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

The equilibrium price is the price at which the volume of demand in the market is equal to the volume of supply. It is expressed as Qd (P) = Qs (P) (see main market parameters).

Service purpose... This online calculator is aimed at solving and checking the following tasks:

Equilibrium parameters of a given market (determination of the equilibrium price and equilibrium volume);
Coefficients of the direct elasticity of supply and demand at the equilibrium point;
Consumer and seller surpluses, net social gain;
The government introduced a per unit subsidy for each unit sold in the amount of N rubles;
The amount of the subsidy allocated from the state budget;
The government introduced a sales tax on each unit sold in the amount of N rubles;
Describe the consequences of the government's decision to fix the price by N above (below) the equilibrium one.

Instruction. Enter supply and demand equations. The obtained solution is saved in a Word file (see an example of finding the equilibrium price). Also presented graphic solution tasks. Qd - demand function, Qs - supply function

An example. The demand function for this product is Qd = 200–5P, the supply function is Qs = 50 + P.

Determine the equilibrium price and equilibrium sales.
Suppose that the city administration decided to set a fixed price at the level: a) 20 den. units per piece, b) 30 den. units a piece.
Analyze the results obtained. How will this affect consumer and producer behavior? The solution is presented graphically and analytically.

Solution.
Let's find the equilibrium parameters in the market.
Demand function: Qd = 200 -5P.
Suggestion function: Qs = 50 + P.
1. Equilibrium parameters of this market.
In equilibrium Qd = Qs
200 -5P = 50 + P
6P = 150
P equals = 25 rubles. is the equilibrium price.
Q is equal to 75 units. - equilibrium volume.
W = P Q = 1875 rubles. - the seller's income.

Consumer surplus shows how much better individuals live on average.
Consumer surplus(or gain) is the difference between the maximum price that he is willing to give for the product and the one that he really pays. If we add up the surplus of all consumers who purchase this product, then we get the total surplus.
Producer surplus(gain) is the difference between market price and the minimum price for which manufacturers are willing to sell their goods.
Seller's surplus (P s P 0 E): (P equals - Ps) Q equals / 2 = (25 - (-50)) 75/2 = 2812.5 rubles.
Buyer's surplus (P d P 0 E): (Pd - P is equal) Q is equal / 2 = (40 - 25) 75/2 = 562.5 rubles.
Net social gain: 2812.5 + 562.5 = 3375
The knowledge of surpluses is widely used in practice, for example, when distributing the tax burden or subsidizing industries and firms.

2) Suppose that the city administration decides to set a fixed price at 20 den. units a piece
P fix = 20 rubles.
Demand volume: Qd = 200 -5 20 = 100.
Supply volume: Qs = 50 + 1 20 = 70.
After fixing the price, the volume of demand decreased by 25 units. (75 - 100), and the deficit of producers decreased by 5 units. (70 - 75). There is a shortage of goods in the market in the amount of 30 pcs. (70 - 100).

Suppose that the city administration decides to set a fixed price of 30 den. units a piece.
P fix = 30 rubles.
Demand volume: Qd = 200 -5 30 = 50.
Supply volume: Qs = 50 + 1 30 = 80.
After fixing the price, the volume of demand increased by 25 units. (75 - 50), and the surplus of producers increased by 5 units. (80 - 75). There is a surplus of goods in the market in the amount of 30 pieces. (80 - 50).

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. Build supply and demand graphs of this product, plotting the quantity of goods on the abscissa axis (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

but) 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 most high 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 draw 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 PC. 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 PC. in year;

Qs = 1000 + 250 NS 3 = 1750 PC. 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 PC. in year;

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

b) at P = 3000 rub. will be sold Q = 1750 PC. in year.

c) at P = 5000 rub. will be sold Q = 1900 PC. 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 will the volume of gas sales change with an increase in the price of oil 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:

3.75P n -5P 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 at 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.

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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 inverse form of the function, on the contrary, is the dependence of the price on the demand value - Pd (Q)).

IN general view 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 the 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.

With 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 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 build 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 if we find the coordinates of the intersection of our lines with the Q and P axes. To do this, we substitute Q = 0 into each function, and then P = 0. This principle works well when constructing linear functions demand, in other cases its application may 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 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.

ANSWER: You must enter the number 1.

Task number 4.

The demand function is given by the equation Qd = 50 - 2P,

and sentences Qs = 5 + 3Р. Identify consumer surplus.

