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In almost all NK models public government consumption (like coastal defences) or private government consumption (like primary and secondary education) is by default 'wasteful'. Government consumption as we measure it is however defined as production of goods and services by the government which are used/consumed by households). But it seems silly to assume that a privately paid nurse is, by definition, productive while a government paid nurse (working for instance at the NHS in the UK) has zero productivity. So, why is government consumption (again: government production which is consumed by households) not included in 'utility'? Same for government investments. Addendum: not including it means that institutional differences between countries (the NHS!) make it impossible to use NK models to compare countries.

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You can include government consumption in the utility function of the representative consumer, but why? When we add exogenous government purchase shocks to a macro model, we enquire about the dynamics of the agent's consumption and labour supply change. This concept is clear and easy to measure.

Let's assume that you are including government spending in an additively separable utility function. For example, the representative consumer is happy because of the government is investing in coastal defences. But he is unhappy about the state's Olympic games project. It's all OK, but how are you going to defend these assumptions empirically speaking? I think you should introduce a few agents plus. How do you describe the preferences of agents about the different government spending projects (defence forces, park building, etc.)? It's a normative assumption, what makes this research Economics no more. This research becomes Political Science. Unless you find or do a reliable research on the preference of agents about government spending projects. But keeping in mind, political preferences are very far-fetched from the so-called rationality. There is always an emotional factor, which makes a spending project likeable or not. How to make these "soft" questions into "hard" science? It's a very hard question.

Anyway, what is the purpose of including include government consumption in the utility function? I can imagine an OLG model with a social choice problem. E.g. children want playground, adults park, and elders hospital.

To answer your sub-question, government purchase is wasteful by default in Macroeconomics, unless this purchase don't effect the production function in a positively. In any other case, the welfare of the economy can be suboptimal. If government buys tablets for every pupils in schools, it may makes the workforce more productive in the long run.

You can measure the effectiveness of government spending considering the fiscal multipliers indicator. It's a stylized fact that the effectiveness of government spending depends on the countries structural features (income, openness, exchange rate regime, etc.).

See also: Enrique G. Mendoza, Carlos A. Vegh , Ethan Ilzetzki - How big are fiscal multipliers? New evidence from new data (01 October 2009)

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