keynasian model

Introduction

Macroeconomics seeks to understand fluctuations in output around its potential level, driven by booms and recessions. This chapter introduces a model that explains these fluctuations, focusing on the interaction between output and spending. Spending determines output and income, which, in turn, determine spending.

The Keynesian Model

The Keynesian model simplifies by assuming fixed prices and that firms are willing to sell any amount of output at those prices. This creates a flat aggregate supply curve. The chapter focuses on developing the aggregate demand schedule.

A key concept is that increases in autonomous spending, like government purchases, lead to further increases in aggregate demand due to feedback between spending and output. More complex models will introduce dynamic links and the effects of changing prices and interest rates.

Aggregate Demand and Equilibrium Output

Aggregate demand (AD) is the total demand for goods in the economy, comprising consumption (C), investment (I), government purchases (G), and net exports (NX):

AD=C+I+G+NXAD = C + I + G + NX

Equilibrium occurs when output equals aggregate demand:

Y=AD=C+I+G+NXY = AD = C + I + G + NX

Unplanned inventory investment (IU) arises when aggregate demand does not equal output:

IU=Y−ADIU = Y - AD

  • If IU > 0, output exceeds demand, leading firms to cut production.

  • If output is less than demand, inventories are drawn down, and firms increase production.

Visual Representation:

  • Aggregate Demand Curve: A graph plotting total spending versus income (output). It slopes upward because higher income leads to higher consumption.

  • Equilibrium Point: The intersection of the AD curve and the 45-degree line, representing where output equals aggregate demand. Any deviation from this point results in unplanned inventory changes, prompting firms to adjust production.

The Consumption Function and Aggregate Demand

Consumption demand is a key determinant of aggregate demand. The analysis simplifies by initially omitting government and foreign trade (G=0G = 0 and NX=0NX = 0).

Consumption increases with income. This relationship is described by the consumption function:

C=C‾+cYC = \overline{C} + cY

where \overline{C} > 0 and 0 < c < 1

  • C‾\overline{C} is autonomous consumption (consumption when income is zero).

  • cc is the marginal propensity to consume (MPC).

Consumption and Saving

Income not spent on consumption is saved. Saving (S) is defined as:

S=Y−CS = Y - C

Substituting the consumption function into the budget constraint yields the savings function:

S=Y−C=Y−C‾−cY=−C‾+(1−c)YS = Y - C = Y - \overline{C} - cY = -\overline{C} + (1 - c)Y

Saving is an increasing function of income, with the marginal propensity to save (MPS) being s=1−cs = 1 - c. For example, if c=0.9c = 0.9, then s=0.1s = 0.1, meaning 10 cents of each extra dollar of income is saved.

Graphical Illustration:

  • Consumption Function: A line showing how consumption (C) increases with income (Y). The vertical intercept is autonomous consumption C‾\overline{C}, and the slope is the MPC (c).

  • Saving Function: A line showing how saving (S) increases with income (Y). The vertical intercept is negative autonomous consumption \/overline{C}, and the slope is the MPS (s).

Consumption, Aggregate Demand, and Autonomous Spending

Investment, government spending and taxes, and foreign trade are added to the model, initially assumed to be autonomous (independent of income). Consumption now depends on disposable income:

YD=Y−TA+TRYD = Y - TA + TR

C=C‾+cYD=C‾+c(Y+TR−TA)C = \overline{C} + cYD = \overline{C} + c(Y + TR - TA)

Aggregate demand is the sum of consumption, investment, government spending, and net exports:

AD=C+I+G+NX=C‾+c(Y−TA+TR)+I+G+NX=[C‾−c(TA−TR)+I+G+NX]+cY=A+cYAD = C + I + G + NX = \overline{C} + c(Y - TA + TR) + I + G + NX = [\overline{C} - c(TA - TR) + I + G + NX] + cY = A + cY

Here, A=C‾−c(TA−TR)+I+G+NXA = \overline{C} - c(TA - TR) + I + G + NX is autonomous spending. Aggregate demand increases with income because consumption increases with income. The aggregate demand schedule is obtained by adding the demands for consumption, investment, government spending, and net exports at each income level.

Equilibrium Income and Output

Equilibrium income occurs where aggregate demand equals output. This is where the AD curve intersects the 45° line (AD=YAD = Y). At any income level below equilibrium, demand exceeds output, inventories decline, and firms increase production. Above equilibrium, inventories pile up, and firms cut production.

The formula for equilibrium output is derived from the equilibrium condition Y=ADY = AD:

Y=A+cYY = A + cY

Y0=11−cAY_0 = \frac{1}{1 - c}A

Thus, equilibrium output is higher with a larger MPC and a higher level of autonomous spending. The change in output related to changes in autonomous spending is:

ΔY=11−cΔA\Delta Y = \frac{1}{1 - c} \Delta A

Graphical Representation:

  • AD-Y Diagram: The AD curve intersects the 45-degree line at the equilibrium output level. An increase in autonomous spending shifts the AD curve upward, resulting in a higher equilibrium output.

