Calculus is one of the great achievements of higher mathematics, which is widely used in all fields of daily life. Using the mathematical quantification of calculus in higher mathematics to analyze and solve the reasons in various fields has become an important part of economics, which has transformed economics from qualitative to quantitative, and made the role of calculus in the economic field more and more obvious.
Calculus; Keywords; Economics; marginal analysis
Calculus is a great achievement of higher mathematics. Calculus comes from production technology and theoretical science, and also affects the development of science and technology.
In the field of economics, some economic reasons are transformed into mathematical reasons by using relevant models, and the economic reasons are studied and analyzed by using mathematical strategies, and the actual reasons in economic activities are quantified by using calculus strategies. The results obtained on this basis have scientific quantitative basis.
1. The application of calculus in economics
1. 1 marginal analysis
Marginal cause in economics refers to the reason why the change of each independent variable leads to the change of the dependent variable, so the marginal function is the dependent variable from which an economic function is derived, and the value of a certain point is the marginal value of that point.
Example 1: Knowing the functional relationship between the income (yuan) and the sales volume (tons) of a certain product in a factory, find the marginal income when the product sells 60 tons, and explain its economic significance.
Solution: according to the meaning of the question, the total revenue function of selling tons of this product is. Therefore, the marginal revenue of selling 60 tons of this product is RMB. Its economic meaning is: when the sales volume of this product is 60 tons, the total income of another ton (i.e. = 1) will be 188 yuan. This truth seems simple, but it plays a great role in real life. Another example is:
Example 2: A factory produces a mechanical product. The functional relationship between the total monthly cost C (thousand yuan) and the output X (pieces) is as follows: If the sales price of each product is 20,000 yuan, find the marginal profit when producing 6 pieces, 9 pieces, 156 pieces and 24 pieces per month, and explain its economic significance.
Solution: According to the meaning of the question, the total revenue function of X pieces of mechanical products produced by this factory every month is. Therefore, the profit function of X product produced by this factory is:, from which the marginal profit function can be obtained. Then the factory produces 6 pieces, 9 pieces, 15 pieces and 24 pieces each month, and the marginal profits are (thousand yuan/piece), (thousand yuan/piece), (thousand yuan/piece) and (thousand yuan/piece) respectively.
The meaning of this economics is: when the monthly output of this factory is 6 pieces, if the output increases by 1 piece, the profit at this time will increase by 18000 yuan; When the monthly output of the factory is 9 pieces, if the output increases by 1 piece, the profit increases by 12000 yuan, and the profit decreases; When the monthly output is increased to 15 pieces, the profit will not increase if the output of 1 piece is increased. When the monthly output is 24 pieces, if the output increases by 1 piece, the profit will correspondingly decrease by 18000 yuan.
From this, we can draw a conclusion that the profit of the product is the largest, not when the quantity is the largest, that is to say, increasing the output will definitely increase the profit, and only by making reasonable overall arrangements for the production capacity of the factory can the profit be maximized.
It can be concluded that when the marginal revenue of the product is equal to the marginal cost of the product, the profit has been maximized at this time, and if the production is expanded, the product will lose money.
Elastic analysis of 1.2
In economics, the degree to which one variable reflects the change of another variable is called elasticity or elastic coefficient [2].
There are many kinds of flexibility in economic work, and the types of flexibility are different for different reasons. If it is a reflection between price changes and demand, this reflection is called demand elasticity. Due to different consumer needs and different commodity attributes, the same price changes have different effects on the demand of different commodities. Some goods are sensitive and flexible, and price changes will cause great sales changes; Some commodities have slow response and little elasticity, and price changes have little effect on them.
① Elasticity of demand. As for the demand function, because the demand function of commodities is monotonous when the price rises, it is a monotonic decreasing function with opposite signs, so the elastic function of demand to price is defined as.
Example 3: Let the demand function of a commodity be, and find the elasticity function of demand; Demand elasticity.
Solution: When the price rises 1%, the demand decreases by 0.6%, and the range of demand change is smaller than that of price change; It shows that when the price rises by 1%, the demand also decreases by 1%, and the range of demand change is the same as that of price change. It shows that the price increased by 1% and the demand decreased by 1.4%, and the range of demand change was greater than that of price change.
② Income elasticity. Income R is the product of the price of a commodity and its sales Q. At any price level, the sum of income elasticity and demand elasticity is always equal to 1. If so, the commodity price increases (or decreases) 1%, and the income increases (or decreases); If yes, the price changes 1%, and the income remains unchanged; If so, the price will increase (or decrease) 1%, and the income will decrease (or increase).
Maximum analysis 1.3
In production theory, the study of long-term production reasons is usually mainly expressed by the production functions of two variable production factors [3]. If an enterprise uses two variable production demands, labor and capital, to produce a product, the production function of the variable production demand is:
In the formula, L is the input of variable required labor, K is the input of variable required capital, and Q is the output of products. Manufacturers who produce products can achieve the best combination of production factors with the largest output at a certain cost by continuously adjusting the production factors of two input variables.
Suppose the labor price recognized in the factor market, that is, the wage rate, is? , the price of the approved capital, that is, the interest rate, is r, and the cost expenditure approved by the product manufacturer is c, then the cost equation can be obtained according to the correlation function: c Under certain conditions, the Lagrange equation is thus established:
The first-level conditions for maximizing product output are:
From the above two formulas, it can be concluded that the principle of factor combination to achieve maximum output under the approved conditions is: that is, the manufacturer of the product constantly adjusts the input of labor and capital, so that no matter which factor of production is used for procurement, the cost of the last unit is the highest, so as to maximize output under the approved cost conditions.
1.4 optimization analysis
Marginal analysis studies the extreme value of a function at the marginal point [4]. In other words, study whether a variable changes from increase to decrease or from decrease to increase at the marginal point. The function value of this marginal point is the maximum or minimum value of the function. The focus of economic research is to study whether the marginal point is the best point, because it is the most reasonable marginal point to make the best decision. So calculus is an indispensable strategy to study the causes of optimization.
Optimization theory is the basis of economic analysis and economic decision-making in economics. To achieve the optimization of economics, all economic activities in economics are required to be at the best peak position, and any deviation must be tilted downward from the peak. This will definitely use differentiated thinking.
Example 4: Suppose the marginal cost of producing a product, its fixed cost is RMB, and the unit price of the product is 500 yuan. Assuming the balance between production and sales, the maximum profit is obtained when the output is the same.
Solution: The total cost function is 0, the total revenue function is 0, and the total profit is 0, 0, 0. Because when the output is 200, the profit is the largest, and the maximum profit L (200) = 400 200-200 200-1000 = 39,000 yuan.
summary
Calculus plays a very important role in economics. Now in the field of economics, many economic studies need quantitative research, so more and more knowledge of calculus is applied, which is not only conducive to the development of calculus, but also helps economics to be more quantitative, precise and precise.
The application of calculus in economics has made great progress in economics and finally led to the formation of microeconomics.
References:
[1] Chen Chaobin. Application of Calculus in Economic Optimization [J]. Journal of Baoshan Normal University, 2009 (5): 34-36.
[2] Zhang Liling. Application of Calculus in Economics [J]. baise university Research, 2009 (5): 49-52.
[3] Cai. Analysis on the Application of Calculus in Economics [J]. Mathematics Learning and Research, 20 10 (9): 99- 100.
[4] Xiang Jumin. Application of Calculus in Economic Analysis [J]. Science and Technology Information, 20 1 1 (26): 57-82.