A, profit maximization and cost minimization test analysis
In the process of economic research, comparative static analysis and dual analysis are commonly used methods in mathematical economic model analysis.
Comparative static analysis of (1) first-order conditions. Let's take two input models as examples to discuss the problem of profit maximization. Let the price of a commodity be p= 1, from which the expressions of profit maximization can be obtained as Maxf(x 1(w 1, w2), x2(w 1, w2))-w1w2-w2.
The first-order condition of profit maximization can be obtained. On the basis of the first-order condition, the second-order condition of profit maximization can be obtained by calculating the partial derivatives of w 1 and w2 respectively:
f 1 1
f 12f
2 1
f[]
22
×? x 1
w 1？ x 1？ ww？ x2？ w 1
x2？ w
2=
10
[]
0 1, which shows how producers follow.
Replace one input with another to cope with price changes.
The same method can also be used to study the cost minimization problem. We are fake.
Suppose that input are two commodities,
The expression of cost minimization can be expressed as L(λ, x 1, x2) = w1x 2+w2x2-λ (f (x1(w1,w2, y))-.
The substitution matrix Dλ(w) with the minimum cost can be obtained similarly.
Dx(w[])=
0Df(x)
Df(x)
T
λD2
f(x[])- 1
[]0
1。
(2) Comparative static analysis of algebra. Theorem 1: (Profit maximization is weak.
Axiom) let the price vector be pt,
The output vector is yt(pt
), the other output vectors are ys, (t, s= 1, …, t), and if producers want to maximize profits, then the price is pt.
Time and producer output are yt.
When the profit level, at least with producers.
When choosing other output levels, the profit level is the same, that is, pty? ptys
(t，
s= 1，…，T).
Theorem 2: (Weak axiom of cost minimization) Let the output vector be yt.
, factor price
The lattice vector is wt and the element level is xt.
, (t, s= 1, …, t), if the producer wishes.
Minimize costs,
Then when the price is wt, the producer's input is xt.
When the cost level is reduced, it should be less than or equal to at least the same output of any other product.
Input cost, namely wtxt? wtxs
t，
S= 1, …, T. From the theorem 1, we can know the vector of price change and its associated network.
The inner product of the vector that produces the change must be nonnegative. Theorem 2 states that the demand vector and the price vector will always develop in opposite directions.
(3) Double analysis method. For a given data set, if the weak axiom of profit maximization or cost minimization is satisfied, a method can always be found. Through this method, we can see whether the producer's choice is to maximize profits or minimize costs, and then determine the true set of a method by constructing the external and internal boundaries of the production set. Through analysis, we can get the convex monotone shells of the inner boundary YI and the outer boundary YO of the production set under the condition of maximizing profits, so that YO and YI constitute the closest inner and outer boundaries of the real production set of this method. Similarly, the convex monotone shell of the inner boundary YI and the outer boundary YO of the generating set can be formed under the condition of minimizing the cost, so that YO and YI can form the nearest outer boundary and inner boundary of the real generating set in this way.
Two. Profit function and cost function (1) Total cost and marginal cost. Marginal cost of different angles and demands
If there are different explanations,
One is the increased cost when the unit output increases 1, which can be intuitively expressed as when the last unit output is Q=6.
The cost of spending; Second, if you want to calculate the marginal cost when the output is Q = 5.5, you can first calculate the marginal cost when Q=7.5, and finally take the average of the two marginal costs as the marginal cost when Q=7. The approximation of the latter marginal cost is better than the former; The third is that marginal cost is the change of total cost TC caused by the slight change of output Q, that is, MC=
dTCdQ = limδQ→0δTC
δQ
. This is a true explanation of marginal cost.
(2) Total income and marginal income. Because under the condition of free competition, the behavior of a single producer will not have any impact on the market price, so the price of a certain commodity will not change for producers in a certain period of time. In order to express the total income and average income of producers, we assume that the price of a product is A and the output of the commodity is Q, so the expression of total income is TR = AQ；; The expression of average income is AR=
Türkiye
Q
. From 2. 1, we can know that the marginal income can be understood as the increase of income per unit output, that is, the total income.
The slope of the benefit curve, so the marginal benefit is the price A of the increased unit output 1, where the marginal benefit is the average benefit, but it should be pointed out that this conclusion is not always true, because there are still many factors that will affect it in reality.
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