Some sort of solution is what surplus value of labor leads to--almost inherently. The Stanford Encyclopedia offers a nice cut-and-paste short-hand:
"Capitalism is distinctive, Marx argues, in that it involves not merely the exchange of commodities, but the advancement of capital, in the form of money, with the purpose of generating profit through the purchase of commodities and their transformation into other commodities which can command a higher price, and thus yield a profit. Marx claims that no previous theorist has been able adequately to explain how capitalism as a whole can make a profit. Marx's own solution relies on the idea of exploitation of the worker. In setting up conditions of production the capitalist purchases the worker's labour power — his ability to labour — for the day. The cost of this commodity is determined in the same way as the cost of every other; i.e. in terms of the amount of socially necessary labour power required to produce it. In this case the value of a day's labour power is the value of the commodities necessary to keep the worker alive for a day. Suppose that such commodities take four hours to produce. Thus the first four hours of the working day is spent on producing value equivalent to the value of the wages the worker will be paid. This is known as necessary labour. Any work the worker does above this is known as surplus labour, producing surplus value for the capitalist. Surplus value, according to Marx, is the source of all profit. In Marx's analysis labour power is the only commodity which can produce more value than it is worth, and for this reason it is known as variable capital. Other commodities simply pass their value on to the finished commodities, but do not create any extra value. They are known as constant capital. Profit, then, is the result of the labour performed by the worker beyond that necessary to create the value of his or her wages. This is the surplus value theory of profit."
The reason why we're often told that Marx offers no solutions is because his solutions are untenable, especially as laid out in Kapital Vol. 2 and 3. He truly believed himself to be doing science, but his economics suffer mightily. Singer's _Very Short Introduction_ lays this out in general terms and succinctly.
I'm not a fan of the following excerpts (which follow the above paragraph quoted), but they give support that he did in fact offer a solution:
"It appears to follow from this analysis that as industry becomes more mechanised, using more constant capital and less variable capital, the rate of profit ought to fall. For as a proportion less capital will be advanced on labour, and only labour can create value. In Capital Volume 3 Marx does indeed make the prediction that the rate of profit will fall over time, and this is one of the factors which leads to the downfall of capitalism. (However, as pointed out by Marx's able expositor Paul Sweezy in The Theory of Capitalist Development, the analysis is problematic.) A further consequence of this analysis is a difficulty for the theory that Marx did recognise, and tried, albeit unsuccessfully, to meet also in Capital Volume 3. It follows from the analysis so far that labour intensive industries ought to have a higher rate of profit than those which use less labour. Not only is this empirically false, it is theoretically unacceptable. Accordingly, Marx argued that in real economic life prices vary in a systematic way from values. Providing the mathematics to explain this is known as the transformation problem, and Marx's own attempt suffers from technical difficulties. Although there are known techniques for solving this problem now (albeit with unwelcome side consequences), we should recall that the labour theory of value was initially motivated as an intuitively plausible theory of price. But when the connection between price and value is rendered as indirect as it is in the final theory, the intuitive motivation of the theory drains away. But even if the defender of the theory is still not ready to concede defeat, a further objection appears devastating. Marx's assertion that only labour can create surplus value is unsupported by any argument or analysis, and can be argued to be merely an artifact of the nature of his presentation. Any commodity can be picked to play a similar role. Consequently with equal justification one could set out a corn theory of value, arguing that corn has the unique power of creating more value than it costs. Formally this would be identical to the labour theory of value.
Although Marx's economic analysis is based on the discredited labour theory of value, there are elements of his theory that remain of worth. The Cambridge economist Joan Robinson, in An Essay on Marxian Economics, picked out two aspects of particular note. First, Marx's refusal to accept that capitalism involves a harmony of interests between worker and capitalist, replacing this with a class based analysis of the worker's struggle for better wages and conditions of work, versus the capitalist's drive for ever greater profits. Second, Marx's denial that there is any long-run tendency to equilibrium in the market, and his descriptions of mechanisms which underlie the trade-cycle of boom and bust. Both provide a salutary corrective to aspects of orthodox economic theory."
http://plato.stanford.edu/entries/marx/
Again, I have no desire to get into the nitty-gritty of his solutions, but they do exist--both in general and in particular.
= new reply since forum marked as read
at 'a harmony of interests between worker and capitalist' 
