Demand. Demand functions. Demand law. Equilibrium price and equilibrium volume Direct and inverse demand functions

Section II. FOUNDATIONS OF THE THEORY OF MICROECONOMICS

This section provides an introduction to the study of microeconomics. The section provides general concepts that describe behavior in a market economy, without which it is impossible to study an advanced course in 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 the study of the foundations of the behavior of the subjects of the modern market economy.

Chapter 5. DEMAND: SUPPLY AND MARKET BALANCE

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

The purpose of the analysis of this chapter is the mechanism of interaction between supply and demand, i.e. the supply-demand 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 in the conditions of modern methods of regulating the 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 at work in the economy and thus reflects important aspects of real life.1

Demand. Demand functions. Demand law

Demands in a market economy appear in the form of demand. Market demand is an indirect reflection of people's need for a given product or service.

Human needs are known to be unlimited. Can we talk about unlimited demand? What is the difference between these concepts? The fact is that demand is a form of expression of the need presented on the market and secured with money, that is, demand is a solvent need. It is not enough to desire to purchase a product, it is necessary that the consumer has a certain amount of money in order to realize his desire. The market does not respond to insolvent needs. More precisely, the category of demand can be expressed by the term magnitude or demand volume.

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

It is important to distinguish between the terms "demand volume" and "actual purchase volume". The volume (value) of demand is determined only by the buyer, and the volume of the actual purchase is determined by both the buyer and the seller. For example, government price caps can cause significant increases in demand. At the same time, the volume of sales (“volume of actual purchase”) is likely to be low as a result of the manufacturer's disinterest in selling at fixed prices.

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

Product price P (price)

Consumer income I (income)

Tastes, fashion T (tastes)

Prices for related goods: interchangeable (substitutes) P S or complementary (complements) P ​​C

Number of buyers N

Expectation of future prices and incomes W

Other factors X

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

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

Attempts to investigate the nature of the change in the value of demand Q d under the influence of all factors at once will not give a positive result. In this case, in order to identify the nature of the change in the value of demand Q d, it is first necessary to fix the values ​​of all variables except one and to study the relationship between Q d and this variable. This method means that we investigate the dependence of the quantity of demand on each variable. all other things being equal.

The amount of demand for a product, first of all, depends on the price. If all factors, except for the price, are taken as unchanged for a given period, then price demand function will look:

The inverse dependence of the price on the amount of demand is called inverse demand function and has the form:

All other things being equal, a decrease in price leads to an increase in the amount of goods purchased by buyers; an increase in price causes a reverse reaction: the purchase of goods is reduced. Thus, the specified property of demand reflects the inverse relationship between the change in price and the amount of demand. The inverse relationship between the price and the amount of demand (other parameters are unchanged) is universal and reflects the operation of one of the fundamental economic laws - the law of demand.

Antoine Augustin Cournot(1801-1877) - the creator of the mathematical theory of demand. A. Cournot was above all a talented mathematician, but he was bored in the world of pure mathematics, and he tried with its help 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, A 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 area of ​​science grew - mathematical economics.

It was A.Courno who was the first to deeply analyze the relationship between demand and prices in various market situations. This made it possible for him to formulate the law of demand and bring economics close to understanding the concept of “elasticity of demand” (A. Marshall picked up the ideas of A. Cournot and brought them to their logical conclusion). Cournot was able to mathematically rigorously prove that the highest sales proceeds are often provided by far from the highest price.

Why does demand behave this way? This happens for a number of reasons that argue the law of demand and take into account the following circumstances:

Common sense and life experience directly influence 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 deliver less and less pleasure, no matter how much the price decreases. After a certain level of saturation of the need, the satisfaction received from the product or service begins to decrease. Economists call this effect the law of diminishing marginal utility. The decline in marginal utility explains why low prices stimulate demand. Goods sold at a high price are usually not bought for the future 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 action of the law of demand can be explained based on two interrelated effects - income effect and substitution effect.

