A typical application of the implicit-function theorem is a gcneral-functional form of the IS-LM model.1 Equilibrium in this macroeconomic model is characteri7ed by an income level and interest rates that simultaneously produce equilibrium in both the goods market and the money market.
A goods market is described by the following set of equations:
Y is the level of gross domestic product (GDP), or national income. In this form of the model, Y can also be thought of as aggregate supply. C, /, G, and T are consumption, investment, government spending, and taxes, respectively.
where
f IS stands for "investment equals savings" and LM stands for "liquidity preference equals money supply."
1. Consumption is assumed to be a strictly increasing function of disposable income (Y - T). If we denote disposable income as Yd = Y - T. then the consumption function can be expressed as
where dCjdY4 = C'(Yd) is the marginal propensity lo consume (0 < C(Yd) < 1).
2. Investment spending is assumed to be a strictly decreasing function of the rate of interest, r:
3. The public sector is described by two variables; government spending (G) and taxes (T). Typically, government spending is assumed to be exogenous (set by policy) whereas taxes dT
are assumed to be an increasing function of income. — = 7 (7) is the marginal tax rate(0 < T!(Y) < 1). dY
If wc Substitute the functions for C, G into the first equation Y = C -f / -f <7, we get
which gives us a single equation with two endogenous variables: Fand r. This equation gives us all the combinations of Y and r that produce equilibrium in the goods market. This equation implicitly defines the IS curve.
If we rewrite the IS equation, which is in the nature of an equilibrium identity, then the total differential with respect to Tand r is dY - C'(Yd)[] - T'( Y)] dY - f(r) dr = 0 dYd
We can rearrange the t/Fand dr terms to get an expression for the slope of the IS curve:
Given the restrictions placed on the derivatives of C, L and L we can easily verify that the slope of the IS curve is negative.
The money market can be described by the following three equations:
Md — L{Y,r) [moneydemand] where Ly > 0 and Lr < 0
where the money supply is assumed to be exogenously determined by the central monetary authority, and
Md = M* [equilibrium condition]
Substituting the first two equations into the third, we get an expression that implicitly defines the LM curve, which is again in the nature of an equilibrium identity.
Slope of the LM Curve
Sincc this is an equilibrium identity, we can take the total differential with respect lo the two endogenous variables, Kand r.
LydY + Lr dr = 0 which can be rearranged to give us an expression for the slope of the LM curve dY Lr
Since Ly> 0 and Lr < 0, we can determine that the slope of the LM curve is positive.
The simultaneous macroeconomic equilibrium state of both (he goods and money markets can be described by the following system of equations:
which implicitly define the two endogenous variables, y and r, as functions of the exogenous variables, Gtq and M^ Taking the total differential of ihe system, we get dY - C{Yd)[[ - T\Y) 1 dY - /'(r) dr = dG{}
or. in matrix form,
"i -a - m)] |
~ï(r)' |
'dY' |
dGtj | ||
Ly |
Lr |
dr J |
_<tM*0_ |
The Jacobiao determinant is
Since |./| ^ 0, this system satisfies the conditions of the implicit-function theorem and the implicit functions and r = r(Go, A/5)
can be written even though wc arc unable to solve for Y* and r* explicitly, Even though we cannot solve for Y* and r* explicitly, we can perform comparative-static exercises to determine the effects of a change of one of the exogenous variables ((7(li M^j on the equilibrium values of Y* and r*. Consider the comparative-static derivatives dY*/3G{) and dr* jdG\) which we shall derive by applying the implicit-function theorem to our system ol total differentials in matrix form
-C'(Yd)[\-T\Y)] —/V) |
dY |
dGi} | ||
Lr Lr |
_drm |
First we set = 0 and divide both sides by (IGq.
~ dY* | ||||
1 _ C" ■ (1 - T') -r'(r)~ |
dGt) |
"1" | ||
Ly Lr |
dr* |
0 | ||
L^gJ |
Using Cramer's rule, we obtain dT
0 Lr dGt
From the implic it-function theorem, these ratios of differentials, dY*fdG$ and dr~ f d Go. can be interpreted as partial derivatives,
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