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I am trying to model a transition path of an economy exposed to a permanent shock to government spending on dynare via matlab.

We start with these model equations (standard dynare code format).

c(+1) = c* beta*(1+alpha*k^(alpha-1)-delta);
y = k(-1)^alpha;
k = (y-c)+(1-delta)*k(-1)-g;

Where alpha, beta are constants, g is exogenous (the shock), and c, y, k are consumption, production, and capital respectively.

We then want to change our utility function to

$$U = \sum_{t=0}^{\infty} \beta^t \frac{C_t^{1-\sigma}}{1-\sigma}$$

Note that the first equation in the model is basically an Euler Equation, so only that will change if the utility model changes.

I get that the derivative of modified $U$ at time $t$ is $b^t*C_t^{-\sigma}$, which when put into the Euler Equation makes the dynare model

c(+1)^(-sigma) = c^(-sigma)*beta*(1+alpha*k^(alpha-1)-delta);

But if I do this, then the steady state values of K and thus Y and C do not change because the two cs become 1 when divided out at steady state…which I feel is somehow wrong.

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