The Product Market

Assume the economy produces only one commodity, y, which is used either for consumption or investment.

Equilibrium condition	y = c + i + g		(1)
Consumption function	c = c0 + cyyd		(2)
Investment behavior	i = i0 + irr		(3)
Disposable income	yd = y - t		(4)
Tax function	t = t0 + tyy		(5)

where
y = real income/output
yd = real income after tax
c = real private consumption
i = real private investment
r = rate of interest
t = tax
and
0 < c_y < 1
0 < t_y < 1
i_r < 0

Substituting (2), (3), (4) and (5) into (1) yields

y =

c0 + cy[y - (t0 + tyy)] + i0 + irr + g

(6)

Rearranging the result, one gets

	1 - cy(1 - ty)		1
r =	------------- y	-	-----	[c0 + i0 + g - cyt0]	(7)
	ir		ir

Equation (7) expresses the relationship between r and y that results in equilibrium in the product market given values of autonomous expenditure (c₀, i₀, g and t₀), and is known as the IS curve.

The slope of the IS curve is [1 - c_y(1 - t_y)] / i_r and takes a negative value (the curve slopes downward from left to right), as 0 < c_y < 1 and 0 < t_y < 1.

The intercept of the IS curve is - [c₀ + i₀ + g - c_yt0] / i_r, which is a positive value, as i_r < 0 and 0 < c_y < 1.

An increase in autonomous expenditure (c₀, i₀, and g) will shift IS curve to the right while an increase in t₀ will shift the curve to the left, as - 1 / i_r > 0.

The Money Market

Money Market Equilibrium	Md	=	Ms	(8)
Demand for Money	Md / P	=	L0 + Lyy + Lrr	(9)
Supply of Money	Ms	=	lB	(10)

Where
M^d = Nominal money demand
M^s = Nominal money supply
P = Price level or implicit price deflator
B = Monetary base
l = Money multiplier

Substituting (9) and (10) into (8), one obtains

L0 + Lyy + Lrr

=

lB/P

(11)

Or

r =

( / Lr)B/P - L0 / Lr-(Ly / Lr)y

(12)

Equation (11) expresses the relationship of r and y that results in equilibrium in the money market, given values of l, B, P and L₀, and is known as the LM curve.

The slope of the LM curve is - L_y / L_r and is positive (ie the curve slopes upward from left to right), as L_y > 0 L_r < 0

And its intercept is (l / L_r)B/P - L₀ / Lr An increase in monetary base, or a reduction in the price level will shift the LM curve to the right as l / PL_r < 0.

Equilibrium in the Product and Money Market

From (7), the equilibrium in the product market is

    r = (1 - c_y(1 - t_y))y / i_r - (c₀ + i₀ + g - c_yt₀) / i_r

From (12), the equilibrium in the money market is

    r = (l/L_r)B/P - (L₀/L_r) - (L_y / L_r)y

Solving these two equations for y and r, one obtains

	ir[lB/PLr - L0/Lr + (c0 + i0 + g - cyt0)/ir]
Y =	---------------------------------------------------	(13)
	1 - cy(1 - ty) + irLy/Lr

	[1 - cy(1 - ty)]lB/PLy - L0/Ly - (c0 + i0 + g - cyt0)
r =	---------------------------------------------------	(14)
	[1 - cy(1 - ty)]Lr/Ly + ir

which is the income and interest rate determination equation respectively. Here, both income and interest rate are determined by real money stock (lB/P), and real autonomous expenditure (c₀, i₀, g and t₀).

An increase in government expenditure (or other autonomous expenditure) would lead to an increase in income and interest rate as

    Dy / Dg = 1 / [1 - c_y(1 - t_y) + (i_rL_y / L_r)] > 0
    Dr / Dg = -1 / [{1 - c_y(1 - t_y)}(L_r/L_y) + i_r] > 0

An increase in monetary base will lead to an increase in income and a fall in the rate of interest as

    Dy / DB = i_rl / [1 - c_y(1 - t_y/) + (i_rL_y / L_r)]L_rP > 0
    Dr / DB = l / [{i_rL_y / (1 - c_y(1 - t_y))} + L_r]P < 0