First, I'm going to make a radical departure from canonical economic theory, and unambiguously denote the three components of economic analysis as cost, value and price. I've concluded that using value to represent exchange value, i.e. price, is not just misleading but intentionally misleading. Instead, I'm going to consistently use price to represent exchange value, and value to represent use value.
The marginal theory of price says first that the price (exchange value) of an item is most strongly related to its value (use value). In other words, when two people exchange commodities, what is equivalent about the exchange is generally the increase (margin) in each party's respective value. If a hamburger adds ten "units of happiness" to my life, and a hammer adds ten units of happiness to your life, (and losing the hammer and hamburger subtracts one unit of happiness from each of our lives), I'll give you a hammer in exchange for a hamburger. Alternatively, if we say that a dollar arbitrarily represents one unit of happiness, I'll pay ten dollars for a hamburger if and only if it adds ten units of happiness to my life.
There are difficulties actually making consistent and comparable measurements of "units of happiness", but the concept has intuitive appeal. We can introspectively determine that possessing and consuming a hamburger gives us more happiness than possessing and (somehow) consuming a turd. We can attach a number to anything that can be ranked, so we're in reasonably good shape... at least so far.
The marginal theory of price assumes that value can be ranked: Suppose there are 100 people, and they want widgets. Alice wants a widget most, Bob wants one second most, ... Zelda wants one the least. (If any person wants two widgets, we'll just add how much they want the second widget to the ranking with no loss of generality.) We'll assume we can not only rank their desires, but we can (roughly) quantify them: Alice is willing to work for 42 hours in exchange for a widget; Bob is willing to work 38 hours; ... Martin is willing to work 21 hours; ... Zelda is willing to work 2 hours.
If we order the people by rank, we have by definition a monotonically decreasing value function: The total extra social use value added by supplying one more widget decreases as we add widgets. Supplying one widget gives us 42 hours of use value, supplying the second gives us an additional 38 hours, etc.
There are two forms of the marginal theory of price. The first assumes diminishing returns on cost: the more widgets we make, the higher the cost of the additional widgets. We can see this if we rank the producers of widgets in order of efficiency: Andrew, the most efficient producer, makes a widget in 6 hours; Betty in 6.2 hours hours, ... Mary in 9.3 hours; ... Zachary in 12.7 hours. Again, we rank them, this time from shortest time to longest time, creating a monotonically increasing function.
It's a theorem of mathematics that a function that is monotonically decreasing will intersect a function that is monotonically increasing at at most one point. (We'll assume that our value function and our cost function do indeed intersect in the effective domain of 0-100 widgets.) We can simplify our analysis by assuming our value and cost functions are linear. (There are other simplifying assumptions that don't affect generality; we have to preserve only monotonicity and opposite slope.) The price of a widget, then, will be the point at which the demand and supply functions intersect.
Therefore, assuming that the demand function by rank for widgets is H = -0.75R + 100 and the cost function is C = 0.45R + 16, price will be H(R) = C(R); -0.75R + 100 = 0.45R + 16; P = H(70) = C(70) = 47.5 hours.
However, this analysis trivially agrees with the labor theory of price! If the socially necessary cost (which can be any rigorously defined statistic of actual cost) is defined to be the maximum cost to create a widget to fulfill demand, and it's worth it for the the 70th ranked producer to take an actual 47.5 hours to create a widget, then the socially necessary cost to create a widget is 47.5 hours, therefore the price of a widget is the socially necessary cost in labor time.
To get a marginal theory of price that differs from the labor theory of price, we have to look not at the cost of production, but the choice of what to produce.
Assume we can produce 100 widgets, 100 doodad, or any combination of 100 widgets and doodads; i.e. for every widget we produce, we're not producing a doodad, and vice versa. More importantly, when we produce a widget for the person who wants one most, we are "taking away" a doodad from the person who wants one least (and vice versa). So we'll rank the demand function for doodads from low to high to make it monotonically increasing. The point where the functions intersect is the price of widgets and doodads.
For example, if Hw = -0.75R + 100 and H-1d = 0.45R + 16*, then the price of widgets and doodads will be 47.5 hours. We have calculated an equilibrium price without reference to how many hours it costs to produce either a widget or a doodad, thus falsifying the the labor theory of price, right?
*Yes, I just recycled the parameters of the cost equation to avoid having to do more algebra.
Wrong. Consider the assumption: "We can produce 100 widgets, 100 doodad, or any combination of 100 widgets and doodads; i.e. for every widget we produce, we're not producing a doodad, and vice versa." This assumption entails that it takes approximately the same amount of time to produce a widget and a doodad. The labor theory of price entails that things that take the same amount of time to produce will trade at an equivalent price, precisely what we have determined the marginal theory of price entails. If a widget takes approximately 1.5 times as long to produce as a doodad, then the marginal theory of price will find a price for widgets that's 1.5 times the price of a doodad, again, exactly what's predicted by the labor theory of price.
Furthermore, I slipped in a subtle equivocation when talking about the diminishing returns form of the marginal theory of price: I had some people consuming widgets (Alice, Bob, etc.) and different people producing widgets (Andrew, Betty, etc.). But this is not the case: Alice, Bob, etc. are producing and consuming widgets and doodads: essentially, each person makes a widget and a doodad, and either keeps it for him- or herself or trades it one for one for the other item. The "price" of 47.5 doesn't mean anything, since the price of widgets and doodads is the same, therefore they trade one for one.
Thus, the marginal theory of price does not differ theoretically from the labor theory of price: it does not make contradictory predictions. The marginal theory of price tells us how much of a commodity will be produced and consumed to achieve the labor theory of price.