the time from R0 to R2. at the point B we have a long-term macroeconomic equilibrium. (троекратное ура!)
Stimulating monetary policy (changing prices)
Economy starts at the point B. Money supply increase moves LM curve to the right to LM which reflects increase in AD to AD. Prices go up from P0 to P1. It decreases real money supply and LM moves to the left to LM. At the point A we have short-term macroeconomic equilibrium. But inflation of expenses leads to AS decrease: AS moves to the left to AS. Increase of prices from P1 to P2 drives LM back to LM. At the point B we have a long-term macroeconomic equilibrium. A conclusion: in a long-run money supply increase provokes only a prices increase and doesnt change real variables. (хватит деньги штамповать, если вкратце)
Classical and Keynesian theory suggest two extreme approaches:
Y=Y*+a(P-P^e), where Y* is potential GDP, P current price level, P^e expected price level, a positive coefficient (alpha).
The slope of AS curve depends on changing in AD and price level. If fluctuations are big the AS curve is steep AS almost doesnt react to demand and prices fluctuations. And vice versa.
Phillips curve shows interdependence between inflation and unemployment. We take an equation of AS Y=Y*+a(P-P^e), where Y* is potential GDP, P current price level, P^e expected price level, a positive coefficient (alpha). Lets rewrite it as P=P^e+1/a(Y-Y*). Lets deduct from the both parts P(-1) (the price of the previous period) and replace the level of prices difference by the inflation rate = P-P(-1)=п P^e-P(-1). Then we replace the difference between current Y and potential by the difference between current level of unemployment and its natural level. (это можно сделать по какому-то закону Оукена. Не знаю, что это и знать, откровенно говоря, не хочу). As a result we will get the Phillips curve equation:
п=п^e-B(u-u*)+e, where п and п^e are current and expected inflation level; u and u* - current and natural unemployment levels; B (beta) a positive parameter; e (epsilon) shows the influence of supply shocks over the price level.
Expected inflation and supply shocks move Phillips curve.
If u=u* and there are no shocks real inflation equals an expected one.
If written like we did, the Phillips curve depicts a short-run perspective. Economic policy targeted at U decrease through AD stimulation is efficient only until economic agents change there inflation expectations.
In the long-run stimulation of AD going along with money supply growth doesnt lead to significant changes in U and real output and mostly causes price increase. Thus, the Phillips curve gets close to a vertical curve at the level of natural U.
Кейс из Мэнкью: Because inflation and unemployment are such important measures of economic performance, macroeconomic developments are often viewed through the lens of the Phillips curve. Lets take an example of US inflation and unemployment since 60s up to the oil shocks. The 1960s showed how policymakers can, in the short run, lower unemployment at the cost of higher inflation.The tax cut of 1964, together with expansionary monetary policy, expanded aggregate demand and pushed the unemployment rate below 5 percent. This expansion of aggregate demand continued in the late 1960s largely as a by-product of government spending for the Vietnam War. Unemployment fell lower and inflation rose higher than policymakers intended. The 1970s were a period of economic turmoil.The decade began with policymakers trying to lower the inflation inherited from the 1960s. President Nixon imposed temporary controls on wages and prices, and the Federal Reserve engineered a recession through contractionary monetary policy, but the inflation rate fell only slightly. The effects of wage and price controls ended when the controls were lifted, and the recession was too small to counteract the inflationary impact
of the boom that had preceded it. By 1972 the unemployment rate was the same as a decade earlier, whereas inflation was 3 percentage points higher. Beginning in 1973 policymakers had to cope with the large supply shocks caused by OPEC.OPEC first raised oil prices in the mid-1970s, pushing the inflation rate up to about 10 percent. This adverse supply shock, together with temporarily tight monetary policy, led to a recession in 1975. High unemployment during the recession reduced inflation somewhat, but further OPEC price hikes pushed inflation up again in the late 1970s.
Adaptive expectations means that people form their expectations about what will happen in the future based on what has happened in the past. For example, if inflation has been higher than expected in the past, people would revise expectations for the future. If inflation expectations are built on this theory, the Phillips curve demonstrates inflation inertia: inflation will continue without unemployment deviation and without shocks just because there exist inflation expectations.
