They literally don't. The last time Republicans cut anything (spending, agency scope, etc) was.... back in Reagan's time? They say they want smaller government, but every time Republicans have control, they grow government, they just grow it differently than Democrats.

For example:

• Bush Sr. - didn't do much other than passing bipartisan bills
• Bush Jr - Patriot Act, TSA, military spending
• Trump - tax cuts and increased spending (at the tail end of an economic boom), stupid tariffs

And Reagan wasn't the epitome of small government either, he just talked about it a lot. According to Mother Jones (strong leftist bias), Reagan grew the national debt substantially and increased the number of government employees (there were far fewer in 2012 than the end of Reagan's presidency).

I used to consider myself Republican back when I believed they were actually in favor of small government, but I have seen no evidence that Republicans actually are. So I switched my registration to Libertarian and now vote for both parties, depending on how I think the candidate views government.