Equilibrium price and equilibrium volume. Theme: “The theory of supply and demand”

1. Direct and inverse demand functions

Condition: It is known that free consumers are willing to purchase 20 units of good; with each increase in price by 1, the value of demand falls by 2 units. Write down the direct and inverse form of the demand function that describes this situation.

Decision: Since a change in price by 1 always changes Q by 2 units, we are dealing with a demand function of a linear form. (The direct form of the demand function is the dependence of the demand value (Q) on the price (P) - Qd (P); and the reverse form of the function, on the contrary, is the dependence of the price on the demand value - Pd (Q)).

In general, the direct linear demand function is written as: Q d (P) \u003d a - bPwhere a and b are the odds we need to find. We know that at P \u003d 0 the demand is equal to 20 units, it follows that a \u003d 20. In this case, the coefficient b \u003d 2. Thus, the direct demand function can be written as Q d(P) \u003d 20 - 2P.

To get the inverse demand function, we express the price from the previously obtained expression: P d(Q) \u003d 10 - 0.5Q.

Answer: Q d (P) \u003d 20 - 2P- direct demand function ; P d (Q) \u003d 10 - 0.5Q- inverse demand function .

Note:both types of demand functions are equally often used in solving problems, however, it does not matter if you forget which of the types is called.

2. Recovery of the linear demand function

Condition: At a price of P 0 \u003d 10, consumers want and can buy 5 units of products. If the price rises by 50%, then the demand will fall by 40%. Write down the demand function for a given good, if it is known that it has a linear form.

Decision: In general, the demand function of a linear form can be written as Q d (P) \u003d a - bPwhere a and b are the odds we need to find. Since we have two unknowns, in order to find them it is necessary to compose a system of at least two equations. To do this, we find the coordinates (Q, P) of two points that correspond to a given demand function.

When P 0 \u003d 10, consumers are ready to buy 5 units of goods, that is, the value of demand Q 0 is 5 - these are the coordinates first point. If the price increases by 50%, the price will be equal to 15; and the value of demand after falling by 40% will be equal to 3 units. So the coordinates second point is (3, 15). We write the system of equations:

5 \u003d a - b * 10

3 \u003d a - b * 15

The system is solved when a \u003d 9 and b \u003d 0.4.

Answer: Q d (P) \u003d 9 - 0.4P.

Note:this is the standard way of finding the coefficients of the linear demand function, which will be required in most tasks that do not give the demand function itself, but it is indicated that it has a linear form.

3. Plotting a linear demand function

Condition: Given the demand functions for some good: Q d1 (P) \u003d 20 - 2P and P d2 (Q) \u003d 5 - Q. Let demand, expressed by the first function, decrease by 5 units. at each price level, and demand expressed by the second function increased by 60%. Build on the graph the initial and modified demand functions.

Decision: To begin with, we write down the demand functions in the direct form, that is, we express Q in terms of P: Q d1 (P) \u003d 20 - 2P and Q d2 (Q) \u003d 5 - P. To construct any linear functions just find the coordinates two points. The farther these points are from each other, the more accurately the line can be drawn. The ideal option is if we find the coordinates of the intersection of our lines with the axes Q and P. For this, we substitute Q \u003d 0 and then P \u003d 0 into each function. This principle works well when constructing linear demand functions, in other cases it can be used limitedly:

Now we find new demand functions calculated taking into account the changes. The first demand decreased by 5 units. at each price value, i.e. Q new d1 (P) \u003d Q d1 (P) - 5: Q new d1 (P) \u003d 15 - 2P. In the graph, a new demand curve is obtained by shifting the initial curve to the left for 5 units - this is red line D 3. Second demand increased by 60% at each price level. So, with P 1 \u003d 5 and Q 1 \u003d 0, no change will occur, since 60% of 0 is 0. Moreover, with P 2 \u003d 0 and Q 2 \u003d 5, the change in demand will be maximum and will be 0.6 * 5 \u003d 3 units So the new demand function will be Q new d2 (P) \u003dQ d2 (P) +Q d2 (P) * 0.6:Q new d2 (P) \u003d8 - 1.6P.We verify the result by substituting the points (0.5) and (8.0) already known to us in the function. Everything is done, on the graph, this demand is displayed blue line D 4.