Quantity Q

Answer options:

Consumer surplus is the difference between the maximum price that the consumer is willing to pay for a unit of goods, and the actual value of the price that he actually paid. The area of the triangle bounded by the demand curve and the equilibrium market price is equal to consumer surplus... Therefore, you need to find the sides AB and AC.

Qs = Qd or 50 - 2P = 5 + 3P, hence 5P = 45 or P = 9,

those. the equilibrium price (or point A) is 9.

Qd = 50 - 2P = 50 - 2 * 9 = 50 - 18 = 32, that is, AC = 32

We find point B by equating Qd = 0 or 50 - 2P = 0, hence P = 25 or point B = 25

AB = 25 - 9 = 16

Area of triangle ABC = ½ × 32 × 16 = 256 Answer : 256

ANSWER: option 2, i.e. 256

Task number 5

The figure shows the consumer indifference curve and his budget line. Write the budget line equation if the price of item Y is equal to P = 6 rubles

Answer options:

1) Qy = 10 - 1.5 Qx

2) Qy = 15 - 0.67Qx

3) Qy = 10 - 0.67 Qx

4) Qy = 15 - 1.5 Qx

SOLUTION:

An indifference curve is a curve showing different combinations of 2 products that have the same utility for the consumer.

The budget line is a curve that shows the different combinations of the quantities of two goods that a consumer can buy based on the budget allocated for the acquisition of these goods and their prices. At the point where the budget line touches the indifference curve, the consumer's optimum is determined, but for this task the point of contact does not matter.

If the consumer spends all the money only on the product Y, then he can buy the maximum amount of 10 units, if he spends all the money on the product X, then he can purchase the maximum amount of 15 units.

A consumer can buy 10 units of product Y by spending his entire budget, which means that his budget is 6 rubles × 10 units = 60 rubles.

Then the price of goods X = 60 rubles / 15 units = 4 rubles. for 1 unit of goods X.

Now you can write the budget line equation

RUB 6 × Qy + 4 rub. × Qx = 60 or in another form Qy = 10 - 0.67Qx

Answer: option 3.

Task number 6

If the production function is defined by the equation Q = 100 + 12 K² + 10L, then the equation for the marginal product of capital has the form

Answer options:

2) MPK = 100 +24 K

SOLUTION:

The marginal product of capital is equal to the first derivative of the production function with respect to capital, i.e. take the derivative of Q:

(Q) "= (100 + 12 K² + 10L)" = 100 "+ (12K²)" = 10 L "= 0 + 12 × 2K + 0 = 24K

You can check this solution with the following reasoning:

Let K1 - the previous value of the capital, and K2 - the subsequent value of the capital after increasing it by one unit., ∆K = K2 - K1; ∆Q = Q2 - Q1.

Then ∆Q = 100 + 12 (K2) ² + 10L - =

12 (K2) ²- 12 (K1) ² = 12 (K2 ─K1) × (K2 + K1);

MRK = ∆Q / ∆K = 12 (K2 ─K1) × (K2 + K1) / (K2 - K1) = 12 (K2 + K1)

Since for an infinitely small increment K2 = K1, then MRK = 24 K

Answer: option 4.

Task number 7

Using the data in the table, calculate the marginal cost of production for the first unit of production:

Production volume, units
Average fixed costs, rubles
Average variable costs, rub.

Enter your answer:

SOLUTION:

Total costs are equal to the sum of fixed and variable: TC = FC + VC

Marginal cost (MC) = TC2 - TC1 = VC2 - VC1, since FC1 = FC2

Since we are talking about marginal costs the first units, then the previous value of the volume of production is 0. At zero volume of production, fixed costs are equal to 60, and variables are equal to 0. For the first (one) unit, the average and total values coincide, therefore MC = 100 - 0 = 100

ANSWER: MC for first unit = 100

Task number 8

The company produces and sells 100 valves per month. If the cost of production is 12,000 cash units and the average profit is 50 cash units. units, then the gross income of the company is equal to:

Enter your answer:

SOLUTION:

IN economic theory gross income (GI) means income from the production and sale of products, i.e. product of quantity products sold on the unit price. (it should be borne in mind that in the Soviet models of cost accounting, gross income was understood as part of the proceeds minus material costs). Gross income includes both cost of production and profit. We find the total profit by multiplying the average profit by the number of products.