Dealers will always seek to make a profit on any car sale. And they will want to hedge their risks of losing money. So their trade-in offers will often be close to auction price, which is going to be less than their asking retail price so they can make a profit. But close to auction so they don't lose money if they have to dump it. If they sell the car on their lot, they make a few grand. If they can't sell it on their lot then they either sell it to a local buy-here-pay-here for a few hundred more than they paid (or same if in a pinch) or they ship it to the auction.

In the states, sales taxes are structured to help car dealers stay in business, so the customer doing the trade is taxed only on the net spend (purchase price - trade in = net) -- so your tax savings on the new car is passed to the dealer in the form of the lower trade-in price.

For dealers of desirable cars -- for example the Lexus dealer in my town -- they lowball trades about USD$3k below most other dealers' trade-in offers. This is both good and bad. Bad if you trade-in for a new purchase, but good if you're going to buy a used car off their lot, because you can low-ball your offer and they'll take it if it's higher than auction price less fees & transport, because they're still making money. Been there and done that myself and saved a bundle!