Section II. BASES OF THE THEORY OF MICROECONOMICS

This section is an introduction to the study of microeconomics. The section gives general concepts describing behavior in a market economy, without which it is impossible to study the advanced course of microeconomics. The section begins with a study of the basic concepts of microeconomics - demand, supply, equilibrium. Further, the concept of elasticity is revealed, which will subsequently be used not only in the course of microeconomics, but also in macroeconomics and the world economy. The section ends with a study of the fundamentals of behavior of the subjects of a modern market economy.

Chapter 5. DEMAND: OFFER AND MARKET EQUILIBRIUM

From the previous chapters it is known that the relationship between producers and consumers in the commodity economy is carried out indirectly, indirectly - through the market. A specific form of the implementation of commodity relations is a market mechanism, the main elements of which include demand, supply, price.

The purpose of the analysis of this chapter is the mechanism of interaction of supply and demand, i.e. the demand-supply model, which performs analytical and descriptive functions and is the most useful and important tool in the arsenal of an economist.

The supply and demand model, on the basis of which prices are formed, has been the core of economic theory for more than a century. Despite the fact that under modern methods of regulating a market economy, equilibrium is achieved not only due to the interaction of market forces, but also with an active economic policy of the state, this model simply and convincingly leads to explicit and unambiguous conclusions that can be used to analyze various economic problems . It describes in a simple form some of the forces operating in the economy and thereby reflects important aspects of real life.

Demand. Demand functions. Law of demand

The needs in a market economy are in the form of demand. Market demand is an indirect reflection of the needs of people in this product or service.

Human needs, as you know, are not limited. Can we talk about unlimited demand? What is the difference between these concepts? The fact is that demand is a form of expression of demand presented on the market and secured by money, i.e., demand is solvent demand. It is not enough to desire to purchase a product; it is necessary that the consumer possesses a certain amount of money in order to fulfill his desire. The market does not respond to insolvent needs. More precisely, the demand category can be expressed by the term value or volume of demand.

The magnitude (volume) of demand - this is the quantity of goods that consumers want and are able to purchase at a certain price from a number of possible over a period of time.

It is important to distinguish between the concepts of “volume of demand” and “volume of actual purchase”. The volume (magnitude) of demand is determined only by the buyer, and the volume of the actual purchase by both the buyer and seller. For example, government price restrictions can cause a substantial increase in demand. At the same time, sales volume (“actual purchase volume”) is likely to be low as a result of the manufacturer’s disinterest in selling at set prices.

What determines the amount of demand? Various factors influence the desires and opportunities of a particular consumer to purchase a certain amount of goods. These include:

· Price of goods P (price)

· Consumer income I (income)

Tastes, fashion T (tastes)

· Prices of related products: interchangeable (substitutes) P S or complementary (complements) P \u200b\u200bC

· Number of customers N

· Expectation of future prices and revenues W

· Other factors X

So, in the most general form, the demand function is written as follows:

Q d \u003d f (P, I, T, P S, P C, N, X)

Attempts to investigate the nature of changes in demand Q d under the influence of all factors at once will not yield a positive result. In this case, to identify the nature of the change in the quantity of demand Q d, it is necessary first to fix the values \u200b\u200bof all variables except one and study the relationship of Q d with this variable. A similar method means that we examine the dependence of demand on each variable ceteris paribus.

The magnitude of the demand for a product, first of all, depends on the price. If all factors, except the price, are assumed to be unchanged for a given period, then demand function from price will look:

The inverse relationship of price to demand is called inverse demand function and has the form:

Other things being equal, a reduction in price leads to an increase in the quantity of goods purchased by customers; an increase in price causes a backlash: the acquisition of goods is reduced. Thus, this property of demand reflects the inverse relationship between the change in price and the magnitude of demand. The inverse relationship between price and demand (other parameters are unchanged) is universal and reflects the action of one of the fundamental economic laws - law of demand.

Antoine Augustin Cournot (1801-1877) - creator of the mathematical theory of demand. A. Cournot was primarily a talented mathematician, but he was bored in the world of pure mathematics, and with his help he tried to take a fresh look at the problems of other sciences and find connections between them.

In 1838, Cournot published his most famous book today, The Study of the Mathematical Principles of Wealth Theory. In fact, this was the first conscious and consistent attempt to apply a serious mathematical apparatus for the study of economic processes. From this sprout a whole branch of science has grown - mathematical economics.

