Aggregate Demand is the total amount of goods demanded in the economy. Distinguishing among goods demanded for consumption(C), for investment (I), by the government (G), and as net exports(NX), aggregate demand (AD) is determined by

AD=E=C+I+G+NX (1)

Output is at its equilibrium level when the quantity of output produced is equal to the quantity demanded. Thus, an economy is at its equilibrium output when

Y=E=AD=C+I+G+NX (2)

The simple form of (2) and of the identities discussed later results from neglecting some complexities in the definitions of GDP and national income.

Depreciation is neglected, so we do not need to distinguish between GDP and Net-National Product. We also assume that GDP and national income are equivalent. This means we do not include items in the model that cause a discrepancy between the two totals (primarily indirect business taxes). A final assumption relates to the units in which each of the variables is measured. We assume that the aggregate price level is fixed.All variables are real variables and all changes are in real terms.

With national product also measuring national income, we can write

YC+S+T (3)

Equation 3 is an accounting definition or identity, stating that national income, all of which is assumed to be paid to households in return for factor services(wages, interest, rents, dividends) is either consumed (C), paid out in taxes (T), or saved(S).

In addition , from the fact that Y is national product, we can write

YC+I

_{r}+G (4)Equation (4) defines national product as equal to consumption plus realized investment plus government spending.

Realised investment is the total that appears in the national income accounts whether or not that investment was desired by the firms.

Using the definitions given in Equations 3 and 4, we rewrite the condition given for equilibrium income given in equation (2) in two lternative ways, which will help us understand the nature of equilibrium in the model.By 2, Y must equal C+I+G in equilibrium, and from 3 Y is defined as C+S+T; in requilibrium therefore,

C+S+TYC+I+G

Or equivalently

S+T=I+G (5)

From equations 2 and 4, we can see that in equilibrium,

C+I

_{r}+GYC+I+GOr,

By canceling terms

I

_{r}=I (6)There are then three equivalent ways to state the condition for equilibrium in the model:

Y=E=C+I+G (2)

S+T=I+G (5)

I

_{r}=I (6)To see how realized and inventory investment can differ, consider what happens when a level of output() is produced that exceeds AD(E=C+I+G).

In this case

YE

C+I

_{r}+GC+I+GWhere I

_{r}-I is the unintended inventory accumulation. The amount by which output exceeds Aggregate Demand(I_{r}-I) will be unsold output over and above the amount of inventory investment the firm desired. The excess is unintended inventory accumulation.In the reverse situation, in which AD exceeds Output, we have

C+I+GC+I

_{r}+G (8)II

_{r}Where I-I

_{r, }is the unintended inventory shortfall. Demand is greater than output and firms sell more than was planned.Inventories end up at less than the desired level.The equilibrium point(I= I_{r}) is a level of production that, after all sales are made, leaves inventory investment at just the level desired by the firms. As can be seen from equation (7) or (8), this is the level at which output is equal to Aggregate Demand and hence is equivalent to the other two ways of expressing the condition for equilibrium.The third way of expressing the condition for equilibrium in this model, shows clearly why there cannot be equilibrium at any other point.If at a given level of output, firms are accumulating undesired inventories or are seeing their inventories depleted, there is a tendency for output to change. I

If production exceeds demand, YE, firms are acumulatinghh unwanted inventories I

_{r}I, and hence there is a tendency for output to fall as firms cut production to reduce the level of inventories.If alternately, demand is outstripping production, EY, there is an inventory shortfall I_{r}<I and a tendency for output to rise as firms try to prevent further fall in inventories.Only when AD equals Output will firms be satisfied with their current level of output.There is neither an unintended inventory build up nor a shortfall and therefore no tendency for output to change. The situation is what is meant by equilibrium.

Graphical Determination of Equilibrium Income.

Income is measured along the horizontal axis, and the components of AD along the ve3rtical axis. The 45 line is drawn to split the positive quadrant of the graph. All points along this line indicate that aggregate expenditure equals aggregate output. The value of the variables measured on the vertical axis(C+I+G), is equal to the value of the variable on the horizontal axis, Y.C=a+bYD and Y=C+I+G, because the autonomous expenditure components I,G do not depend directly on income, the C+I+G schedule lies above the consumption function by a constant amount.

The equilibrium level of income is shown at the point where the C+I+G scheduler crosses the 45 line, and AD is therefore equal to income.

This intersection illustrates the equilibrium condition expressed in Equation 2

Y=E=C+I+G (2)

The understanding of the 45 line and of the properties of an equilibrium level of income is aided by considering why other points on the graph are not points of equilibrium.Consider a level of income below Y, for example Y

_{L}.A level of income equal to Y

_{L}generates consumption as shown along the consumption function.When this level of consumption I is added to the auitonomous expernditures I+G, aggregate demand exceeds income, the C+I+G schedule is above the 45 line. It follows that with demand outstripping production, desired investment will exceed actual investment at points such as Y_{L,}C+I+G>Y C+I_{r}+G, therefore I> I_{r}.There will be an unintended inventory shortfall at such points below and therefore a tendency for output to rise. Conversely, at levels of income above , output will exceed demand the 45 line is above the C+I+G schedule, and unintended inventory investment wil bev taking place(Y C+I_{r}+G>C+I+G; therefore I_{r}>I), and there will be a tendency for output to fall.It is only at that output is equal to AD; there is no unintrended inventory shortfall or accumulation, and consequently no tendency for output to change.