It's not really an either/or option having a mobile responsive or mobile first website. You cannot have a mobile first website without it being responsive...
The question you need to ask your developer to rank higher in Google is 'Do I having a Mobile First website?'
Many designers will not be able to answer this, especially if they have used a template to build your website. Even adding a couple of lines of custom code could make all the difference. The difference is subtle, but makes all the difference to website loading speeds on mobile devices.
A mobile responsive website
If you know what a mobile responsive website is then you may want to skip this part.
A mobile responsive website is one that scales and re-positions content to work on different sized screens. This makes it much easier to build and manage websites where you edit one page which will work on all devices from you mobile to tablet without having to create separate websites.
It used to be that a separate website was created for mobile visitors on a sub-domain such as mob.yourwebsite.com. However, it's now common practice to have one page that works across different display sizes.
Mobile Website Speed
Before we begin to explain what mobile first means, it's important to understand the reason why having a mobile first website is becoming so important. Websites have become very complex as internet speeds have increased with ultra fast broadband and WiFi hot-spots everywhere. In some cases, websites have become as complex as many desktop applications.
However, mobile phone signals are not always great and therefore internet on mobile devices can be very slow. As more than half of Google searches are on mobile devices, the mobile market is very important! You may be aware that Google rank websites on page loading speeds as they want to speed up the internet so people get answers to their questions sooner.
When your browser loads a page it will grab images and resources in the order they're served in the HTML. If your website is mobile responsive it will most likely serve two versions of the same image. One larger for desktops and another smaller file for mobiles.
Many people who develop websites build them to work on desktops and then optimise them to work on mobile. This means the browser will load the larger desktop image and then if it's mobile will load the smaller image. This may seem wrong to you which leads me on to...
A Mobile First Website
A mobile first website works by the browser loading the smaller mobile images and resources and then, if the screen is bigger, loading the larger image size.
It sounds simple really, but you will be amazed at how many websites miss this step. Quite simply, a website should be built to work on mobiles first and then scale up to larger screen sizes if required.
Websites loading on desktops with ultra fast broadband can scale up quickly without you even noticing it. If however, you're on a slow mobile data connection while commuting to work you will be grateful that a website loaded the smaller images ignoring the larger images.
So please don't ask for a mobile responsive website. Instead, ask for a Mobile First Website!
If you're serious about your website ranking highly on Google and value your mobile visitors then you should only settle for a Mobile First website and not just a Mobile Responsive one