Obviously, at a lower price, the buyer can afford to buy more of a given product without giving up purchasing other products. He feels richer because a price cut increases his real purchasing power, or real income with the same value of his monetary income. it income effect.

Income effect(as a result of price changes) - 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 extent of the income effect depends mainly on how much of the income is spent on the purchase of a given product. The more income is spent on a product, the more the effect of price increases on the consumer's real income will influence and the more consumption will decrease.

On the other hand, the consumer is inclined to replace more expensive goods with cheaper analogs, which leads to an increase in the value of demand for these goods. it substitution effect.

Substitution effect- the desire of consumers to buy a product in greater quantities when its relative price decreases (replacing this product with others) and to 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 goods.

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

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

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

Table 5.1

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

GUIDELINES

Example 1. There are three demand functions and their corresponding supply functions:
a) QD = 12 - P, Qs = - 2 + P;
b) QD = 12 - 2P, Qs = - 3 + P;
c) QD = 12 - 2P, Qs = - 24 + 6P.
The state introduces a subsidy to producers in the amount of 3 den. units for each piece. When will consumers receive most of the subsidy? Why?
Solution:
Let's determine the equilibrium price and volume of sales in each case. To do this, let's equate the supply and demand function:
a) 12 - P = -2 + P => P = 7, Q = 5;
b) 12 - 2P = -3 + P => P = 5, Q = 2;
c) 12 - 2P = -24 + 6P => P = 4.5, Q = 3.
If a producer subsidy is introduced, sellers will be able to reduce the bid price by the amount of the subsidy. We express the offer price taking into account the subsidy:
a) Ps = Qs + 2 - 3 = Qs - 1;
b) Ps = QS + 3 -3 = Qs;
c) Ps = QS / 6 + 4 - 3 = Qs / 6 + 1.
Hence the new suggestion function:
a) Qs = 1 + P;
b) Qs = P;
c) Qs = - 6 + 6P.
We find a new state of equilibrium:
a) 12 - P = 1 + P => P = 5.5; Q = 6.5;
b) 12 - 2P = P => P = 4, Q = 4;
c) 12 - 2P = -6 + 6P => P = 2.25, Q = 7.5.
Answer: Thus, consumers will receive most of the subsidies in option c) supply and demand functions: the price will decrease by 2.25 den. units, i.e. by 50% of the original value, while the volume of sales will grow 2.5 times.
Example 2. The equilibrium grain price on the world market is P = $ 1.5 per pound. Q = 720 million pounds of grain is sold annually. The price elasticity of grain demand is EP (D) = -0.8. Determine the linear function of grain demand.
Solution:
It should be noted that the coefficient of price elasticity of demand is the tangent of the slope of the demand graph to the abscissa axis. Considering the above, we will draw up a linear equation for the dependence of demand on price. The linear relationship model looks like this:
QD = a + EP (D) × P,
where QD is demand, P is price, EP (D) is the linear coefficient of price elasticity of demand.
Knowing that P = $ 1.5 per pound, q = 720 units. (million pounds), EP (D) = -0.8, we find the unknown parameter in this model:
720 = a - 0.8 × 1.5; a = 721.2.
Thus, the model of the dependence of demand on price is as follows: QD = 721.2 - 0.8P.
Example 3. The cross-elasticity between the demand for kvass and the price of lemonade is 0.75. What products are we talking about? If the price of lemonade increases by 20%, how will the demand for kvass change?
Solution:
Kvass and lemonade are interchangeable goods, since the coefficient of cross-elasticity of demand EA, B has a positive value (0.75).
Using the formula for the cross-elasticity coefficient EA, B, we will determine how the demand for kvass will change with an increase in the price of lemonade by 20%.
If we take the change in demand for kvass as x, and the change in the price of lemonade for y, then we can write the equation EA, B = x / y; whence x = EA, B × y or
x = 0.75y = 0.75 × 20% = 15%.
Thus, with an increase in the price of lemonade by 20%, the demand for kvass will increase by 15%.
Example 4. The functions of supply and demand for goods are given:
QD = 150 - 3P, QS = - 70 + 2P.
The state introduced a commodity tax in the amount of 7.5 USD. from each unit of products sold. Determine the equilibrium price and equilibrium volume before and after the introduction of the tax. How much of the tax will be paid by the manufacturer and the buyer?
Solution:
The initial market equilibrium will be at point E (Pe, Qe), where QD = QS. 150 - 3P = -70 + 2P; 220 = 5P; Pe = 44 c.u.
Substitute the equilibrium price (Pe) into the supply or demand function and find the equilibrium sales volume Qe = -70 + 2 × 44 = 18 units.
After the introduction of the tax, the market equilibrium will move to point E1 (the intersection point of the old demand function Qd = 150 - 3P and the new supply function QS1 = - 70 + 2 (P - t) = -70 + 2P - 15 = -85 + 2P.
Thus, the new equilibrium is calculated as follows:
QD = QS1: 150 - 3P = -85 + 2P; 235 = 5P; Pe1 = 47 USD
The new equilibrium sales volume is Qe1 = 150 - 3 × 47 = 9 units.
The amount of tax paid by the buyer:
tD = Pe1 - Pe = 47 - 44 = 3 c.u.
The amount of tax paid by the seller:
tS = Pe - (Pe1- t) = 44 - (47 - 7.5) = 4.5 c.u.
Since demand is more elastic than supply, in this case the tax burden will fall more on the shoulders of the seller than the buyer.

