Supply Curve from Piecewise Linear Cost Function

Suppose we know that a firm has a piecewise cost function of the following form: if and if , where and where is constant and stands for a given quantity where the cost curve breaks between the and pieces (we may change it with the respective control). This outset differs from the usual case with in the following point: the firm should define what quantity to produce if the market price is in the range (i.e. it has two options: to use or pieces). In more general terms, the firm should define its supply curve, which does not simply coincide with the cost curve in this case. Note that if the firm produces according to the first piece of the cost curve, , there is nothing special to analyze. But if the firm switches to the piece of the cost curve, then there is a challenge: is it profitable or not? Indeed, to move to the piece, given the price, there are first some losses before the "breaking" quantity level of production, and then some profits. (See this by changing the P slider and observing the red and green triangles.) The trick is to integrate those triangles to define the break-even price level , where the producer is indifferent between two pieces of the cost curve. This price is found by solving the following equation:

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This price level allows us to define the horizontal "break" level of the supply curve (can be shown by the checkbox on the graph). This also means that the red and green triangles on the graph have equal areas. In economic terms, red corresponds to losses and green reflects profits. When profits and losses are equal, the producer is indifferent between the two alternatives, so the supply curve (derived from the marginal cost curve lying above the average cost curve) has a discontinuity between the two indifferent solutions. In this "problematic" range of prices , one cannot switch to the piece of the cost curve without suffering some losses that should be compensated by additional profits. (You can see the supply curve on the graph by checking the checkbox of the respective control.) As a result, we may define the supply curve that becomes discontinuous in this model.