The manufacturer pursues the maximization of profit (π) under the constraint conditions.

Profit is equal to total revenue minus total cost, that is π=TR-TC.

When the first derivative of profit is obtained from the first extreme condition, MR-MC=0.

So we come to the conclusion that whether it is a completely competitive market or a monopoly market.

The first-order condition of equilibrium is MR=MC, that is, marginal revenue = marginal cost.

In a perfectly competitive market

Horizontal demand curve faced by manufacturers

Its price average income and marginal income coincide with the demand curve.

That is, P=AR=MR, so it satisfies P=MC according to the equilibrium condition.

In a monopoly market

Manufacturers are faced with a demand curve that slopes downward to the right.

Assume that the demand curve is linear and P=a-bQ.

Total income tr = pq = AQ-bq 2

Average income AR=a-bQ obviously, the average income curve coincides with the demand curve.

Marginal income MR=a-(b/2)Q Obviously, the marginal income curve is lower than the demand curve, that is, MR.

Assuming that the demand curve is nonlinear, the right side of the equation P=P(Q) indicates that the price p is a function of the demand q.

Total income TR=P(Q)Q

Average income AR=P(Q) Obviously, the average income curve coincides with the demand curve.

Marginal income MR=P+Q( 1- 1/e)(e is the price elasticity of demand, and e is not negative in economics).

Therefore, from the equation of marginal revenue, we can see that Mr.

Therefore, under monopoly conditions, whether the demand curve is linear or nonlinear, MR is less than P.

According to the equilibrium condition, MC = Mr

Well, the last and most crucial step.

Is to explain why MC=P is to achieve the optimal allocation of resources and MC.

Because price p is the price that consumers are willing to pay for the last unit of goods, it objectively represents the social evaluation of the value of a commodity.

Marginal cost MC is the cost that manufacturers must pay to produce the last unit of goods, which objectively represents the social evaluation of the cost of a commodity.

When MC

Of course, production will continue until the value of the last unit product is equal to its cost, that is, MC=P (because the marginal cost curve has a positive slope and the demand curve is not negative, there must be an intersection, that is to say, if production is increased, MC=P will inevitably appear in the end).

So the problem is proved.

Finally, according to my usual style, of course, I have to talk nonsense, haha

Economics is full of charm, not because it has strict logic in mathematics, but because it really explains real life with its powerful explanatory power.

Mathematics is just a tool, and the significance of a tool lies in its assistance, not in the tool itself.

So behind the equation, we should think more about the economic significance expressed by the equation.

To add:

In a monopoly market, if manufacturers can achieve the first level of price discrimination (that is, complete price discrimination)

It can also realize the optimal allocation of resources.

Because under the first kind of price discrimination, the manufacturer's marginal revenue curve will coincide with the demand curve.

The equilibrium condition is P=MR=MC.

Its economic significance lies in the fact that the consumer surplus is completely deprived by the manufacturer.

In other words, it increases the interests of manufacturers without losing the interests of consumers.

From the perspective of the whole society, Pareto optimality has been achieved.

Finally, the optimal resource allocation state in Pareto sense is achieved.