ECONOMIC THEORY

1. The demand for the product is represented by the equation P = 5 - 0.2Q d, and the supply P = 2 + 0.3Q s. Determine the equilibrium price and the equilibrium quantity of the product on the market. Find the elasticity of supply and demand at the equilibrium point.

Solution:

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

Let's make calculations and determine the equilibrium price and equilibrium quantity of goods on the market: Q E = 6; P E = 3.8.

By the condition of the problem, P = = 5 - 0.2Q d, hence Q d = 25 - 5P. Derivative of the demand function (Q d) / = -5.

At the equilibrium point P e = 3.8. Let's define the elasticity of demand at the equilibrium point: E d (3.8) = - (3.8 / 6) · (-5) = 3.15.

The elasticity of the sentence at the point is determined in a similar way: Е s = - (P 1 / Q 1) · (dQ s p / dP), where dQ s p / dP is the derivative of the supply function at the point Р 1.

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

At the equilibrium point P e = 3.8. Let's calculate the elasticity of the supply at the equilibrium point: E s (3.8) = - (3.8 / 6) · (10/3) = 2.1.

Thus, the equilibrium price is P e = 3.8; equilibrium amount - Q e = 6; elasticity of demand at the equilibrium point - E d (3.8) = 3.15; supply elasticity at the equilibrium point - E s (3.8) = 2.1.

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

Solution:

At the equilibrium point Q d = Q s. Let's equate the supply and demand functions: - 2P + 44 = -20 + 2P. Accordingly, P e = 16. Substitute the resulting equilibrium price into the demand equation: Q d = - 2 · 16 + 44 = 12.

Substitute (for verification) a certain equilibrium price in the supply equation: Q s = - 20 + 2 16 = 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 = - (P 1 / Q 1) · (ΔQ d p / ΔP), where ΔQ d p / ΔP is the derivative of the demand function at point Р 1.

Since Q d = -2Р + 44, the derivative of the demand function (Q d) / = -2.

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

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

Solution:

According to the condition of the problem, the demand function: Q d 1 = 20 - 6P. An increase in the price of product Y causes a change in demand for product X by 20% at each price. Accordingly, Q d 2 = Q d 1 + ΔQ; ΔQ = 0.2Q d 1.

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

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

Solution:

At the equilibrium point Q d = Q s. Accordingly, 92 - 2P = -20 + 2P. Let's make calculations and determine the equilibrium price and equilibrium quantity: P e = 28; Q e = 36.

