Keynesian Cross and Planned Expenditure

Why Planned Expenditure Matters

In the short-run Keynesian Cross model, firms produce output and households, businesses, and government plan purchases of goods and services. Planned expenditure summarizes those intended purchases at each level of output. The central diagram compares planned expenditure with output: their intersection identifies the level at which the two are equal.

This is a deliberately simplified way to organize aggregate demand. It provides a clear starting point for analyzing expenditure and output without requiring a full account of prices, interest rates, or international transactions. The treatment here is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: In the simplified closed-economy model, planned expenditure, denoted $PE$, is the sum of planned consumption $C$, planned investment $I$, and government purchases $G$: $$ PE=C+I+G. $$ The model treats these quantities as measured over the same period and in the same units. Net exports are omitted in this closed-economy setup.

What the Components Include

Consumption is planned household spending on goods and services. Investment is planned spending on newly produced capital goods, such as equipment or structures, as well as inventory investment in the model’s accounting. The term does not mean the purchase of an existing financial asset. Government purchases are government spending on currently produced goods and services. A transfer payment is not itself a purchase of output, so it is not added directly as a separate component of $PE$.

The components can vary with economic conditions, and the model may describe their totals as functions of output. For this tutorial, a schedule or a set of observations is enough: it tells us the planned total at different levels of output. The detailed behavior of consumption is a separate topic.

It is important to distinguish planned expenditure from actual output. Output $Y$ is the value of goods and services produced. Planned expenditure is what economic decision-makers intend to buy. The model’s equilibrium condition is that these quantities are equal. At other output levels, planned expenditure can be greater or less than output.

The 45-Degree Line and Equilibrium

In a Keynesian Cross diagram, output $Y$ is on the horizontal axis and planned expenditure $PE$ is on the vertical axis. The 45-degree line consists of all points whose vertical and horizontal coordinates are equal: on that line, $PE=Y$. An expenditure schedule plots planned expenditure at each output level. Its intersection with the 45-degree line is therefore the output level at which planned expenditure equals output.

Equilibrium condition: A Keynesian Cross equilibrium output $Y^*$ satisfies $$ PE(Y^*)=Y^*. $$ Graphically, this is the intersection of the planned-expenditure schedule and the 45-degree line. It is an equality condition, not a claim that planned expenditure equals output at every output level.

The equality can also be seen algebraically. Suppose, for illustration, that the planned-expenditure schedule is a straight line:

$$ PE=A+sY, $$

where $A$ is the schedule’s intercept and $s$ is its slope. At equilibrium, the point on this schedule must also lie on the 45-degree line. Substituting the condition $PE=Y$ gives:

Y=A+sY \quad\Longrightarrow\quad (1-s)Y=A \quad\Longrightarrow\quad Y^*=\frac{A}{1-s}, \qquad s\ne 1.

This is simply the algebraic solution for the intersection of two lines. For a positive intercept and a slope between zero and one, the intersection is positive and unique. If a schedule is not linear, one instead finds where its plotted values meet the 45-degree line; the straight-line expression should not be applied to an arbitrary schedule.

Finding the Intersection in Practice

A reliable method is to assemble the planned components, express or estimate planned expenditure at the output levels of interest, and then impose the equality with output. When a problem gives two points and says the schedule is straight between them, first calculate its slope. Use that slope and one known point to write the schedule, then solve $PE=Y$. The intersection should be checked by substituting the resulting output back into the expenditure schedule.

Identify the model and units.
Confirm the period, units, and included expenditure components. In the simplified closed economy used here, add consumption, investment, and government purchases.

Construct the planned-expenditure schedule.
Use the given component values, table, graph, or equation. If interpolation is required, calculate the schedule’s slope from the supplied points.

Set expenditure equal to output.
Solve $PE(Y)=Y$, or locate the point where the schedule crosses the 45-degree line.

Check the result.
Substitute the candidate output into the expenditure schedule and verify that the resulting planned expenditure is the same number.

Worked Example: Assemble Components and Find Equilibrium

A hypothetical closed economy reports these planned expenditure components at two output levels, all in billions of dollars. At $Y=400$, planned consumption is 360, investment is 90, and government purchases are 100. At $Y=800$, the corresponding values are 520, 90, and 100. Assume total planned expenditure changes linearly between these observations. Find the equilibrium output.

Method. First add the components at each output level:

PE(400)=360+90+100=550, \qquad PE(800)=520+90+100=710.

The slope of the total expenditure schedule is the change in planned expenditure divided by the change in output:

s=\frac{710-550}{800-400} =\frac{160}{400} =0.4.

Using the point at $Y=400$, the straight schedule can be written as $PE=550+0.4(Y-400)$, or $PE=390+0.4Y$. Set this equal to output:

390+0.4Y=Y \quad\Longrightarrow\quad 390=0.6Y \quad\Longrightarrow\quad Y^*=650.

Check. At $Y=650$, planned expenditure is $390+0.4(650)=390+260=650$. Thus the schedule and the 45-degree line intersect at $Y=650$ billion dollars. The common mistake avoided here is treating the component total at one observed output as if it were itself the equilibrium.

