Question

1a. Derive government spending and tax multiplier in the Keynesian-cross model using calculus

1b. Consider the model of Keynesian cross with fixed planned investment expenditure, government spending and taxes. Assume that consumption function is given by C=a+mpc*(Y-T), where the parameter a >0 is called autonomous consumption, and the marginal propensity to consume satisfies 0< mpc <1. Compute equilibrium output (income) as a function of parameters (a and mpc) and exogenous variables. How does equilibrium output depend on a? mpc? Government spending? Taxes?

1c. Suppose that the real interest rate is 2% and expected inflation rate is 4%. What does Fisher equation predict nominal interest rate t

Answer 1

Answer.The consumer, firm and government demands can be represented graphically. Although
several graphical presentations are possible, Figure 1 below is the most common. It is known as the
"Keynesian cross" because of the upward sloping AD and 45o
lines. It is important to know that the
intersection of these two lines yields the equilibrium level of national income, but it is much more
important to know why.
In Figure 1, the slope of line C (the consumption function) is constant and less than one
reflecting the assumed properties of the MPC and the tax rate. The lines I and G reflect the
exogenously-determined investment and government demands for output. AD is a simple sum of
the three demands for each given level of income, Y. Note that (1) AD is parallel to C because
the other two demands have a zero slope and that (2) the distance between AD and C equals the
sum of I and G. The slopes tell us how the components of demand change as income changes. A
zero slope for I means that investment spending, for example, does not change as income changes.
The fact that the slope of the consumption function is less than one means that a one-dollar increase
in national income leads to less than a one dollar increase in consumption.
I
G Income ($)
Output and the
Components of
Agg. Demand ($) 45o
C
AD = C + I + G
AD
GDPa
AD
GDPa
2
2
1
1
GDP AD a
e= e
Y Y Y 2 e 1
Figure 1: Equilibrium Income and Output via the Keynesian Cross
A common source of confusion concerns the units and variables on the axes. The
y-axis is measured in dollars per year. C, I, G, AD, and actual GDP (GDPa
) can all be read off the
y-axis. The x-axis is also measured in dollars per year. Actual GDP can be easily measured on the
y-axis by use of the 45o
line (where it can be compared to AD). Because of the first assumption,
(see page 2) actual GDP equals Y. As an example of how to read the figure, when national income
is Y2, investment equals I0, government spending equals G0, actual output (GDPa
2 ) equals national
income (Y2). Aggregate Demand equals AD2. The vertical distance above the C line up to the AD
line is equal to the sum of I0 and G0. Because C and AD are parallel, this vertical distance is
always the same.
Equilibrium occurs at Ye because this is the only level of output at which there is no
tendency to change. The 45o
degree line translates actual output on the horizontal axis to the
vertical axis. The 45o
line allows us to compare a given level of actual GDP and AD on the
vertical axis. If they are not equal, we know from the discussion above that firms will adjust