When the price is set at 25 currency units, a deficit is formed in the market.

Let's determine the size of the deficit. With P const = 25 monetary units, Q d = 92 - 2 · 25 = 42 units. Q s = -20 + 2 25 = 30 units.

Consequently, when the price is set at 25 monetary units, the deficit in the market for this product will be Q s - Q d = 30 - 42 = 12 units.

5. The functions of supply and demand are given:

Q d (P) = 400 - 2P;

Q s (P) = 50 + 3P.

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

Solution:

The equilibrium price is set under the condition Q d = Q s. According to the condition of the problem, P const = 50 thousand rubles.

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

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

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

Solution:

Equilibrium condition in the market: Q d = Q s. Let us equate the supply and demand functions: 41 - 2 Q d = 10 + 3Q s. Let's make the necessary calculations and determine the equilibrium amount of goods on the market: Q e = 6.2. Let us determine the equilibrium price of a product in the market by substituting the obtained equilibrium quantity of a product into the supply equation: P = 10 + 3Q s = 28.6.

Let us substitute (for verification) the obtained equilibrium quantity of goods in the equation of demand P = 41 - 2 · 6.2 = 28.6.

Thus, in the market for this product, the equilibrium price (P e) will be 28.6 currency 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 = 700 - 35Р. Determine the elasticity of demand at a price equal to 10 currency units.

Solution:

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

Let's make calculations: ΔQ d p / ΔP = (Q d) /? = 35. Let's define the elasticity of demand at a price equal to 10 monetary units: E d p = 10 / (700-35 10) 35 = 1.

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

8. Calculate the income elasticity of demand for a product if, with an increase in income from RUB 4,500 to RUB 5,000 per month, the volume of purchases of a product decreases from 50 to 35 units. Round your answer to the third decimal place.

Solution:

We define the income elasticity of demand using the following formula: E d I = (I / Q) × (ΔQ / ΔI) = (4500/50) × (15/500) = 2.7.

Consequently, for these buyers, this product has the status of a normal or high-quality product: the coefficient of the elasticity of demand for the product with respect to income (E d I) has a positive sign.

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

Solution:

The maximum market capacity can be defined as the market volume for a given product (Q d) when the price for this product is equal to zero (P = 0). The free term in the linear equation of demand characterizes the value of the maximum demand (market capacity): Q d = 900.

10. Market demand function Q d = 10 - 4Р. The increase in household income led to an increase in demand by 20% at each price. Define a new demand function.

Solution:

Based on the condition of the problem: Q d 1 = 10 - 4P; Q d 2 = Q d 1 + ΔQ; ΔQ = 0.2Q d 1.

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

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

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

Solution:

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

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

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

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

Solution:

The opportunity cost of Paul's education is equal to the cost of an annual college tuition and the cost of lost opportunity. 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 = 18,000 den. units in year.

Since Pavel receives additional income that he could not receive if he worked, this income must be deducted from the opportunity cost of his decision.

Therefore: 18,000 den. units - 4,000 den. units = 14,000 den. units in year.

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

2-1p. Population demand function for this product: Qd = 7-P. Suggestion function: Q s = -5 + 2P,where Qd - demand volume in million units per year; Qs - supply volume in million units per year; R - price in thousands of rubles. Plot supply and demand graphs for a given product, plotting the quantity of the product on the abscissa (Q) and on the ordinate - the unit price (R).

Solution

Since the given functions reflect a linear relationship, each of the graphs can be plotted using two points.

2-2p. Determine the market demand function based on individual demand data:

Q (1) = 40-8P at P ≤ 5 and 0 at P> 5,

Q (2) = 70-7P at P ≤ 7 and 0 at P> 7,

Q (3) = 32-4P at P ≤ 8 and 0 at P> 8.

a) Derive the equation of the demand curve analytically.

b) Which of these consumer groups do you think is richer? Is it possible to draw an unambiguous conclusion?

