Dependence of consumption volume on income. Dependence of consumption volume on income Consumption and demand
The volume of consumption of a certain good by a household (q) depending on income (I) is described by the equality:
Determine at what values of income a product for a given household is
a) the lowest good;
b) a normal good;
c) a necessary good;
d) a luxurious good.
TASK No. 2
An individual consumes two goods in quantities x and y, respectively. Are the utility functions below consistent with the axioms of consumer preferences? (Not really)
a) U(x, y) = yjx2 + y2 ;
c) U(x, y) = - +
TASK No. 3
An individual's preferences are characterized by marginal rates of substitution MRSxy = 2, MRSxz = 0.8. Find the marginal rates of substitution a) MRS, b) MRS, c) MRS, d) MRS.
A 1 / yx7 / zx7 " yz7 " zy
TASK No. 4
A household consumes two goods in quantities x and y; his preferences are described by the utility function U(x, y). Find the household demand function if
a) U(x, y) = x3y2;
b) U(x, y) = xaye.
TASK No. 5
The preferences of two individuals are described by utility functions
U-(x, y) = --; U2(x, y) = ln x + ln y ln(x + y).
Do these individuals have different preferences? TASK No. 6
Consider a model in which consumer preferences relate not to products, but to the characteristics that products have (Lancaster model). Let us assume that we are considering a set of products that have two characteristics (X and Y).
| Product (0 | ||
Let us denote (x., y) the quantitative measures of the corresponding characteristics in the unit of the i-th product, and for the sake of simplicity, the quantity of the product purchased for one monetary unit is taken as the unit of each product. We will assume that preferences in the space of characteristics satisfy the same axioms as preferences in the space of goods in the traditional theory.
The table (above) shows data for six different products. Which of them have no prospects of being sold on the market?
TASK No. 7
A household consumes two goods, X and Y, in quantities x and y; his income I = 60, and his preferences are described by the utility function U(x, y) = xy.
benefit pX = 9, pY = 4.
from the benefits of prices and income.
TASK No. 8
A household consumes two goods, X and Y, in quantities x and y; his preferences are described by the function
utility U(x, y) = l/x + yfy . Income is known: I = 60.
a) Find the volume of demand for each of the goods at prices
benefit pX = 10, pY = 5.
b) Determine the dependence of the demand volumes for each
from the benefits of prices and income.
c) Determine the nature of the interdependence of goods in consumption.
TASK No. 9
A household consumes two goods, X and Y, in quantities x and y; his preferences are described by the function1
utility U(x, y) = y, prices of goods are equal to pX = 16, pY = 25.
x X a) Find the volume of demand for each of the goods at income values I = 70; I = 15.
b) Determine the dependence of the volume of demand for each of the goods on income.
TASK No. 10
An individual consumes two goods, X and Y, in quantities x and y, respectively. Individual's utility function: U = ax + by + xy, a > 0, b > 0.
a) Let a = 10, b = 25. Determine the volume of consumption of goods,
if the prices of goods are pX = 5, pY = 2 with the individual’s income I = 200;
b) the same for the individual’s income I = 100;
c) at what ratios of income and prices will the consumer’s optimum be internal (x > 0, y > 0)?
TASK No. 11
A household acquires good X, produced by a natural monopoly, at a price pX = 10 in an amount x = = 5. The state, which regulates the price of the product of a natural monopoly, considered it appropriate to increase the price to p"X = = 14 and pay compensation to the household in the amount of (p "X pX) x = 20.
a) Has the household's wealth changed?
and if so, in which direction?
b) Check the statement using the following example: in addition to the good X, the household consumes another good, Y, the price of which pY = 1 has not changed; household income
economy I = 100, and the utility function U(x, y) = -Jxy.
TASK No. 12
The classification of goods based on Engel curves takes into account changes in the share of income allocated to the purchase of the good in question, depending on changes in income. Prove the following statements:
if the share of income allocated to the purchase of a given good increases with income, then the income elasticity of consumption is greater than one;
If the share of income allocated to the purchase of a given good decreases with increasing income, then the elasticity of consumption with respect to income is less than one.
TASK No. 13
A household consumes three goods, X, Y and Z. Their shares in expenses are sX = 50\%, sY = 30\%, sZ = 20\%, respectively. The income elasticities of consumption volumes of goods X and Y are known: EI[x] = 2, E^y] = 0.6.
a) Find the elasticity of the volume of consumption of good Z with respect to
b) Determine what type each of the goods belongs to.
TASK No. 14
Prove the statement: if among the goods consumed by a household there is at least one inferior one, then among them there is also at least one luxurious one.
TASK No. 15
The telephone company offers service consumers a choice of two tariff options: (a) 4 units/min without a subscription fee; (b) 2 units/min and subscription fee 20 units. Which tariff will each of the following consumers choose:
utility function U1 = x0.5y0.5, income 11 = 100 units;
utility function U2 = x0.25y0.75, income 12 = 100 units;
utility function U3 = x0.25y0.75, income I = 200 units. Here x is the number (in minutes) of consumed services
telephone company, y is the volume of consumption of all other goods whose price is equal to 1 unit.
2.2 SOLUTIONS
SOLUTION TO PROBLEM No. 1
The graph shows that as income increases from zero to a certain level, the volume of consumption of the good increases, so that the benefit is normal; with a further increase in income, this product is replaced by some substitute, the volume of its consumption decreases and the product becomes inferior.
Let's find the boundaries of the area of increasing consumption;
To do this, we differentiate the volume of consumption by income:
dq = _ 2I (I +10)8 8I2(I +10)2 = _ 20I -12
dI (I +10)6 (I +10)4.
