# The Aggregate Expenditure Model The aggregate expenditure model (also known as the Keynesian cross diagram) is a graph that compares the level of aggregate expenditures in an economy with that economy's real GDP.
The aggregate expenditure model is a visual representation of the relationship between aggregate expenditures and the real gross domestic product (real GDP), which is the total output of the economy adjusted for inflation. This relationship is generally shown by a simple graph, where aggregate expenditures is represented on the vertical axis and real GDP is represented on the horizontal axis. This graph is also known as a Keynesian cross diagram. On this graph, there are two observed components. One is the aggregate expenditure function, which shows how aggregate expenditures increase as real GDP increases. The aggregate expenditure function component looks like a line on the graph with a positive slope. The second component to the aggregate expenditure model is a 45-degree line that shows where aggregate expenditures (AE) equals real GDP. Graphically speaking, the coordinates to this line start at the origin (aggregate expenditures and real GDP both equal zero), and extend outward with a constant slope of 1 (this way, the coordinates on the line are always equal each other). The intersection of these two components, the aggregate expenditure function and the 45-degree line, is the equilibrium point or equilibrium GDP. The intersection of these two lines must be the equilibrium because the point of intersection is the definition of GDP (the value, or total expenditure, of all final goods and services produced within a country's borders in a given year). This equilibrium GDP is where $\text{AE}=\text{GDP}$.

#### The Keynesian Cross The Keynesian cross diagram shows that aggregate demand increases as output, measured as real GDP, increases. This demonstrates the principle of effective demand. The equilibrium is the point where aggregate expenditures equal real GDP, which is show when the aggregate expenditures line crosses the 45 degree line.
With regard to equilibrium GDP, the aggregate expenditures model shows that an economy will move toward equilibrium GDP over time, as changes to demand influence output. Recall that the equilibrium GDP is where aggregate expenditures equals real GDP. An economy can be above or below this equilibrium level, but it will have a natural tendency to move back to equilibrium. For example, when output is greater than planned aggregate expenditures, the economy is at a point on the aggregate expenditures curve to the right of the equilibrium point. This situation is known as an unplanned inventory accumulation, an excess of unsold goods caused by an unplanned event. Inventories of unsold products build up, which results in businesses reducing their orders of goods. This, in turn, reduces overall output, pushing the economy to the left on the model, down the AE curve, until it reaches the equilibrium GDP. Similarly, when the output is less than planned aggregate expenditures, the economy is at a point on the AE curve to the left of the equilibrium point. This situation is called a unplanned negative inventory, or inventory rundown, which is a shortage of stored materials or goods caused by increased demand. Inventories of unsold product begin to decrease, causing businesses to increase their orders of goods. This situation then increases overall output, pushing the economy to the right on the cross diagram, or up the AE curve. This again moves the economy toward the equilibrium GDP, as illustrated by the aggregate expenditures model. In these ways, the actions of businesses to bring their inventories to equilibrium drive the entire economy toward equilibrium as well. This graph demonstrates the relationship between expenditure and income, and shows that either side of the equilibrium point, inventories will either decrease or accumulate.
The equation for the aggregate expenditure function is $\text{AD}=\text{C}+\text{I}+\text{G}+(\text{X}-\text{M})$, where C is consumption, I is investment, G is government spending, and $(\text{X} - \text{M})$ is net exports (or exports minus imports). The aggregate expenditure function shows how aggregate expenditures increase as real GDP increases. For example, consider an economy (following figures are given in billions of dollars). On a graph, the 45-degree line is a set of points where AE is equal to real GDP, so AE is $Y$. Suppose C is $150 + 0.85(Y-\text{T})$. T is taxes, equal to $0.25Y$. Investment, or I, is 500. Government spending, G, is 850. Exports are 500, and imports are $0.1Y$, so net exports are $500 - 0.1Y$. Therefore:
\begin{aligned}Y&=150+0.85(Y-0.25Y)+500+850+(500-0.1 Y)\\ Y&=150+0.85 Y-0.2125 Y+500+850+500-0.1 Y\\ Y&=2\text{,}000+0.5375 Y\\ Y-0.5375 Y&=200\\0.4625 Y&=2\text{,}000\\Y&=2\text{,}000/0.4625\\Y&=4\text{,}324.324\end{aligned}
Investment, government spending, and net exports are not functions of real GDP in the current year and are expressed as constants in the aggregate expenditure function. For example, investment spending tends to be more forward-looking, based on future expectations and not dependent on current real GDP. Government spending is based on congressional decision-making and not dependent on real GDP. Exports are shaped by other nations' real GDP and demand for foreign goods. This is not to say that these three components do not fluctuate, but they are not a function of real GDP and are thus constant in the model. However, consumption is a function of real domestic production. The consumption function is generally expressed as the following linear equation:
$\text{C}=a+\text{MPC}(\text{Y}-\text{T})$
where $a$ is the baseline level of consumption ($y$-intercept), MPC is marginal propensity to consume, $Y$ is real GDP, and T is taxes. Consumption has a positive relationship with real GDP and increases at a rate consistent with MPC.

The intersection of the aggregate expenditure function and the 45° line (where $\text{AE}=\text{GDP}$) is the equilibrium point where no incentive exists to shift away from that outcome. The optimal equilibrium point is one that occurs when the economy is at full employment GDP. In this case, unemployment is low and there is no recession. If the equilibrium point occurs at a level of output lower than full employment GDP, a recessionary gap exists. If the equilibrium point occurs at a level of output above full-employment GDP, then an inflationary gap occurs.