I realize this will be only an estimation based on passages from the book.
In addition, if his wealth were to be redistributed across Middle Earth, would the influx of currency cause a resurgence of the economy?
The current Forbes "Fictional 15" ranks him at number 2 (behind Scrooge McDuck), with an estimated wealth of $54.1 billion. See http://www.forbes.com/special-report/2013/fictional-15/smaug.html for this year's estimate.
There's a writeup available (from a couple of years ago) outlining how they came about this figure, so I'll provide some quotes, first of all to establish some working criteria:
We know from the novel that Smaug’s wealth comes down to three primary components, the mound of silver and gold that he sleeps on, the diamonds and other precious gemstones encrusted in his underbelly, and the “Arkenstone of Thrain,” which is depicted as something like the Hope Diamond on steroids.
Then, using the D&D rules (a dubious source IMO) they estimate Smaug's body length as 19.2 feet, and:
Dragons are long and narrow, so we can safely assume that Smaug can curl comfortably up on a treasure mound with same diameter as his body length – 19.2 feet ... Assuming the mound is twice the height of Bilbo, we can say that the mound has a height of approximately 6 feet.
I won't take you through the rest of their calculations, including estimates of how much gold and silver coins there are, the value of diamonds, valuing the Arkenstone itself, etc, but it's all available at the link provided, coming out at 2.8 million gold coins and 1.7 million silver coins, 38.9 thousand diamonds, and the Arkenstone valued as a fourteenth share.
IMO there are quite a few leaps of faith in the logic of this estimation (and the final figure is different from the current estimation, being from a couple of years previous), but it does serve to demonstrate one way of deriving a value.
If this were distributed around Middle-earth it's difficult to say what kind of impact it would have on the economy. We're talking about a world where pearl-strewn beaches were once a feature, and where the current setup is one of distributed pockets of civilization trying to survive against great evil. Amassing personal wealth seems to be something that would be quite low on any concievable list of priorities (unless you're someone like the Master of Lake Town, of course).
Forbes greatly underestimated Smaug's size and thus the size of his pile of gold, silver, and jewels. When I read the Hobbit at the age of 13 I calculated that Smaug should be about 200 feet long, about ten times their estimate. and my more recent calculations really don't contradict it. Although possibly Smaug can magically change his size to be bigger or smaller when he wants to.
In Middle-earth the value of gold did not depend on rarity. According to Tolkien Morgoth put his evil power into everything in the world, so that the world hates everybody. Morgoth especially concentrated his evil power into gold, so that all gold has an evil power to make people lust after it.
If I may add my own speculation, Morgoth might have finally wised up and stopped putting more evil power evenly into all the gold in the world and might have mined vast amounts of gold, many times all the gold which has been mined in recorded history, and then put a lot of evil power into that gold which he actually had possession of, to make it easier to bribe dragons, for example, with it.
Most of Morgoth's super extra cursed gold would have been lost when Angband was destroyed, but the tiny remaining amount would still have been far more than all the gold available today. Important dwarves, especially dwarf kings, would have gradually built up large hoards of that enchanted gold over the centuries.
Then about Second Age 1600 Sauron gave the seven dwarf rings to the seven kings of the seven races of the dwarves. It turned out the dwarves could not be controlled by their rings but the rings did fill them with an even greater lust for gold. So the dwarf kings would have all gradually changed the economic and tax and trade policies of their realms to acquire more and more gold from their subjects and outsiders. So the seven great dwarf hoards got bigger and bigger for thousands of years. And outsiders would value gold more and more as it became harder and harder to get enough gold to buy things from the dwarves.
And then after Third Age 1000, dragons reappeared and began to attack the dwarf kingdoms and steal the seven great dwarf hoards.
I expect that when Smaug conquered Erebor and Dale, most of their gold was newly mined from the mountain and only had the average amount of gold lust enchantment on it, though some of it no doubt was the extra enchanted type which had been extra cursed by Morgoth. And then Smaug slept and brooded on the gold for about 150 years and filled the gold with his own dragonish lust for it, so that all the gold became super cursed.
So even if Smaug's hoard increased the world's gold supply by ten times or more, it would not cause inflation or cause gold to fall in value, because Morgoth's evil power and the lust of dwarves and dragons over thousands of years made everyone lust for gold far more than its actual scarcity or economic value justified.
And centuries or millennia after Lord of the Rings, some terrible disaster caused mountains to fall thousands of feet and become seabeds, and seabeds to rise up into dry land, and changed the shape of lands and seas to that of the modern world. And no doubt all the enchanted gold which was being slept on by dragons in deep caverns, or stored in vaults and treasuries deep below ground, was swallowed up and buried.
And the much smaller gold supply in the world today which has been mined in the thousands of years since that catastrophe would have only the average amount of gold lust enchanted in it. But of course it would be possible to write fan fiction about uncovering vast amounts of the super enchanted gold from the time of Lord of the Rings and how everyone would kill to get their hands on it.
Basing Erebor on an actual world economy of the present, particularly California seems more sensible then trying to estimate wealth based on dragon size and piles of treasure. The fact that always seems to be ignored, particularly by Forbes Fictional 15, is that Smaug usurped a city-state/nation and its entire treasury, which means it is not based on an individual’s ability to amass wealth in a lifetime; but, a nation’s ability to amass a grand treasury over generations (all under the power of magic via the seven rings given to the Dwarf Lords). The economy of California is the sixth largest in the world at $2.46 trillion dollars (http://www.sacbee.com/news/business/article83780667.html). For simplicity sake I am not including the effects of inflation/deflation, or fluctuating gold prices. If a meager 0.02% (0.0002) of the annual revenues where “profits” accumulated into the royal treasury over the history of Erebor that would equate to a national treasury valued at $379,332,000,000 (that’s 379 billion, 332 million US$; Erebor founded in 1999 and Smaug usurping the kingdom in 2770 = 771 years). As in the aftermath of the overthrow/death of Muammar Gaddafi of Libya, with accounts valued at more than $200 billion which was not personal wealth but embezzled national funds; a nation’s ability to accumulate wealth is exponential to an individual ability to accomplish the same task.