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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.

A H
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  • If $Y$ is truly a function of $3$ variables, then it is not at all clear to me what is meant by "the rate of growth of $Y$." – Gerry Myerson Apr 26 '13 at 13:37
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    For some basic information about writing math at this site see e.g. here, here, here and here. – Kasper Apr 26 '13 at 13:39
  • It is an economic growth model, where growth in one period is determined by the stocks in the last period. So Y in period 2 is determind by L,B,K and g in period 1. That is what I mean by rate of growth of Y. – A H Apr 26 '13 at 13:47
  • I should say that g in period 2 is also determined by period 1 L, B and K.

    In simulations the model definitly converges to to a steady state where the change in Y is equal to g, (e.g. (Y2-Y1)/Y1 = g). And I have the equations for Y and g. I just can't figure out the exact form of the steady state.

     – A H Apr 26 '13 at 13:56

2 Answers2

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

AnilB
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  • Sorry for being ignorant but I am not sure how to calcualte a time differential like dL/dt.

    Would it just be the rate of change of L over one time period?

     – A H Apr 26 '13 at 19:38
  • Yes exactly, it is the rate of change – AnilB Apr 26 '13 at 20:14
  • $dL/dt$ is the instantaneous rate of change. When you write, "rate of change of $L$ over one time period," I think you are talking about $L(t+1)-L(t)$, which is not $dL/dt$. – Gerry Myerson Apr 27 '13 at 04:40
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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.

Trurl
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