Solution

a) Q = Q (1) + Q (2) + Q (3) = 142-19P at 0 ≤ P ≤ 5,

Q = Q (2) + Q (3) = 102-11Р at 5 < Р ≤ 7 ,

Q = Q (3) = 32-4P at 7 < P ≤ 8 ,

Q = 0 at P> 8.

b) The third group of consumers agree to pay the highest prices. For example, for P = 7.5 the first two groups will stop buying, and the buyers of the third group will buy 2 units. (32-4x7.5 = 2). But it is impossible to make an unambiguous conclusion that the third group includes the richest buyers, since we do not know either their income or other direct and indirect signs of wealth.

2-3p. The demand for VCRs is described by the equation:

Qd = 2400-100R, and the offer of video recorders - by the equation Qs = 1000 + 250Р, where Q - the number of VCRs bought or sold during the year; R - the price of one VCR (in thousand rubles).

a) Determine the equilibrium parameters in the VCR market.

b) How many VCRs would have been sold at a price of 3000 rubles?

c) How many VCRs would have been sold at a price of 5,000 rubles?

Solution

a) In order to determine the parameters of equilibrium, let us equate the volume of demand to the volume of supply:

Qd = Qs, or 2400-100P = 1000 + 250P.

Solving the equation, we find the equilibrium price:

1400 = 350P; Pe = 4000 rub.

Substituting the found price into the equation describing the demand, or into the equation describing the supply, we find the equilibrium quantity Qe.

Qe = 2400-100 x 4 = 2000 PCS. in year.

b) To determine how many VCRs will be sold at a price of 3000 rubles (i.e. at a price below the equilibrium one), you need to substitute this price value in both the demand equation and the supply equation:

Qd = 2400 - 100 NS 3 = 2100 PCS. in year;

Qs = 1000 + 250 NS 3 = 1750 PCS. in year.

This shows that at a price below the equilibrium price, consumers will want to buy more VCRs than manufacturers will agree to sell (Qd> Qs). In other words, consumers will want to buy 2,100. VCRs, but they can buy exactly as much as the sellers will sell them, i.e. 1750 pcs. This is the correct answer.

c) Substitute the price of 5,000 rubles into each of these equations:

Qd = 2400 - 100 NS 5 = 1900 PCS. in year;

Qs = 1000 + 250 NS 5 = 2250 PCS. in year.

If the price is higher than the equilibrium price, the producers will want to sell 2,250 units. VCRs, but consumers will only buy 1,900. VCRs, therefore, only 1900 pieces. VCRs and will be sold at a price of 5,000 rubles.

Answer: a) equilibrium parameters: Pe = 4000 rubles, Qe = 2000 PCS. in year.

b) at P = RUB 3000 will be sold Q = 1750 PCS. in year.

c) at P = RUB 5000 will be sold Q = 1900 PCS. in year.

2-4p. The gas demand function has the form: Qd g = 3.75P n -5P g, and the function of its proposal is: Qs g = 14 + 2P g + 0.25P n,where R n, R g- oil and gas prices, respectively.

Define:

a) at what prices for these energy carriers the volumes of gas demand and supply will be equal to 20 units;

b) by what percentage the volume of gas sales will change with an increase in oil prices by 25%.

Solution

A) To determine at what prices for these energy carriers the volumes of gas demand and supply will be equal to 20 units. we solve the system of equations:

3.75R n -5R g = 20

14 + 2P g + 0.25P n = 20Þ R n = 8; P g = 2.

Since from the first equation P n = (20 + 5P g) / 3.75, substitute this expression into the second equation.

14 + 2P g +0.25 (20 / 3.75) +0.25 (5P g / 3.75) = 20,

2P g +0.25 (5P g / 3.75) = 20-14-0.25 (20 / 3.75),

2P g + 0.33P g = 6-1.33,

2.33P g = 4.67,

P g = 2.