Rational expectations is a hypothesis in economics which states that agents' predictions of the future value of economically relevant variables are not systematically wrong in that all errors are random. Equivalently, this is to say that agents' expectations equal true statistical expected values. Economic agents build their expectations on the base of available information, not only basing on their previous experience. That is why they can quickly change their inflation expectations and as a result they might require increasing nominal salary. As a result, there is no increase in real output even in a short term and increase in AD leads only to prices increase.
Anti-inflation policy of the government in a short-run leads to increase of U and decrease of output, according to Phillips curve. A decrease in G or in money supply decreases price level, while salary remains the same. Firms profit goes down, they try to decrease their costs and U as a result goes down. Supporters of rational expectations theory believe that such loses can be decreased by announcing about such an anti-inflation policy implementation before economic agents form their expectations.
Anti-inflation policy might be handled both as a shock therapy (monetary measures liberalization of economy, prices liberalization, limitation of governmental activity, limitation of money supply growth, budgetary expenses decrease) or gradually (active regulatory role of government: support of industries, tax stimulation of enterprises, partial regulation of pricing, market infrastructure development).
Кейс из Мэнкью: The early 1980s saw the largest and quickest reduction in inflation in recent U.S. history. By the late 1970s inflation had reached the double-digit range; in 1979, consumer prices were rising at a rate of 11.3 percent per year. In October 1979, only two months after becoming the chairman of the Federal Reserve, Paul Volcker announced that monetary policy would aim to reduce the rate of inflation. This announcement began a period of tight money that, by 1983, brought the inflation rate down to about 3 percent. How does such a monetary tightening influence interest rates? According to the theories we have been developing, the answer depends on the time horizon. The Fisher effect in suggests that in the long run Volckers change in monetary policy would lower inflation, and this in turn would lead to lower nominal interest rates.Yet the theory of liquidity preference predicts that, in the short run when prices are sticky, anti-inflationary monetary policy would lead to falling real money balances and higher nominal interest rates. Both conclusions are consistent with experience. Nominal interest rates did fall in the 1980s as inflation fell. But comparing the year before the October 1979 announcement and the year after, we find that real money balances (M1 divided by the CPI) fell 8.3 percent and the nominal interest rate (on short-term commercial loans) rose from 10.1 percent to 11.9 percent. Hence, although a monetary tightening leads to lower nominal interest rates in the long run, it leads to higher nominal interest rates in the short run
Supporters of this theory believe taxes decrease to be a factor of AS increase. They use the Laffer curve to prove it.
Taxes decrease can create additional incentives to labor, investment, savings, production expansion, and as a result, to tax base increase. Tax avoidance will fall down. It will be beneficial for budget U increase will decrease unemployment relief, income increase will raise total tax proceeds. However, critics of this approach say that it works only in a long-run and the quality of positive changes largely depends on the sensitivity of the economy to taxation changes. Moreover, in practice you never know at which point on the curve the economy starts.
Measures of AS stimulation:
Врезка из Мэнкью: In the early 60th the USA experienced a remarkable growth. Kennedy executed a tax cut. The tax cut was intended to stimulate expenditure on consumption and investment and thus lead to higher levels of income and employment. Economists continue to debate the source of this rapid growth in the early 1960s. A group called supply-siders argues that the economic boom resulted from the incentive effects of the cut in income tax rates. According to supply-siders, when workers are allowed to keep a higher fraction of their earnings, they supply substantially more labor and expand the aggregate supply of goods and services. Keynesians, however, emphasize the impact of tax cuts on aggregate demand. Most likely, both views have some truth: Tax cuts stimulate aggregate supply by improving workers incentives and expand aggregate demand by raising households disposable income.
