National Income and Output Determination

Keynesian cross and consumption function

The Keynesian cross is a device in which planned expenditure E = C + I + G is set against income Y, and equilibrium occurs where output equals planned expenditure, Y = E. The core of the model is the linear consumption function C = \alpha + \beta Y, where \alpha > 0 is autonomous consumption, the part of spending that does not depend on income, and \beta is the marginal propensity to consume (MPC) with 0 < \beta < 1. The MPC is \Delta C / \Delta Y, and it is constant in the linear form.

The average propensity to consume (APC) is consumption divided by income, APC = C/Y. Dividing the linear function by Y gives APC = \alpha/Y + \beta. The MPC stays fixed, while autonomous consumption is spread over a larger income, so APC falls as income rises whenever \alpha > 0. APC is therefore not constant and does not rise with income.

Because each unit of income is either consumed or saved, APC + APS = 1 and MPC + MPS = 1, where APS = S/Y and MPS = \Delta S/\Delta Y.

Saving is S = Y - C, so a change in income changes saving by \Delta S = (1 - MPC)\,\Delta Y. If the MPC is high, most of any income rise is spent and only a small part adds to saving.

With no government and no taxes, disposable income equals output, so Y_d = Y, and equilibrium requires Y = C + I. For C = a + bY and a given investment I, the equilibrium is Y^* = (a + I)/(1 - b). The simple expenditure multiplier is k = 1/(1 - MPC). For example, an MPC of 0.75 gives k = 4.

Since \Delta Y / \Delta G = 1/(1 - MPC) in this model, the MPC can be recovered from the ratio of a change in equilibrium income to the change in government purchases, MPC = 1 - \Delta G/\Delta Y. Two equilibria with different government purchases can be used to solve for it.

IS-LM model

The IS curve gives all combinations of output and the nominal interest rate i at which the goods market clears, Y = C(Y) + I(z) + \bar{G}, where z = i - \pi^e is the real interest rate and \pi^e is expected inflation. Investment falls as z rises, so I' < 0. At a given nominal rate, a higher expected inflation lowers z, raises investment, and shifts the IS curve to the right.

The LM curve gives the combinations at which money demand equals real money supply, \bar{M}/\bar{P} = kY - li. Expected inflation does not appear in this relation, so the LM curve does not move when expectations change. In the short run, a rightward shift of IS with LM fixed raises equilibrium output.

To find the slope of IS, write the goods market with investment I = a - br and saving S = sY, where s is the marginal propensity to save. Equilibrium gives sY = a - br + \bar{G} (in a closed economy with no taxes), so the IS curve in the (Y, r) plane has slope dr/dY = -s/b. Its steepness is s/b. The IS curve is steeper when investment is less responsive to the interest rate (small b) and when the MPS is large. A very high interest elasticity of investment makes it flatter. The interest elasticity of money demand does not affect IS; it determines the slope of LM.

Equilibrium output and interest rate are found by solving the IS and LM equations together. Express r from one equation and substitute into the other. For instance, if the LM equation gives r in terms of Y and the real money supply, substituting this into the IS equation yields a single linear equation in Y. A change in government spending then shifts the solution through the IS curve.

With T = T_0 + tY, C = C_0 + c(Y - T), I = I_0 - br, and LM given by r = (l_1 Y - M^S/P)/l_2, eliminating r gives Y\left[1 - c(1-t) + \frac{l_1 b}{l_2}\right] = C_0 - cT_0 + I_0 + \frac{b}{l_2}\frac{M^S}{P}. The investment multiplier is therefore \frac{\partial Y}{\partial I_0} = \frac{1}{[1 - c(1-t)] + \frac{l_1 b}{l_2}}. The term \frac{l_1 b}{l_2} reflects the interest-rate feedback through LM, so the multiplier is smaller than the goods-market multiplier 1/[1 - c(1-t)].