Saving and Investment

Without government and foreign trade, equilibrium occurs where planned investment equals saving. With government and foreign trade, the relationship is:

I=S+(TA−TR−G)−NXI = S + (TA - TR - G) - NX

Investment equals private savings plus the government budget surplus minus net exports.

The Multiplier

The multiplier effect explains how a $1 increase in autonomous spending raises equilibrium income by more than $1. The initial increase in spending leads to increased output and income, which in turn cause further induced spending as consumption rises. This process continues in successive rounds, with each round generating smaller increases in spending.

The cumulative change in aggregate spending is:

ΔAD=ΔA+cΔA+c2ΔA+c3ΔA+…=ΔA(1+c+c2+c3+…)\Delta AD = \Delta A + c\Delta A + c^2\Delta A + c^3\Delta A + … = \Delta A(1 + c + c^2 + c^3 + …)

This geometric series simplifies to:

ΔAD=11−cΔA=ΔY0\Delta AD = \frac{1}{1 - c} \Delta A = \Delta Y_0

The multiplier, denoted by α, is the amount by which equilibrium output changes when autonomous aggregate demand increases by one unit:

α=11−c\alpha = \frac{1}{1 - c}

A larger MPC leads to a larger multiplier. The multiplier effect explains why output fluctuates when autonomous spending changes.

The Multiplier in Practice

The multiplier is the formal way of describing a commonsense idea: If the economy experiences a shock that reduces income, people whose incomes have gone down will spend less, thereby driving equilibrium income down even further. One empirical estimate of the multiplier, by Robert Hall, is around 1.7.

Key Points:

  • An increase in autonomous spending raises the equilibrium level of income.

  • The increase in income is a multiple of the increase in autonomous spending.

  • The larger the marginal propensity to consume, the larger the multiplier.

The Government Sector

The government affects equilibrium income through government purchases (G) and taxes and transfers, which influence disposable income (YD). Disposable income is income plus transfers minus taxes, Y+TR−TAY + TR - TA. The fiscal policy of the government includes government purchases, transfers, and the tax structure.

Fiscal policy is defined with:

G=G‾G = \overline{G}, TR=TR‾TR = \overline{TR}, and TA=tYTA = tY

Where tt is the tax rate. Consequently, the consumption function is:

C=C‾+c(Y+TR‾−tY)=C‾+cTR‾+c(1−t)YC = \overline{C} + c(Y + \overline{TR} - tY) = \overline{C} + c\overline{TR} + c(1 - t)Y

Income taxes lower consumption spending. The marginal propensity to consume out of income is now c(1−t)c(1 - t).

Combining the aggregate demand identity:

AD=C+I+G+NX=[C‾+cTR‾+c(1−t)Y]+I+G+NX=(C‾+cTR‾+I+G+NX)+c(1−t)Y=A+c(1−t)YAD = C + I + G + NX = [\overline{C} + c\overline{TR} + c(1 - t)Y] + I + G + NX = (\overline{C} + c\overline{TR} + I + G + NX) + c(1 - t)Y = A + c(1 - t)Y

Where A=C‾+cTR‾+I+G+NXA = \overline{C} + c\overline{TR} + I + G + NX. The AD schedule is flatter because of income taxes.

Equilibrium Income with Government

The equilibrium condition for the goods market, Y=ADY = AD, is used to include government:

Y=A+c(1−t)YY = A + c(1 - t)Y

Y0=11−c(1−t)(C‾+cTR‾+I+G+NX)=A1−c(1−t)Y_0 = \frac{1}{1 - c(1 - t)}(\overline{C} + c\overline{TR} + I + G + NX) = \frac{A}{1 - c(1 - t)}

The government sector raises autonomous spending and lowers the multiplier.

Income Taxes and the Multiplier

Income taxes lower the multiplier because they reduce the induced increase of consumption out of changes in income.

Income Taxes as Automatic Stabilizers

A proportional income tax is an automatic stabilizer, which reduces the amount by which output changes in response to a change in autonomous demand, without intervention.

Effects of a Change in Fiscal Policy

An increase in government purchases shifts the aggregate demand schedule upward. The change in equilibrium income is:

ΔY0=ΔG+c(1−t)ΔY0\Delta Y0 = \Delta G + c(1 - t)\Delta Y0

ΔY0=11−c(1−t)ΔG=αGΔG\Delta Y0 = \frac{1}{1 - c(1 - t)} \Delta G = \alpha_G \Delta G

Where αG=11−c(1−t)\alpha_G = \frac{1}{1 - c(1 - t)}

Thus, a $1 increase in government purchases leads to an increase in income in excess of a dollar.

Implications

Changes in government spending and taxes affect the level of income, so fiscal policy can be used to stabilize the economy. When the economy is in a recession, taxes should be cut or spending increased to get output to rise. And when the economy is booming, taxes should be increased or government spending cut to get back down to full employment.

The Budget

The budget surplus (BS) is the excess of government revenues (taxes) over total expenditures (purchases and transfers):

BS=TA−G−TRBS = TA - G - TR

Assuming a proportional income tax, TA=tYTA = tY:

BS=tY−G−TRBS = tY - G - TR

The budget deficit depends on government policy choices and the level of income. An increase in investment demand increases output, reducing the deficit. We should, accordingly, not be