Help with Log-linearization

General DSGE Modeling
1

Dear All,

I need to do an implementation in my model but I need some help with a log-linearization (i.e. log-deviation of the variables from SS as in Uhlig method’s (1995)) of a utility function in CRRA version.

Precisely, having a function of this kind:
U_{i,t}=\left[q_{1}^{i} \frac{C_{i,t}^{1-\eta_{i}}}{1-\eta_{i}}-\frac{N_{i,t}^{1+\theta_{i}}}{1+\theta_{i}}+(1-q_{1}^{i})(G_{i,t})\right]
how it would be its log-linear version? Trying to find some similarities in Uhlig (1995) but I’m finding some difficulties in doing this.

Thank you in advance for your help
Best

2

Your equation is:
U_{i,t}=\left[q_{1}^{i} \frac{C_{i,t}^{1-\eta_{i}}}{1-\eta_{i}}-\frac{N_{i,t}^{1+\theta_{i}}}{1+\theta_{i}}+(1-q_{1}^{i})(G_{i,t})\right]

Doing Uhlig’s substitution X_t = Xe^{\tilde{X}_t}, where \tilde{X}_t =\ln{X_t/X} (note that dropping t subscript means steady state), yield:

U_{i,t}=\left[q_{1}^{i} \frac{(C_{i}e^{\tilde{C}_{i,t}})^{1-\eta_{i}}}{1-\eta_{i}}-\frac{(N_{i}e^{\tilde{N}_{i,t}})^{1+\theta_{i}}}{1+\theta_{i}}+(1-q_{1}^{i})(G_{i}e^{\tilde{G}_{i,t}})\right]

U_{i,t}=\left[q_{1}^{i} \frac{C_{i}^{1-\eta_{i}}e^{(1-\eta_i)\tilde{C}_{i,t}}}{1-\eta_{i}}-\frac{N_{i}^{1+\theta_{i}}e^{(1+\theta_{i})\tilde{N}_{i,t}}}{1+\theta_{i}}+(1-q_{1}^{i})(G_{i}e^{\tilde{G}_{i,t}})\right]

From this point we perform the approximation by acknowledging that, up to first order: e^{a\tilde{X}_t}\approx 1+a\tilde{X}_t. Replacing this approximation gives:

U_{i,t}=\left[q_{1}^{i} \frac{C_{i}^{1-\eta_{i}}(1+(1-\eta_i)\tilde{C}_{i,t})}{1-\eta_{i}}-\frac{N_{i}^{1+\theta_{i}}(1+(1+\theta_{i})\tilde{N}_{i,t})}{1+\theta_{i}}+(1-q_{1}^{i})(G_{i}(1+\tilde{G}_{i,t}))\right]

(more precisely
U_{i,t}\approx\left[q_{1}^{i} \frac{C_{i}^{1-\eta_{i}}(1+(1-\eta_i)\tilde{C}_{i,t})}{1-\eta_{i}}-\frac{N_{i}^{1+\theta_{i}}(1+(1+\theta_{i})\tilde{N}_{i,t})}{1+\theta_{i}}+(1-q_{1}^{i})(G_{i}(1+\tilde{G}_{i,t}))\right]
)

From here the expression is already log-linearized, clearly you can do some algebra to simplify things out and express it more cleanly.

3

Thank you very much for your answers @dsge_modeling
From your last expression, doing some algebra the final version, to implement eventually in the code, it would be in this way?
U_{i,t}\approx\left[q_{1}^{i} \frac{(1-\eta_{i})C_{i}\tilde{C}_{i,t}}{1-\eta_{i}}-\frac{(1+\theta_{i})N_{i}\tilde{N}_{i,t}}{1+\theta_{i}}+(1-q_{1}^{i})(G_{i}\tilde{G}_{i,t})\right]
Sorry if there are mistakes but the presence of the exponential and the parameter at the denominator confuse me a lot