It was A. Kurno who for the first time deeply analyzed the relationship between demand and price in various market situations. This gave him the opportunity to formulate the law of demand and bring economics closely to the understanding of the concept of “elasticity of demand” (A. Marshall picked up the ideas of A. Cournot and brought them to its logical conclusion). Cournot was able to prove mathematically rigorously that the highest sales revenue is often provided by far from the highest price.

Why is demand behaving this way? This occurs for a number of reasons that argue the law of demand and take into account the following circumstances:

Common sense and life experience directly affect the volume of purchases depending on the price. The lower the price, the more purchases - this is a psychological moment.

Of course, at low prices, the volume of purchases increases, but sooner or later the consumer reaches the limit when each next unit of goods will provide less and less pleasure, no matter how much the price decreases. After a certain level of saturation of needs, the satisfaction received from a product or service begins to decrease. Economists call this effect the law of diminishing marginal utility. Decreasing marginal utility helps explain why low prices drive demand. Products sold at a high price are usually not bought for future use or “at random”. But if the price is low and affordable, then most likely the buyer purchases this product even a little more than he needs.

The effect of the law of demand can be explained on the basis of two interrelated effects - income effect and substitution effect.

Obviously, at a lower price, the buyer can afford to buy more of this product without refusing to purchase other goods. He feels richer because lower prices increase his real purchasing power, or real income at a constant value of his cash income. it income effect.

Income effect (as a result of a change in price) - a change in the value of demand for a product, due to the fact that a change in its price leads to a change in the real income of the consumer.

The degree of effect of the income effect depends mainly on how much of the income is spent on the purchase of a given product. The more part of the income is spent on the product, the more the effect of price growth on the real income of the consumer will affect and the more consumption will be reduced.

On the other hand, the consumer tends to replace more expensive products with cheaper counterparts, which leads to an increase in the demand for these products. it substitution effect.

Substitution effect - the desire of consumers to buy a product in larger quantities when its relative price decreases (replacing others with this product) and consume it in smaller quantities when its relative price rises (to replace this product with others). It is this effect that determines the negative slope of the demand curve.

The magnitude of the substitution effect depends mainly on the quantity and availability of substitute products.

The income effect combined with the substitution effect form a common effect price changes.

The functional relationship between the magnitude of demand and price can be expressed in various ways:

1. Tabular - in the form of a table or a demand scale (table 5.1):

Table 5.1

The ratio of the price of goods X and the quantity X for which demand is presented.

Typical Tasks

Task 1

The population demand function for this product has the form: Q D \u003d 7 - –P.

The supply function of this product: Q S \u003d 5 + 2P, where Q D and Q S are, respectively, the volume of demand and supply in million units per year, P is the price per day. units

a) Determine the equilibrium price and the equilibrium sales volume.

b) Suppose a tax is introduced on this product, paid by the seller in the amount of 1.5 den. units for a unit.

Determine the equilibrium sales volume and equilibrium prices for

the buyer (Pe +) and the seller (Pe +).

Decision

Hence: 7 - P \u003d - 5 + 2P, Pe \u003d 4.

Q D \u003d 7 - 4 \u003d 3, Q S \u003d - 5 + 2 × 4 \u003d 3, Q e \u003d 3.

b) Since the seller pays the tax, the price for it will be P - \u003d P + - 1.5.

Hence: Q D \u003d 7 - P +,

Q S \u003d - 5 + 2P - \u003d - 5 + 2 (P + - 1,5).

therefore: 7 - P e + \u003d - 5 + 2P e + - 3.

Hence: P e + \u003d 5; P e - \u003d 3.5; Q e \u003d 2.

In the figure, this position can be represented as follows:

Task 2

In the market for this product equilibrium was established at a price of 4 den. units per unit and sales volume of 18 thousand units per day. In this case, the coefficient of direct elasticity of demand (e D) is 0.05, and supply (e S): + 0.1.

Determine the equilibrium price of a product in the event of a 10% reduction in demand for it, based on the assumption that the supply and demand functions are linear.

Decision

In the case of linear supply and demand functions:

Q D \u003d a - bP; Q S \u003d m + nP,

hence,
a

As
a
then in equilibrium:

- 0.05 \u003d - b × 4/18, i.e. b \u003d 0.225.

0.1 \u003d n × 4/18, i.e. n \u003d 0.45.

