$3.$ Preference Shock in a RBC Model $(30$ points$)$ Assume in a two$-$period real inter$-$temporal model, the utility function has the following separable form:

U$($C$,$ C$^\text{'},$ N$,$ N$^\text{'})=[$log $($C$)+\backslash$eta log $(1-$N$)]\_$Period $1+\backslash$beta $[$log$($C$^\text{'})+\backslash$eta $^$'log$(1-$N$^\text{'})]\_$Period $2$

where C$,$ C$^\text{'}$ are consumption and N$,$ N$^\text{'}$ are labor. Parameter $\backslash$eta $($or $\backslash$eta $^\text{'})$ indicates consumer's preference for leisure $($or dislike for labor$)$ while $\backslash$beta indicates the degree of impatience. Now, analyze the general equilibrium effects of a decrease in $\backslash$eta and an increase in $\backslash$beta in the real business cycle model. First, consider the effects of a decrease in $\backslash$eta $.$ a Explain how it changes the consumer's problem intuitively. How does that affect the labor supply N$^$s$.(5$ points$)$ b How does that change affect the good market $($supply and demand$).$ Using these results, can you determine the effects of the change on the current output Y$,$ employment N$,$ consumption C$,$ investment I, wage w and interest rate r$.$ Use graphs to illustrate. $(5$ points$)$ c Consider the effects of this change on the credit market. Suppose after $\backslash$eta decreases, consumers are smoothing consumption and increasing their savings. Determine the effect on interest rate r and investment I. Use graphs to illustrate. $(5$ points$)$ Second, consider the effects of an increase in $\backslash$beta $.$ d Explain how it changes consumer's problem intuitively. How does that affect the labor supply N$^$s and the consumption demand C$^$d$.$ Write the related first order conditions to support your argument. $(7$ points$)$ e How does the increase of $\backslash$beta influence the labor market and good market $($supply and demand$).$ Determine the effects of this change on all equilibrium quantities and prices as above $($N$,$ Y$,$ C$,$ I, r$,$ w$).$ Use appropriate graphs to illustrate. $(8$ points$)$