This symbol is more speciﬁc to integrals. The integral sign, ∫ has a bit more of a story for it. This really cannot be used for anything other than indeﬁnite integrals (which is what antiderivatives are). The letters here should also be italicized, but Khan Academy does weird things with the formatting if I were to try to do that.Īnyways, the antiderivative of f(x) is often written as F(x).

Generally, the lowercase letter is used for functions e.g.: f(x).

Determine your relevant marginal cost of producing Type A bolts and your profit-maximizing production of Type A bolts.Well, there are two ways of representing the antiderivative. A call to a reputable source has revealed that unit costs for producing Type B bolts are identical to those for producing Type A bolts, and that Type B bolts can be sold at a constant price of $4.75 per unit. Since the firm also makes Type B bolts, this means that each Type A bolt produced reduces the number of Type B bolts that can be produced by one unit the total number of Type A and B bolts produced cannot exceed 5 units. In addition to the above information, you know that the firm’s assembly line can produce no more than five bolts. The reported unit overhead costs reflect the $10 spent last month on machines, divided by the projected output of 2 units that was planned when the machines were purchased. Materials and labor are obtained in a competitive market on an as-needed basis, and the reported costs per unit for materials and labor are constant over the relevant range of output. The accounting and marketing departments have provided you with the following information about the per-unit costs and demand for Type A bolts: You are a manager at Spacely Sprockets-a small firm that manufactures Type A and Type B bolts. How much more profit will each firm earn as a result of privatization? Compare the price-quantity combination before and after privatization. Explain why the price elasticity makes sense at the profit-maximizing price-quantity combination. Compute the price elasticity of demand at the profit-maximizing price-quantity combination. Determine the number of kilowatt-hours of electricity each firm will produce and supply to the market, and the per-kilowatt-hour price. Once privatized, each firm will have incentive to maximize profits. Furthermore, to supply electricity to its particular region of France, it costs each firm C ( Q ) = 120 + 0. 13 and that the inverse demand for electricity in each of these two regions of France is P = 1. Suppose that prior to privatization, the price per kilowatt-hour of electricity was n 0. Union workers are concerned that privatizing power utilities would lead to large-scale job losses and power outages similar to those experienced in parts of the eastern coast of the United States and parts of Italy in 2003. BBC News reported that France's CFT union responded by organizing a mass strike, which triggered power outages in some Paris suburbs. The French government announced plans to convert state-owned power firms EDF and GDF into separate limited companies that operate in geographically distinct markets.

Given that we have two facilities and therefore two marginal costs, the relationship MR=MC must be applied for each facility to obtain the quantities of each one. The rule tells us that Q = Q 1 + Q 2so we can substitute the value for each level of production in the previous equation: Now the first derivative is applied to obtain the marginal income: To obtain the marginal revenue, we derive total revenue (TR) first. Therefore, the demand and cost functions are derived to obtain the marginal revenues and costs. In the monopolistic competition market, as in the monopoly, the same function is fulfilled, that production and profits are maximized when marginal income is equal (MR=MC). Inverse demand function: P = 1, 200 + 4 QĬost Function Facility 1: C ( Q 1 ) = 8, 000 + 6 Q 1 2Ĭost Function Facility 1: C ( Q 1 ) = 6, 000 + 3 Q 2 2 An energy company in a monopolistically competitive market has the following demand and cost functions for the two facilities it produces:

Power utility function inverse