Now we can determine the parameters a and m:

a \u003d 18 + 0.9 \u003d 18.9; m \u003d 18 - 1.8 \u003d 16.2.

Hence,

Q D \u003d 18.9 - 0.225P; Q S \u003d 16.2 + 0.45P.

With a decrease in demand by 10%, the equilibrium condition in the market for this product takes the form: 0.9Q D \u003d Q S; i.e

0.9 (18.9 - 0.225P) \u003d 16.2 + 0.45P.

Hence: P \u003d 1.23.

Tasks

1. The demand function for this product has the form: Q D \u003d 5 - P. Supply function: Q S \u003d - 2 + P.

Determine the equilibrium price and volume of sales, as well as surplus seller and buyer. Sellers received a subsidy of 3 den. units for the sold piece. Calculate the equilibrium volume, price and surplus. How was the subsidy distributed between sellers and buyers?

Determine the equilibrium price and sales volume. The state introduced a subsidy for consumers in the amount of 3 den. units per unit. Define new equilibrium sales volume and price. How much subsidy should the state allocate?

3. The demand function for this product has the form: Q D \u003d 12 - P. Supply function: Q S \u003d - 3 + 4P.

Determine the equilibrium price and sales volume. An excise tax has been introduced on sellers in the amount of 20% of sales. Define new equilibrium sales volume and price. How much tax will the state receive?

4. The coefficient of elasticity of demand at a price is - 0.2, and the coefficient of elasticity of supply at a price of + 0.2. In equilibrium, 10 units of good are sold at a price of 5 den. units

Determine the equilibrium volume and price when introducing a commodity tax of 1 den. units paid by manufacturers. (Assume that the supply and demand functions are linear).

5. In what situation will most of the tax burden fall on

manufacturers?

a) Q D \u003d 5 - 2P, Q S \u003d P + 1;

b) Q D \u003d 5 - P, Q S \u003d 1 + P;

c) Q D \u003d 5 - P, Q S \u003d 1 + 2P.

6. The demand function for this product has the form: Q D \u003d 8 - 2P. Suggestion function: Q S \u003d 4 + P.

Determine the amount of the production subsidy that needs to be allocated to producers so that the goods begin to spread as a “free good." How much product will be distributed?

7. The demand function for this product has the form: Q D \u003d 2 - 3P. Suggestion function: Q S \u003d - 0.5 + 2P.

Determine the public benefit arising from the production and sale of goods (the amount of surplus buyers and sellers).

8. The demand function for this product has the form: Q D \u003d 7 - 2P. Suggestion function: Q S \u003d P - 5.

Determine the equilibrium price and sales volume. Calculate the amount of commodity subsidies necessary to promote the product on the market and achieve a sales volume of 3 units.

9. The demand function for this product has the form: Q D \u003d 12 - P. Supply function: Q S \u003d - 3 + 4P.

Determine the equilibrium price and sales volume. Introduced tax on the manufacturer in the amount of 2 den. units per unit sold. Calculate the new equilibrium sales and price, as well as net social losses.

10. The demand function for this product has the form: Q D \u003d 12 - P. Supply function: Q S \u003d - 3 + 4P.

Determine the equilibrium price and sales volume. An excise tax has been introduced on buyers in the amount of 20% of sales. Define new equilibrium sales volume and price. How much tax will the state receive?

11. The demand function for this product has the form: Q D \u003d 5 - P. Supply function: Q S \u003d - 1 + P.

Determine the equilibrium price and sales, as well as surplus

sellers and buyers. A customer tax of 3 den. units per unit. Determine the equilibrium volume, price, surplus of sellers and buyers, as well as the net loss of society.

12. There are three demand functions and their corresponding functions.

offers:

a) Q D \u003d 12 - P, Q S \u003d - 2 + P;

b) Q D \u003d 12 - 2P, Q S \u003d - 3 + P;

c) Q D \u003d 12 - 2P, Q S \u003d - 24 + 6P.

The state introduces a subsidy to producers in the amount of 3 den. units for each piece. In which case will consumers receive most of the subsidies? Why?