Worked Example: Locate Equilibrium From Two Schedule Values

A hypothetical expenditure schedule has $PE=620$ when $Y=500$ and $PE=780$ when $Y=900$, with a straight schedule between those points. Find the equilibrium output and verify it.

Method. Find the slope:

s=\frac{780-620}{900-500} =\frac{160}{400} =0.4.

Using the first point, the schedule is $PE=620+0.4(Y-500)=420+0.4Y$. At the intersection, $PE=Y$:

420+0.4Y=Y \quad\Longrightarrow\quad 420=0.6Y \quad\Longrightarrow\quad Y^*=700.

Check. At $Y=700$, the schedule gives $PE=420+0.4(700)=420+280=700$. The equilibrium is $700$. Avoid confusing a point on the expenditure schedule with its intersection: $PE=620$ is the value at $Y=500$, not the equilibrium output.

Worked Example: Interpolate Before Testing Equality

A hypothetical table gives $PE=560$ at $Y=500$ and $PE=640$ at $Y=700$. Assume a straight schedule over this interval. Find the equilibrium and state whether the answer falls within the interval used for interpolation.

Method. The schedule’s slope is

s=\frac{640-560}{700-500} =\frac{80}{200} =0.4.

Starting from the first observation, $PE=560+0.4(Y-500)=360+0.4Y$. Set planned expenditure equal to output:

360+0.4Y=Y \quad\Longrightarrow\quad 360=0.6Y \quad\Longrightarrow\quad Y^*=600.

Check. At $Y=600$, $PE=360+0.4(600)=360+240=600$. Since $500\le600\le700$, the answer lies within the stated interval, so using the straight schedule between the two observations is consistent. The common mistake avoided here is solving for an intersection without checking whether the result is in the range where the assumed interpolation applies.

Interpreting the Result and Avoiding Traps

The equilibrium condition gives a precise way to compare output with planned expenditure. If $PE>Y$, planned purchases exceed current output at that point. If $PE<Y$, planned purchases are below output. If $PE=Y$, the point lies on the 45-degree line. These comparisons identify the position relative to equilibrium; they do not change the definition of equilibrium.

A frequent exam error is to calculate the correct slope and then mishandle the schedule’s starting value. For example, suppose a schedule has $PE=500$ at $Y=400$, and its slope is $0.4$. The schedule is $PE=500+0.4(Y-400)=340+0.4Y$. A wrong calculation might keep the 500 as an intercept and write $PE=500+0.4Y$, which changes the schedule. The correct equilibrium is found from $340+0.4Y=Y$, so $Y=566.67$ to two decimal places. Checking gives $PE=340+0.4(566.67)=566.668$, which rounds to $566.67$. The small difference before rounding is due to using the rounded output; the exact solution is $Y=1700/3$, at which planned expenditure equals output exactly.

Also keep the model’s accounting clear. Transfers are not directly added to $C+I+G$ as purchases of current output, although they may affect household income and consumption. A purchase of an existing asset is not investment in the national-income sense used here. If a problem includes net exports, it is no longer using the closed-economy formula stated above; follow the expenditure components specified in that problem rather than silently omitting or adding a term.

Key takeaway: Planned expenditure in the simplified closed-economy Keynesian Cross is $C+I+G$. Equilibrium is not merely a total of these components at a chosen output: it is the output $Y^*$ for which the planned-expenditure schedule gives $PE(Y^*)=Y^*$, the intersection with the 45-degree line.

Check Your Understanding

For each problem, use the stated model and verify the equality condition where possible.

A hypothetical economy has planned consumption of 390, investment of 125, and government purchases of 185 at output $Y=650$, with amounts in billions. Calculate planned expenditure and state whether it is above, below, or equal to output. (Answer: $PE=700$ billion; $PE$ is 50 billion above output.)
A straight planned-expenditure schedule is $PE=480+0.25Y$, with output and expenditure measured in the same units. Find the equilibrium output and check the expenditure value there. (Answer: $Y^*=640$, and $PE=640$.)
A schedule gives $PE=520$ at $Y=400$ and $PE=680$ at $Y=800$, and is straight between the observations. Calculate its slope, write the schedule in intercept form, and find its equilibrium. (Answer: slope $0.4$; $PE=360+0.4Y$; $Y^*=600$, where $PE=600$.)
Between $Y=300$ and $Y=700$, a hypothetical straight schedule rises from $PE=510$ to $PE=670$. Find the schedule and its intersection with the 45-degree line, then check that the intersection is in the given interval. (Answer: slope $0.4$; $PE=390+0.4Y$; $Y^*=650$, which is within the interval.)
A straight schedule has $PE=560$ at $Y=400$ and $PE=760$ at $Y=900$. Find the slope and equilibrium, then calculate planned expenditure at $Y=700$ and compare it with output. (Answer: slope $0.4$; $PE=400+0.4Y$; $Y^*=666.67$ to two decimals; at $Y=700$, $PE=680$, which is 20 below output.)

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