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Marginal Product (MP) refers to the additional output produced from using an additional unit of input (usually labor). It is calculated as the change in total product divided by the change in the quantity of inputs. Marginal Cost (MC) is the additional cost incurred in producing an additional unit of output. It is calculated as the change in total cost divided by the change in the quantity of output.

The Marginal Product curve initially rises and then starts to decline due to the law of diminishing marginal returns. Meanwhile, the Marginal Cost curve is U-shaped, initially falling and then rising again. This is because as more units of output are produced, it becomes more expensive to produce each additional unit due to factors like increasing input prices, a higher rate of equipment wear and tear, and a need for more supervision.

There is an inverse relationship between Marginal Product and Marginal Cost. When the Marginal Product is increasing, the Marginal Cost is decreasing and vice versa. This relationship comes from the fact that when more units of the input are added, the extra output obtained from each additional input decreases (diminishing marginal returns), so the cost of producing each additional unit of output increases. In other words, as the Marginal Product curve declines, the Marginal Cost curve rises.

In the short run, at least one input is fixed (e.g., capital), which means that the increasing cost of additional production is tied to the fixed input. This gives rise to the U-shaped Marginal Cost curve and the inverse relationship between Marginal Product and Marginal Cost. In the long run, all inputs are variable, and firms can adjust their inputs to minimize their costs of production. As a result, the relationship between Marginal Product and Marginal Cost observed in the short run may not hold. In the long run, firms can use economies of scale, optimum factor proportions, or technological advancements to increase their output without necessarily incurring a higher cost per unit. This results in a flatter Marginal Cost curve and a weaker relationship between Marginal Product and Marginal Cost. To sum up, the relationship between Marginal Product and Marginal Cost is inverse, which can be observed in their respective curves. In the short run, this relationship holds due to factors such as diminishing marginal returns and fixed inputs. However, in the long run, the relationship may not be as strong due to firms' ability to adjust all inputs and take advantage of various factors that contribute to lowering the cost of production.