13. The demand function for this product has the form: Q D \u003d 6 - 2P. Suggestion function: Q S \u003d - 2 + 2P.

Determine the excess demand at a price of 1 den. units Determine the equilibrium sales volume if the state sets a fixed price a) 1.5 den. units; b) 2.5 den. units

14. The demand function for this product: Q D \u003d 7 - P, the supply function of this product: Q S \u003d - 5 + 2P.

Determine the equilibrium price and the equilibrium sales volume. Suppose a fixed price is determined at the level of: a) 5 den. units for a unit; b) 3 den. units for a unit. Analyze the results. In which of the indicated cases will the volume of consumption be the highest?

15. The demand function for this product: Q D \u003d 16 - 4P, the supply function of this product Q S \u003d - 2 + 2P.

Find the equilibrium price and the equilibrium sales volume. Determine the sales tax rate at which the equilibrium sales volume will be 2 units.

16. The demand function for this product: Q D \u003d 7 - P, the supply function of this product: Q S \u003d - 5 + 2P.

At what tax rate (in den. Units per unit of goods) will the total amount of the tax charge be the maximum?

17. In a state of equilibrium in the market 120 pieces of goods A are for sale

at the price of 36 den. units It is known that the demand and supply function of this good are straightforward and at the same time e D \u003d - 0.75, and e S \u003d + 1.5.

Determine what the price of the good A will be equal to if its supply is reduced by 25%.

18. The market is characterized by the following supply and demand functions: Q D \u003d 12 - P; Q S \u003d 2P - 3.

Determine how much the equilibrium price will change if a 50% turnover tax (on sales) is introduced.

19. Suppose an excise tax on cigarettes of 25 den. units per pack, which caused a shift in the supply curve from S 1 to S 2, as shown in the figure. Answer the following questions:

a) What are the budget revenues from tax if the demand curve is D 1? D 2?

b) Explain why the equilibrium price of cigarettes does not increase by 25 den. units?

c) At what demand (D 1 or D 2) does the introduction of a tax lead to the greatest reduction in the number of smokers?

d) Suppose that instead of introducing a tax, the government decided to limit cigarette sales in the country to 4 million packs per period. Where it leads?

20. The gas demand function has the form: Q r D \u003d 3.75P n - 5P g, where P n, P g are the oil and gas prices, respectively, the gas supply function is: Q g S \u003d 14 + 2P g + 0.25P n

At what prices for these energy carriers will gas demand and supply be balanced at 20 units?

21. The product demand function has the form: Q D \u003d 5 - P, the product supply function has the form: Q S \u003d - 1 + 2P. Suppose that a quota for the production of this product is set at 2 thousand units.

What will be the consequences of this decision? Calculate the excess seller and buyer before and after the introduction of quotas.

22. In region I, the demand function for a certain product has the form: Q D1 \u003d 50 - 0.5P 1, the supply function: Q S1 \u003d - 10 + P 1, where Q D1, Q S1 are the demand volume and the supply volume in region I, respectively P 1 - market price in region I (den. Units / kg). For region II

demand function for the same product: Q D2 \u003d 120 - P 2, supply function: Q S2 \u003d - 20 + P 2.

a) Suppose the transportation of this product between two regions is prohibited.

Determine market prices, sales in each region. Determine the surplus of consumers, the surplus of producers for each region, the total surplus for each region, the total surplus for two regions.

b) Suppose transportation is permitted. Shipping costs are negligible. Define the same as in paragraph a. In addition, determine the volume of production in each region, the volume of traffic.

Who benefits from the lifting of the ban on transportation, who does not benefit from it? Does the overall benefit of lifting the ban increase or not?

c) Transportation is allowed. Transportation costs are 10 den. units per 1 kg transported from one region to another.

Define the same as in paragraph b.

d) Transportation is allowed. Shipping costs are negligible. The government of Region I has established an “export” duty of 10 den. units per 1 kg of exported products.

Define the same as in paragraph c. In addition, determine the total surplus of each region, including the tax received.

e) What will change if the duty is established not by the government of the first region, but by the government of the second region (an “import” duty of 10 den. units per 1 kg of imported products)?

23. Below are data on the volumes of supply and demand for various values \u200b\u200bof the price of this product:

a) If the price of the goods is 6 den. units per unit, how many units of goods

will be offered for sale? How many units will people want to buy? How much will the actual product be sold?

b) If the price of the goods rises to 12 den. units per unit, how many units will be offered for sale? What will be the volume of demand? How many units will be sold in this case?

c) Determine the equilibrium price and the equilibrium sales volume.

d) Display all options graphically.

