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Why do many papers write the intertemporal maximisation problem for the household as

\begin{equation}E_{0} \sum_{t=0}^{\infty} \beta^{t} U\left(C_{t}, N_{t}\right)\end{equation}

and then do not write the actual utility function that could be something like \begin{equation}U\left(C_{t}, N_{t}\right)=\frac{C_{t}^{1-\sigma}}{1-\sigma}-\frac{N_{t}^{1+\varphi}}{1+\varphi}? \end{equation}

Many papers state that they are following woodford (2003) for the microfundations. However also in that book there is no explicit utility function.

How is it possible to get the Euler equation or the IS curve without the actual functon?

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This is done in order to make the solution more general. If you use specific utility function you prove your result only for that specific utility function and not for other ones.

You can still derive Euler equation. For example in a simple 2-period consumption model where agents maximize their expected utility

$$E[u]=E[\sum_{t=1}^T u(C)_t]$$

where $t=1,2$. Given this, the Euler equation would be simply:

$$u'(C_1)=u'(E[C_2])$$

you could insert some specific utility function into the problem. For example, you could use quadratic utility $E[u] = E[\sum_{t=1}^T C_t-\frac{a}{2} C^2_t]$ (as in Romer's Advanced Macroeconomic handbook) which would give you Euler equation $C_1=E_1[C_2]$ and continue using that but in the first case you prove your result for whole range of utility function whereas in the second case only for single 1 quadratic utility. In science we like to make results as general as possible so usually the first approach would be preferred.

In a typical IS-LM model you will actually want to substitute some utility function because for derivation of the model its important to know what optimal consumption $C^*$ is. However, you might not substitute it right away. You are completely vague about your citation of Woodford (2003) which could refer to multiple sources. For example, in his 2003 book Interest and Prices, on PP 145 where he builds New-Keynesian style macro model he actually makes explicit assumptions about consumption and in these macro models utility is usually kept simple such as $U=C-\frac{1}{\gamma}L^{\gamma}$ (See for example again above mentioned Romer's handbook).

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    $\begingroup$ Just to add to @1muflon1's answer: It is hard to come by any paper that doesnt impose any structure on the utility function. Even if the paper doesn't make the functional form explicit, it will make assumptions like continuity, risk-neutrality, quasi-linear in money, satisfying limit conditions etc. The idea is that any real valued function that satisfy the conditions mentioned in the paper would work as a legitimate candidate for a utility function. P.S. some assumptions like (positive MU) are often too standard to be explicitly mentioned. $\endgroup$ – Tomcat Jul 7 at 14:22
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    $\begingroup$ @Tomcat yes that’s completely correct and excellent addition. By the way, that’s why I mentioned in my answer that it will be valid for a range of utility functions instead of all but I should have been more explicit. $\endgroup$ – 1muflon1 Jul 7 at 14:24
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Because authors are lazy.

Also, it does help make the result more general, but it is worthwhile to explain the factors that make it less general.

In the example muflon gave the result is not general. Whether the utility function is quadratic or not has massive implications, it will literally give opposite results at extreme values. That is a poor way to make a function general by not giving it.

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