The term business cycle (or economic cycle) refers to economy-wide fluctuations in production or economic activity over several months or years. These fluctuations occur around a long-term growth trend, and typically involve shifts over time between periods of relatively rapid economic growth (an expansion or boom), and periods of relative stagnation or decline (a contraction or recession).Business cycles are usually measured by considering the growth rate of real gross domestic product, employment level. There are four phases of business cycle which are generally labelled as Peak, Recession, Trough and Recovery. There are four phases of the business cycle:
The explanation of fluctuations in aggregate economic activity is one of the primary concerns of macroeconomics. The main framework for explaining such fluctuations is Keynesian economics. In the Keynesian view, business cycles reflect the possibility that the economy may reach short-run equilibrium at levels below or above full employment. If the economy is operating with less than full employment, i.e., with high unemployment, Keynesian theory states that monetary policy and fiscal policy can have a positive role to play in smoothing the fluctuations of the business cycle.
Neoclassical: The notion of growth as increased stocks of capital goods (means of production) was codified as the Solow-Swan Growth Model, which involved a series of equations which showed the relationship between labor-time, capital goods, output, and investment. According to this view, the role of technological change became crucial, even more important than the accumulation of capital.
Salter cycle: economic growth is enabled by increases in productivity, which lowers the inputs (labour, capital, material, energy, etc.) for a given amount of product (output). Lowered cost increases demand for goods and services, which also results in capital investment to increase capacity. New capacity is more efficient because of new technology, improved methods and economies of scale. This leads to further price reductions, which further increases demand, until markets become saturated due to diminishing marginal utility.
Endogenous growth theory: includes a mathematical explanation of technological advancement. This model also incorporated a new concept of human capital, the skills and knowledge that make workers productive. Unlike physical capital, human capital has increasing rates of return. Therefore, overall there are constant returns to capital, and economies never reach a steady state. Growth does not slow as capital accumulates, but the rate of growth depends on the types of capital a country invests in.
Useful work growth theory: claims that physical and chemical work performed by energy, or more correctly exergy, has historically been a very important driver of economic growth. Key support for this theory is a mathematical model showing that the efficiency of electrical generation is a good proxy for the Solow residual, or technological progress, that is, the portion of economic growth that is not attributable to capital or labor.
13. Definition and factors of economic growth.
Economic growth long-term tendency of real output increasing in the economy. Growth is measured by real GDP or GDP per capita increment. Economic growth is called extensive if it is realized by involvement of additional resources and does not change average labor productivity. Intensive growth is related to usage of more developed production factors and technologies. Thus, it is executed not by increase of resources expenditures, but by increase of efficiency of their usage. Factors of growth can be:
1) extensive (increase of capital and labor expenditures), 2) intensive (technological progress, economy of scale, improvement of professional level of workers, increase of resource mobility and their allocation efficiency, improvement of operation management, relevant legislation development etc.) Sometimes AD is distinguished as a separate factor of economic growth as a main catalyst of the production broadening.
Among the factors restraining growth we have resource and ecological constraints, social costs, non-efficient governmental policy.
Креатив (приводим в пример экстенсивного роста Россию-матушку): The early development of the USSR was primarily of the extensive sort. Increased application of labor inputs came from reduced unemployment, use of women previously engaged within the household, diminished leisure (e.g., communist sabbaticals or subotniki), and forced or prison labor. Now national economy mostly has been developing by increase in minerals extraction and additional labor resources.
Keynesian growth models usually use the same instruments which are used in the model of short-term equilibrium. However, here the analysis from the demand side is to be joined with factors defining the supply dynamics. Strategic variable here is investment.
The simplest Keynesian model is the Domar model. The assumptions: 1) There is an excessive labor supply which provides the price level stability. 2) K/Y is constant 3) the output depends largely on capital. The factor of demand increase is investment. If in the current period I grew by delta I, AD will grow with a multiplier effect:
Delta Yad = delta I*(1/1-MPC)=delta I*1/MPS. Delta Yas = alpha*delta K, alpha is marginal capital productivity. Delta K provides a corresponding investment, so we can rewrite delta Yas=alpha*I. An equilibrium is reached if delta I/MPS=alpha*I or delta I/I=alpha*MPS. Since I=S we can write deltaY/Y=deltaI/I=alpha*MPS. Thus, there is an equilibrium pace of growth of real income in the economy which implies a full usage of production capacities.