R n = (20 + 5 NS 2)/3,75=8.

b) If the price of oil rises to 10 den. units, then the equilibrium in the gas market will be subject to the following equality:

3,75 NS 10 - 5P g = 14 + 2P g + 0.25 NS 10 Þ

37.5-5P g = 14 + 2P g + 2.5Þ

-5P g - 2P g = 14 + 2.5-37.5Þ

-7P g = -21,

P g = 3, Q g = 37.5 - 5 NS 3 = 22,5.

those. gas sales will increase by 12,5%.

Answer: a) if the volumes of demand and supply of gas are equal, 20 units. oil and gas prices will be equal respectively R n = 8; P g = 2.

b) with an increase in the price of oil by 25% , the volume of gas sales will increase by 12,5%.

2-5p. There are three sellers and three buyers in the real estate market. The functions of the offer at the price of sellers are known:

Qs 1 = 2P-6; Qs 2 = 3P-15; Qs 3 = 5P.

and the demand function for the buyers' price:

Qd 1 = 12-P; Qd 2 = 16-4P; Qd 3 = 10-0.5R.

Determine: the parameters of the market equilibrium, as well as the volume of the transaction of each participant in the trade at the equilibrium price.

Provide a graphical and analytical solution.

1. Direct and inverse demand functions

Condition: It is known that consumers are ready to purchase 20 units of a good for free; with each increase in price by 1, the amount of demand decreases by 2 units. Write down the forward and backward views of the demand function describing the given situation.

Solution: Since a price change by 1 always changes Q by 2 units, we are dealing with a linear demand function. (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, a direct linear demand function is written as: Q d (P) = a - bP, where a and b are the coefficients that we need to find. We know that for P = 0 the value of demand is 20 units, from which it follows that a = 20... Moreover, the coefficient b = 2... Thus, the direct demand function can be written as Qd(P) = 20 - 2P.

To obtain the inverse demand function, we express the price from the previously obtained expression: Pd(Q) = 10 - 0.5Q.

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

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

2. Reconstruction of the linear demand function

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

Solution: In general, the linear demand function can be written as Q d (P) = a - bP, where a and b are the coefficients that we need to find. Since we have two unknowns, 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 the given demand function.

When P 0 = 10, consumers are ready to buy 5 units of the good, that is, the value of demand Q 0 is equal to 5 - these are the coordinates first point... If the price rises 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). Let's write down the system of equations:

5 = a - b * 10

3 = a - b * 15

The system is solved when a = 9 and b = 0.4.

Answer: Q d (P) = 9 - 0.4P.

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

3. Plotting a linear demand function

Condition: The functions of demand for some good are given: Q d1 (P) = 20 - 2P and P d2 (Q) = 5 - Q. Let the demand expressed by the first function decreased by 5 units. at each price level, and demand, expressed by the second function, increased by 60%. Plot the original and modified demand functions.

Solution: To begin with, we write down the demand functions in direct form, that is, we express Q through P: Q d1 (P) = 20 - 2P and Q d2 (Q) = 5 - P. To construct any linear function, it is enough to find the coordinates two points. The further 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 Q and P axes. To do this, we substitute in each function first Q = 0, and then P = 0. This principle works well when constructing linear demand functions; in other cases, its application can be limited:

Now we will find new demand functions, calculated taking into account the changes. The first demand decreased by 5 units. at each price value, that is Q new d1 (P) = Q d1 (P) - 5: Q new d1 (P) = 15 - 2P. On the graph, the new demand curve is obtained by shifting the original curve to the left for 5 units. - this is red line D 3... The second demand increased by 60% at each price level. So, with P 1 = 5 and Q 1 = 0, no change will occur, since 60% of 0 is 0. At the same time, with P 2 = 0 and Q 2 = 5, the change in demand will be maximum and will be 0.6 * 5 = 3 units Thus, the new demand function will be Q new d2 (P) =Q d2 (P) +Q d2 (P) * 0.6:Q new d2 (P) =8 - 1.6P. Let us check the result obtained by substituting the already known points (0.5) and (8.0) into the function. Everything is being done, this demand is displayed on the graph blue line D 4.

 

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