24. The following supply and demand functions are given:

a) Q D \u003d 10 - P, Q S \u003d 2P - 2;

b) Q D \u003d 10 - P, Q S \u003d 2 + P;

c) Q D \u003d 10 - P, Q S \u003d 4 + 0.5P;

What situation corresponds to stable equilibrium, unstable equilibrium, uniform fluctuations in a cobweb-like model.

25. Four consumers are ready to purchase this product at individual prices equal to 8, 7, 5 and 2 den. units The supply prices of goods from four manufacturers (each produces a unit of this product) are: 6, 4, 3 and 2 den. units

What is the maximum possible total surplus? How much product and at what price will be produced? (The problem is solved graphically.)

26. The equilibrium in the market of this product was established at P \u003d 5 and Q \u003d 15. The coefficient of direct elasticity of demand at a price is –0.05, and the coefficient of direct elasticity of supply at a price of +0.2.

a) What will be the price of a product if demand for it increases by 15%, and supply - by 10%, provided that the demand and supply functions are linear.

b) Present the solution to the problem on the graph.

27. The supply and demand functions for this product are of the form: Q D \u003d 32 - 2P, Q S \u003d –2 + 3P.

a) What is the maximum amount of tax that can be collected if it is levied on each unit of goods sold? The tax is paid by the seller.

b) Build a Laffer curve.

28. The demand function for the services of a hairdresser has the form: Q D \u003d P 2 - 4P + 10. The function of the service offer: Q S \u003d 6P - P 2.

a) Determine the equilibrium values \u200b\u200bof price and volume.

b) Build the dependence of the quantity of service offer on its price.

c) Determine at what price the total income of the hairdresser will be maximum.

29. At price 3, the demand for good is 30, and at price 4 it is 12. The demand function is linear.

Determine the maximum bid price.

30. The demand function for cabbage has the form: Q Dt \u003d 200 - P t. The offer function has the form: Q St \u003d - 10 + 0.5 where - the price of cabbage in the period t, "expected" by farmers at the time they make decisions on the size of production. Suppose: \u003d P t-1.

a) Determine the sales volume and prices of cabbage in periods 1, 2, ..., 6, if P 0 \u003d 200.

31. The demand function for carrots has the form: Q Dt \u003d 200 - 0.5 . The offer function has the form: Q St \u003d - 10 + 0.5 where \u003d P t - 1.

a) Determine the sales volumes and prices of carrots in periods 1, 2, ..., 6, if P 0 \u003d 145.

b) Determine the equilibrium price and the equilibrium sales volume. Can this equilibrium be called stable? Make a drawing.

c) What, in your opinion, changes can occur in the mechanism of formation of expectations?

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.

a) Determine the sales volumes and prices of carrots in periods 1, 2, ..., 10, if P 0 \u003d P t -1 \u003d 250.

b) Depict the dynamics of price changes in the figure.

c) Determine the equilibrium price and the equilibrium sales volume. Can this equilibrium be called stable?

ECONOMIC THEORY

1. Demand for goods is represented by the equation P \u003d 5 - 0.2Q d, and supply P \u003d 2 + 0.3Q s. Determine the equilibrium price and the equilibrium amount of goods in the market. Find the elasticity of supply and demand at the equilibrium point.

Decision:

At the equilibrium point, Q d \u003d Q s. Therefore, 5 - 0.2Q d \u003d 2 + 0.3Q s.

We make calculations and determine the equilibrium price and the equilibrium amount of goods on the market: Q E \u003d 6; P E \u003d 3.8.

By the condition of the problem, P \u003d \u003d 5 - 0.2Q d, hence Q d \u003d 25 - 5P. The derivative of the demand function (Q d) / \u003d -5.

At the equilibrium point, P e \u003d 3.8. We define the elasticity of demand at the equilibrium point: Е d (3,8) \u003d - (3,8 / 6) · (-5) \u003d 3,15.

Similarly, the elasticity of a sentence at a point is determined: Е s \u003d - (P 1 / Q 1) · (dQ s p / dP), where dQ s p / dP is the derivative of the function of the sentence at the point P 1.

By the condition of the problem, P \u003d 2 + 0.3Q s, hence Q s \u003d 10P / 3 - 20/3. The derivative of the sentence function (Q s) / \u003d 10/3.

