Solving for steady state in macro model, probably just simple calc problem...

Question

I am building a macroeconomic model and I am having trouble calculating the steady state.

GDP in the model is determined by

Y(L,B,K) = x*L+y*B+z*g*K

where (x,y,z) are known constants, L is the stock of loans, B is the stock of bonds, K is the stock of capital and g is the growth rate of capital.

g is given by known function g(L,B,K)

The steady state of the model is reached when the rate of growth of Y is equal to g. I want to find the relation between L B K that can reach a steady state.

So I guess this is Y'(L,B,K) = g(L,B,K), but I am about 10 years away from my last calculus class, and I can't figure out the right way to fit the parital derivatives together, or if I should be trying a different way to solve the problem.

Any help would be appreciated.

Answer 1

I think you also need to specify what determines $B$ and $L$. Then the general solution is to set up a matrix Z=AZ, where Z is the vector $(Y,L,B,K)$, and $A$ is the coefficients in the equations, such as $x$ and $y$,though if $g$ is non-linear you may need to linearlize the equations. Also, check to see if there is a worked out answer in the classics, such as Dixit's Theory of Equilibrium Growth or Barro's Economic Growth. Hope this helps.

Answer 2

If you want to use the time differential of Y such as $\dot Y=g$ $$\frac{dY}{dt}=x*\frac{dL}{dt}+y*\frac{dB}{dt}+z\,K\bigg(\frac{\partial g}{\partial L}\frac{dL}{dt}+\frac{\partial g}{\partial B}\frac{dB}{dt}+\frac{\partial g}{\partial K}\frac{dK}{dt}\bigg)+z\,g\frac{dK}{dt}=g$$