The extension of this model Harrod model (там столько математики с двухэтажными дробями, что вы, друзья мои, точно не хотите это видеть). He formalized an equilibrium growth pace as follows: delta Y/Yt-1=MPS/v-MPS, where v is accelerator. Both models come to a conclusion that under current technical conditions growth rate depends on MPS and dynamic equilibrium can exist under non-full employment condition.
1. Production function Y = f (K, L)
zY = f (zK, zL) constant return to scale
y = f (k) per worker y = Y/L, k = K/L
MPK = f (k+1) f(k) marginal productivity of capital
2. Consumption and Investment function
y = c + i , c = (1-s) y ; i = sy
i = s f(k), c = f(k) sf(k)
d - depreciation rate, or proportional fraction of capital that wears out every year
Change in Capital stock ∆ k = i dk = sf(k) dk
Steady state level of capital k*, when ∆ k = 0 represents the long-run equilibrium in the economy
n-population growth rate; d- depreciation; k-capital per worker; y- output/income per worker; L- labor force; s - saving rate (allocation of output between consumption and investment)
Saving per worker is now greater than population growth plus depreciation, so capital accumulation increases, shifting the steady state from point A to B. As can be seen on the graph, output per worker moves from y0 to y1. Initially the economy expands faster, but eventually goes back to the steady state rate of growth
the population growth rate has now increased from n to n1, this introduces a new capital widening line (n1 + d)
This model assumes that countries use their resources efficiently and that there are diminishing returns to capital and labor increases. The role of technological change became crucial, even more important than the accumulation of capital.
Predictions: 1) Increasing capital relative to labor creates economic growth, since people can be more productive given more capital. 2) Poor countries with less capital per person will grow faster because each investment in capital will produce a higher return than rich countries. 3) Economies will eventually reach a point at which no new increase in capital will create economic growth ("steady state“)
Креатив: Japan and Germany are two success stories of economic growth.Although today they are economic superpowers, in 1945 the economies of both countries were in shambles. World War II had destroyed much of their capital stocks. In the decades after the war, however, these two countries experienced some of the most rapid growth rates on record. Are the postwar experiences of Japan and Germany so surprising from the standpoint of the Solow growth model? Consider an economy in steady state. Now suppose that a war destroys some of the capital stock. (That is, suppose the capital stock drops from k* to k1 in Figure 7-4.) Not surprisingly, the level of output immediately falls. But if the saving ratethe fraction of output devoted to saving and investmentis unchanged, the economy will then experience a period of high growth. Output grows because, at the lower capital stock, more capital is added by investment than is removed by depreciation.This high growth continues until the economy approaches its former steady state. Hence, although destroying part of the capital stock immediately reduces output, it is followed by higher-than-normal growth. The “miracle of rapid growth in Japan and Germany, as it is often described in the business press, is what the Solow model predicts for countries in which war has greatly reduced the capital stock.
Kondratiev waves are described as sinusoidal-like cycles in the modern capitalist world economy. Averaging fifty and ranging from approximately forty to sixty years in length, the cycles consist of alternating periods between high sectoral growth and periods of relatively slow growth. Unlike the short-term business cycle, the long wave of this theory is not accepted by current mainstream economics. Kondratiev identified three phases in the cycle: expansion, stagnation, recession. More common today is the division into four periods with a turning point (collapse) between the first and second phases. The saturation of major markets or infrastructures (canals, railroads) creates an economic stagnation. However, stagnation does not always mean that markets are mature. Markets were temporarily oversupplied. The stagnation phase is characterized by a lack of good investment opportunities that leads to low interest rates. The long cycle supposedly affects all sectors of an economy, and concerns mainly output rather than prices (although Kondratieff had made observations focusing more on prices, inflation and interest rates). According to Kondratiev, the ascendant phase is characterized by an increase in prices and low interest rates, while the other phase consists of a decrease in prices and high interest rates.