At the equilibrium point, P e \u003d 3.8. We calculate the elasticity of the proposal at the equilibrium point: Е s (3,8) \u003d - (3,8 / 6) · (10/3) \u003d 2,1.

Thus, the equilibrium price is P e \u003d 3.8; equilibrium amount - Q e \u003d 6; the elasticity of demand at the equilibrium point - E d (3,8) \u003d 3,15; elasticity of supply at the equilibrium point - E s (3,8) \u003d 2,1.

2. The demand function for this product is given by the equation Q d \u003d - 2P + 44, and the supply function Q s \u003d - 20 + 2P. Determine the elasticity of demand by price at the equilibrium point of the market for this product.

Decision:

At the equilibrium point, Q d \u003d Q s. We equate the supply and demand functions: - 2P + 44 \u003d -20 + 2P. Accordingly, P e \u003d 16. Substitute the obtained equilibrium price in the demand equation: Q d \u003d - 2 · 16 + 44 \u003d 12.

Substitute (for verification) a certain equilibrium price in the equation of supply: Q s \u003d - 20 + 2 · 16 \u003d 12.

Thus, in the market for this product, the equilibrium price (P e) will be 16 monetary units, and 12 units of the product (Q e) will be sold at this price.

The elasticity of demand at a point is determined by the formula of point price elasticity and is equal to: Е d \u003d - (P 1 / Q 1) · (ΔQ d p / ΔP), where ΔQ d p / ΔP is the derivative of the demand function at point P 1.

Since Q d \u003d -2P + 44, the derivative of the demand function (Q d) / \u003d -2.

At the equilibrium point, P e \u003d 3. Therefore, the price elasticity of demand at the equilibrium point of the market for a given product will be: E d (16) \u003d - (16/12) · (-2) \u003d 2.66.

3. Demand for goods X is given by the formula Q d \u003d 20 - 6P. The increase in the price of goods Y caused a change in demand for goods X by 20% at each price. Define a new function of demand for goods X.

Decision:

By the condition of the problem, the demand function: Q d 1 \u003d 20 - 6P. The increase in the price of goods Y determines the change in demand for goods X by 20% at each price. Accordingly, Q d 2 \u003d Q d 1 + ΔQ; ΔQ \u003d 0.2Q d 1.

Thus, the new demand function for goods X: Q d 2 \u003d 20 - 6P + 0.2 (20 - 6P) \u003d 24 - 4.8P.

4. Supply and demand for goods are described by the equations: Q d \u003d 92 - 2P, Q s \u003d -20 + 2P, where Q is the quantity of this product, P is its price. Calculate the equilibrium price and quantity of goods sold. Describe the consequences of setting a price of 25 monetary units.

Decision:

At the equilibrium point, Q d \u003d Q s. Accordingly, 92 - 2P \u003d -20 + 2P. We will make calculations and determine the equilibrium price and the equilibrium amount: P e \u003d 28; Q e \u003d 36.

When the price is set at 25 monetary units, a shortage is formed in the market.

Define the size of the deficit. When P const \u003d 25 monetary units, Q d \u003d 92 - 2 · 25 \u003d 42 units. Q s \u003d -20 + 2 · 25 \u003d 30 units.

Therefore, when the price is set at 25 monetary units, the deficit in the market of this product will be Q s - Q d \u003d 30 - 42 \u003d 12 units.

5. Given the functions of supply and demand:

Q d (P) \u003d 400 - 2P;

Q s (P) \u003d 50 + 3P.

The government introduced a fixed price for goods at the level of 50 thousand rubles. for a unit. Calculate the market deficit.

Decision:

The equilibrium price is established under the condition Q d \u003d Q s. By the condition of the problem, P const \u003d 50 thousand rubles.

Define the volume of supply and demand at P \u003d 50 thousand rubles. for a unit. Accordingly, Q d (50) \u003d 400 - 2 · 50 \u003d 300; Q s (50) \u003d 50 + 2 · 50 \u003d 150.

Thus, when the government sets a fixed price for goods at the level of 50 thousand rubles. per unit, the deficit in the market will be: Q d - Q s \u003d 300 - 150 \u003d 250 units.

6. Demand for goods is represented by the equation P \u003d 41 - 2Q d, and supply P \u003d 10 + 3Q s. Determine the equilibrium price (P e) and the equilibrium quantity (Q e) of the product on the market.

