Macroeconomic model equation question?

Question

A standard macroeconomic model consists of the system of equations

(i) M = kPy + L(r)

(ii) S(y,r,g) = I(y,r)

Here, M, k, and P are positive constants, whereas, L, S, and I are differentiable functions.

a) By using the counting rule, explain why it is reasonable to assume that system, in general, defines y and r as differntiable functions of g.

b) Differentiate the system and find the expression for dy/dg and dr/dg.

Please help, I have an assignment coming up soon that is going to have similar questions to this, thanks.

Answer 1

Since, these two equations have three variables, g, y and r (i) M = kPy + L(r)

(ii) S(y,r,g) = I(y,r)

with M, P, k and constants and L, S and I are function notations, there will one equation short to solve for the three variables. The alternative is to solve for r and Y in terms of g. This means we treat y anbd r as functions of g.. The suolutions of  r and y will come as expressions where g is a variable.

Take equation (i) first, you get  y= [M-L(r)]/kP.= W(r), where W is a a function to denote the expression

Use this in the second equation, you get [M-L(r)]/kP

S([M-L(r)]/kP, r,g) - I([M-L(r)]/kP,r) = 0.

Now apply the implicit differentiation rule using the concept of partial differentials. In this facility, I am unable to type the symbols. For this refer,

http://en.wikipedia.org/wiki/Implicit_fu...

and,

http://library.thinkquest.org/C0110248/c...

and,

http://www.ugrad.math.ubc.ca/coursedoc/m...

and

http://metric2.ma.ic.ac.uk/differentiati...