Креатив размышления по поводу этой чудной теории: Long wave theory is not accepted by most academic economists, but it is one of the bases of innovation-based, development, and evolutionary economics, i.e. the main heterodox stream in economics. Among economists who accept it, there has been no universal agreement about the start and the end years of particular waves. This points to another criticism of the theory: that it amounts to seeing patterns in a mass of statistics that aren't really there. A problem with analyzing Kondratiev waves is that there is little data before 1870 and then it is only for a few countries in Europe and the U.S. Also, the early data consisted mostly of prices, trade statistics and limited information on industrial production. Pre 20th century depressions were periods of depressed prices and profits and not necessarily associated with high unemployment or falling industrial production. Falling prices typically increased real purchasing power and the standard of living was maintained or actually rose. With the end of the gold standard prices were no longer depressed with economic contractions and such periods were called recessions. To date the only true depression with very high unemployment for a number of years was the Great Depression of the early 1930s.
To understand fully the process of economic growth, we need to go beyond the Solow model and develop models that explain technological progress. Models that do this often go by the label endogenous growth theory, because they reject the Solow models assumption of exogenous technological change.
Y = AK production function
where Y is output, K is the capital stock, and A is a constant measuring the amount of output produced for each unit of capital. As before, we assume a fraction s of income is saved and invested. We therefore describe capital accumulation with an equation similar to those we used previously:
DK = sY -dK. This equation states that the change in the capital stock (dK) equals investment
(sY ) minus depreciation (sigmaK). Combining this equation with the Y = AK production function, we obtain, after a bit of manipulation DY/Y = DK/K = sA - sigma. This equation shows what determines the growth rate of output DY/Y. As long as sA > sigma, the economys income grows forever, even without the assumption of exogenous technological progress. Thus, a simple change in the production function can alter dramatically the predictions about economic growth. In the Solow model, saving leads to growth temporarily, but diminishing returns to capital eventually force the economy to approach a steady state in which growth depends only on exogenous technological progress. By contrast, in this endogenous growth model, saving and investment can lead to persistent growth.
Sustainable development (SD) is a pattern of economic growth in which resource use aims to meet human needs while preserving the environment so that these needs can be met not only in the present, but also for generations to come. The concept of sustainable development is often broken out into three constituent parts: environmental sustainability, economic sustainability and sociopolitical sustainability.Environmental sustainability is the process of making sure current processes of interaction with the environment are pursued with the idea of keeping the environment as pristine as naturally possible based on ideal-seeking behavior.An "unsustainable situation" occurs when natural capital (the sum total of nature's resources) is used up faster than it can be replenished. Economic sustainability efficient usage of constraint resources and use of eco-technologies, recycling. Social sustainability saving of cultural systems, distracting conflicts elimination, fair goods allocation.
Examples of sustainable development: 1)A sustainable city considers the natural environment in its design and aims to reduce the input of energy, water and other resources, as well as minimising the generation of waste and other environmental disturbances. Vitoria-Gasteiz in Spain is one example of a sustainable city or eco-city. It has implemented a policy of mixed land use and high density development along its major transport routes. An upgraded public transport system allows more residents to live there, while remaining green belts still provide habitat for wildlife and recreational areas for people. 2) Eco-industrial parks are areas where industries are placed together to co-operatively manage the use of resources and environmental impacts caused by their operations. By sharing resources they improve efficiency and create less waste. An eco-industrial park in Kalundborg, Denmark has a number of businesses that utilise the by-products of other manufacturers. The waste created by a power station in the park is used to make cement by another firm. Other businesses use heat generated by the power plant and cement factory for some of their processes.
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|50942.||Вирішення системи нелінійних рівнянь методом ітерацій||36.5 KB|
|Мета. Навчитися вирішувати систему нелінійних рівнянь методом ітерацій Устаткування: папір формату А4, ручка, олівець, ПЗ С ++.|
|50943.||Метод Ейлера вирішення задачі Коші||38 KB|
|Мета: Навчитися будувати розвязок задачі Коші по методу Ейлера. Скласти програму. Устаткування: папір формату А4, програмне забезпечення Borland С++, ПК.|
|50944.||Метод Рунге-Кутта вирішення задачі Коші. Складання алгоритму||42 KB|
|Мета. Навчитися вирішувати задачу Коші методом Рунге-Кутта; скласти алгоритм. Устаткування: папір формату А4, ПК, програмне забезпечення Borland С++. Хід роботи Правила техніки безпеки Теоретичні дані Індивідуальне завдання.|