Decision:

Equilibrium condition in the market: Q d \u003d Q s. We equate the supply and demand functions: 41 - 2 Q d \u003d 10 + 3Q s. We make the necessary calculations and determine the equilibrium amount of goods on the market: Q e \u003d 6,2. We determine the equilibrium price of the goods in the market by substituting the resulting equilibrium quantity of the goods in the supply equation: P \u003d 10 + 3Q s \u003d 28.6.

We substitute (for verification) the obtained equilibrium quantity of the goods in the demand equation P \u003d 41 - 2 · 6.2 \u003d 28.6.

Thus, in the market for this product, the equilibrium price (P e) will be 28.6 monetary units, and 6.2 units of the product (Q e) will be sold at this price.

7. The demand function has the form: Q d \u003d 700 - 35Р. Determine the elasticity of demand at a price equal to 10 monetary units.

Decision:

The elasticity of demand at the equilibrium point is determined by the formula of point price elasticity and is equal to: Е d p \u003d - (P 1 / Q 1) · (ΔQ d p / ΔP), where ΔQ d p / ΔP is the derivative of the demand function.

We perform the calculations: ΔQ d p / ΔP \u003d (Q d) /? \u003d 35. Let us determine the elasticity of demand at a price equal to 10 monetary units: E d p \u003d 10 / (700-35 · 10) · 35 \u003d 1.

Therefore, the demand for this product at a price equal to 10 monetary units is elastic, so 1< Е d p < ∞ .

8. Calculate the elasticity of demand for a product by income, if with an increase in income from 4,500 rubles to 5,000 rubles per month, the volume of purchases of goods decreases from 50 to 35 units. Round up the answer to the third character.

Decision:

We determine the income elasticity of demand by the following formula: E d I \u003d (I / Q) × (ΔQ / ΔI) \u003d (4500/50) × (15/500) \u003d 2.7.

Therefore, this product for these customers has the status of a normal or high-quality product: the coefficient of elasticity of demand for a product by income (E d I) has a positive sign.

9. The demand equation has the form: Q d \u003d 900 - 50P. Determine the maximum demand (market capacity).

Decision:

The maximum market capacity can be defined as the market volume of a given product (Q d) when the price of this product is zero (P \u003d 0). A free member in the linear equation of demand characterizes the value of maximum demand (market capacity): Q d \u003d 900.

10. The function of market demand Q d \u003d 10 - 4P. The increase in household incomes led to an increase in demand by 20% at each price. Define a new demand function.

Decision:

Based on the conditions of the problem: Q d 1 \u003d 10 - 4P; Q d 2 \u003d Q d 1 + ΔQ; ΔQ \u003d 0.2Q d 1.

Therefore, the new demand function Q d 2 \u003d 10 - 4P + 0.2 (10-4P) \u003d 12 - 4.8P.

11 . The price of the goods changes as follows: P 1 \u003d $ 3 .; P 2 \u003d $ 2.6. The range of changes in the volume of purchases in this case is: Q 1 \u003d 1600 units; Q 2 \u003d 2000 units

Determine E d p (price elasticity of demand) at the equilibrium point.

Decision:

To calculate the price elasticity of demand, we use the formula: E d P \u003d (P / Q) · (ΔQ / ΔP). Accordingly: (3/1600) · (400 / 0.4) \u003d 1.88.

Demand for this product is elastic, since E d p (price elasticity of demand) at the equilibrium point is more than one.

12. Refusing to work as a joiner with a salary of 12,000 den. units per year or work as a referent with a salary of 10,000 den. units per year, Pavel entered college with an annual tuition of $ 6,000. units

Determine what the opportunity cost of his decision in the first year of study is, if Pavel has the opportunity to work in a store for 4,000 den in his free time. units in year.

Decision:

The opportunity cost of Paul’s tuition is equal to the cost of annual college fees and missed opportunities. It should be borne in mind that if there are several alternative options, then the maximum cost is taken into account.

Therefore: 6,000 den. units + 12,000 den. units \u003d 18,000 den. units in year.

Since Pavel receives additional income, which he could not have received if he had worked, this income must be deducted from the opportunity cost of his decision.

Consequently: 18,000 den. units - 4,000 den. units \u003d 14,000 den. units in year.

Thus, the opportunity cost of Paul’s decision in the first year is 14,000 den. units