I would never recommend buying something to represent your business for less than the cost of a dinner at a decent restaurant. It's easy for me to imagine what a $20 logo would look like, because there are a lot of $20-caliber logos out there. It probably has a typeface that comes preloaded on most computers (cough, Helvetica) and it probably has a swoosh in or around it.
$200 is getting closer to the price you can expect from an inexperienced designer. You can expect a better eye for style and maybe a more unique symbol. $200 is still considerably cheaper than most logos so I would expect the design process to be minimal, affording less striking results.
4 digits goes further less and less in the 21st century. Fortunately, when it comes to logo design $1000-2000 will get you the logo you need that will last the life of your business. For the price, you can expect the design process to turn out some exciting options that utilize symbolism, typography and hidden meanings to construct a memorable and unique logo. Beyond the logo itself, most designers will provide additional deliverables, such as business card designs, that will begin to build the brand's personality.
A lot of my logo projects have included business cards and websites which allowed me to develop the brand and offer a brand guide to the client. Brand guides set expectations for the use of the logo on different materials and establish rules for font and color usages. Businesses that have brand guides have less trouble keeping up the look and feel of their brand for internal documents and other projects that don't require the attention of a designer. Brand guides also help when the logo designer is unable to design the rest of the materials needed. It gives the next designer a guide to using the logo and styling effectively.
Big budget logos are more common than most people think. Big brands in retail, food and technology place a high value on their visual identity because of the amount of visibility the brand has. If you have a product that is going to be seen by millions of consumers, an investment in your appearance is a no-brainer. A $20,000 budget would pay for a team of designers at a design firm, collaborating and developing concepts that would strategically place the brand. Similar strategies can be found at lower price points but not with the expertise and tools that would be available to a design firm at a 5 digit level.