I am trying to solve for the steady state (in the context of a DSGE economic model) and one of the equations is reffering to capital accumulation. Particularly:
$$K_t=(1 - \Delta)K_{t-1} + \left( 1 - \frac{k_i}2 \left( \frac{I_t}{I_{t-1}} - 1 \right)^2 \right)I_t$$
With
- $K_t ={}$ capital
- $d = {}$ depreciation
- $k_i = {}$ adjustments costs
- $I_t = {}$ Investment
I am substituting $I_t = \Delta K_t$ (a general calibration advise) and try to solve in respect for $K$'s steady state. I am concluding $K=K$ which means that there is an infinite number of solutions. Is this statement correct? For example can I use $K=1$ and try to solve the rest system of equation for the steady state. Is this approach correct?
You are approaching this incorrectly. In the steady state all variable should be constant. So you should impose that $K_t=K_{t+1}=K$ and $I_t=I_{t+1}=I$. If you plug this into your equation, you obtain $K=\delta I$. So the conclusion is that in the steady state, the investment just covers the depreciation of existing capital. But you still need to solve for $K$.
To solve for $K$ you have to consider the whole system of equilibrium conditions. This will typically be a non-linear system which will have to be solved numerically.