Would a Job Guarantee impact Cost of Living? What effect would a job guarantee, a method of achieving full employment often suggested by modern monetary theorists, have on the cost of living?

In the case of student loans offered by the government, all this has really done is push up the price of tertiary education (simplification for the purpose of example, I'm not really looking for discussion on this topic). Would a job guarantee inflate the cost of living? I suppose this question would also apply to universal basic income.
The answer depends on whether the provision of full employment is a public good that corrects for a negative externality (poverty). Suppose it is and we wish to create full employment.

First, we should consider what makes government provision of jobs different from policies like unemployment insurance? Instead of hedging against business cycles causing cyclical unemployment, we are now trying to effectively eliminate it. Then, we should consider if the government providing jobs as an employer of last resort would be any different than normal public sector jobs. Presumably, providing such undesirable jobs shouldn't crowd out the private labor market, but at the same time, what is to say that people wouldn't just reject these and continue to seek out jobs in the private sector?

The main route I see for making the government an employer of last resort would be to replace black markets, or to provide competition in inefficient markets. The former option has a prohibitively expensive political cost (and if allowed would be blockaded to the point of gross inefficiency). The latter option is harder to parse through. Most monopolies or inefficient markets we think of today are in technology or other places where monopolies benefit from large economies of scale. Treating the provision of goods in those markets as utilities and then subsidizing them is the go-to option. The other thing that could happen is that the government abuses its power to become a monopoly itself in the market it tries to provide employment for.



All that said, let's put those complications aside. Say the government tries to provide more public sector jobs (of some low perceived quality perhaps). The quantity is fixed to always provide jobs for every applicant at exactly the price where the worker would be indifferent between working and taking the public job. Thus, everyone searching can find employment, and in fact, everyone will be induced to join the labor market in this scenario. Note that the government has changed the number of people in the labor market.

You can form an economy with one representative private good, and one collective (public) good. The government levies taxes on consumers to freely to fund their jobs program if needed. The private sector produces the private good, while the government can produce the private or collective good. The government has the same technology as the private sector.

Thus, we have the different components of our economy:

$$
\begin{align}
\text{Utility:} &amp; \qquad U(x_p, x_c, e) \\
\text{Prices:} &amp; \qquad p(\sum x_p), p(\sum x_c) \\
\text{Production:} &amp; \qquad f_p(\sum e), f_c(\sum e) \\
\text{Private Sector Profit:} &amp; \qquad \pi(x_p) = f_p \cdot p_{x_p} - c(w, e)\\
\text{Government Profit:} &amp; \qquad \pi(x_p, x_c, t) = f_p \cdot p_{x_p} + f_c \cdot p_{x_c} - c(w, e) + t\\
\end{align}
$$

You can add in the appropriate subscripts for the collective good to indicate what you desire about how the benefits or costs of the collective good are captured. We have $e$ as effort, $c(w, e)$ cost given wages and effort, $t$ as the tax, in this a case a simple lump sum. Solving for the equilibrium in the case with or without any government intervention here and then applying Shephard's Lemma would get you the consumer's expenditure function, which you can use to compare the cost of living in each case. If utility for each agent increases with the government provision or stays the same, then although the cost of living may increase, we would say in this scenario it is a Pareto improvement to have the government provision. Obvious extensions to the model could be changing the type of tax, making it so that you can levy a tax on production rather than consumption, or changing the competition so that you can use game theory to analyze price setting.

If this all seems rather theoretical (we have not specified the functional form of these equations), it's because your question is very broad. There is no way to succinctly analyze the effects of full employment on the cost of living. I gave my opinion above that I think it sounds infeasible, but if you wanted to set up a model and investigate for yourself, that might be a noble effort. My official answer thus is that it's unclear what effect a government provision of jobs would be on the cost of living.

(Boring safe answer, I know.)