The derivative vanishes at I = 20; at lower values of income the derivative is positive and the volume increases, at larger values it decreases. Thus, the product is normal at I< 20 и низшим - при I > 20.
In order to find out at what levels of income a product is a necessary good and at what levels it is a luxury one, it is advisable to use the income elasticity of consumption: 1 q dl I +10
For a luxury good, the income elasticity of consumption is greater than one. The last equality shows that EI[q] > 1 at 0< I < 5. Если 5 < I < 20, то потребление растет с доходом, но медленнее, чем доход, BI[q] < 1, и рассматриваемый товар является необходимым благом.
So, the product in question is an inferior good for I > 20 and a normal good for I< 20; при 0 < I < 5 он является роскошным благом, при 5 < I < 20 - необходимым.
Comments.
The sign of the derivative always coincides with the sign of elasticity. Therefore, answers to all the questions of the problem could be obtained by considering the ranges of income levels within which the values of BI[q] exceed one, lie between zero and one, and turn out to be negative.
The modern classification of consumed goods originates from the research of E. Engel, carried out in the middle of the 19th century. and, naturally, did not use the concept of elasticity of functions. Having analyzed the structure of consumer budgets, Engel found that as income increases, the amount of food expenses increases, but their share in the income distribution falls. If we consider a particular good consumed in quantity q and purchased at price p (which we assume here to be constant), then expenditure is equal to pq. The share attributable to this product is equal to pq/I; if it decreases with income growth, then BI< 0. Воспользовавшись свойствами эластичности (см. Приложение) и учитывая неизменность цены, представим это соотношение в виде BI[q] 1 < 0, или BI[q] < 1. При этом абсолютная сумма расходов возрастает, EI = BI[q] >0. Thus, Engel’s law in relation to a necessary good (like food) is formulated as a double inequality 0< EI[q] < 1.
SOLUTION TO PROBLEM No. 2
Axioms of consumer preferences:
completeness (comparability of any consumer sets);
transitivity;
unsatiability (“more is better than less”, preference for a set containing a larger volume of any good without reducing the volumes of the rest);
continuity;
the convexity of a set of sets that are preferable to any given one.
If the consumer's preference system is specified by the utility function, then axioms 1 and 2 are thereby satisfied. Axiom 4 holds if the utility function is continuous. In all options a) - c) the utility functions are continuous, so the requirements of axioms 1, 2 and 4 can be considered fulfilled.
Axiom 3 is satisfied if the utility function increases with respect to each argument. The function of option a) obviously satisfies this requirement, option c) does not, it is decreasing with respect to each argument. Because
i.e. the values of functions b) and c) are mutually inverse quantities, function b) is increasing (which can be verified in any other way).
Axiom 5 requires that every indifference curve
bounded the convex region from below. It means that
the marginal replacement rate MRS should decrease with increasing
x and increase with y. Function a) does not meet this requirement
answers: the corresponding indifference curves are 90-degree arcs of circles centered at the origin.
For function b) ^ ^2 ґ 2
dU/dx = -2- I; dU/dy =
so 2 MRS ==Udx = (U1.
Thus, function b) satisfies all preference axioms. Answer:
a) no; b) yes; c) no. SOLUTION TO PROBLEM No. 3
If a unit of good x is replaced by a units of good y while maintaining the level of utility, then a unit of good y is replaced by 1/a units of good x. Therefore MRS = 1/MRS.
If, in addition, a unit of good y is replaced by b units of good z under the same condition, then a unit of good x is replaced by ab units of good z and therefore
MRS MRS = MRS .
This allows you to find all unknown marginal rates of substitution using known MRS and MRS.
A comment.
A more formalized approach relates marginal rates of substitution to derivatives of the utility function:
MRS = Udx, etc.,
whence the above relations follow. Note that the preference system defines the utility function ambiguously: if the function U(x, y, ...) describes the preferences of a given consumer, then the function U1(x, y, ...) = cp(U(x , y, ...)), where φ is an arbitrary monotonically increasing function. But
dU1 / dx = (dp / dU) ■ (dU / dx) = dU / dx
dU1/dy ~ dp / dU) ■ (dU / dy) "dU / dy" so that the ratio of partial derivatives does not depend on the quantitative scale in which utilities are displayed, but only on the preferences of the individual.
a) 0.5; b) 1.25; c) 0.4; d) 2.5.
SOLUTION TO PROBLEM No. 4
a) First of all, let’s determine the marginal rate of substitution
as a function of x and y:
U = 3x2y2; U = 2x3y, hence MRS = Ux = 3y.
For goods prices p , p at the consumer optimum point, the price ratio p/p is equal to the marginal rate of substitution, so
Note that p x and p y are the consumer’s expenses for the first and second goods, respectively. From here it is clear how a given consumer distributes his budget: he must spend a share of 0.6 of his income on the purchase of the first good, and a share of 0.4 on the purchase of the second. If his income is equal to I, then the volumes of demand for the first and second goods are equal:
x = 0.6 -; y = 0.4 -.
Each of the above equalities describes the demand function for the corresponding good.
b) The same reasoning applied to a more general
case lead to the relation:
from: RuU R
a + p px a+p py
A comment.
In the above problems, the volume of demand for each good depended on income and on the price of this good and did not depend
on the price of another good, and the share of expenses for this good in the amount of income depended only on the parameters of the utility function and did not depend on either income or prices.
The constancy of the share of expenses (independence of income) means that both goods occupy a borderline position between necessary and luxurious goods. The independence of the volume of demand for each good from the price of another good means that the goods are independent in consumption.
The shares of expenditures for each benefit did not depend on the absolute values of the parameters a and p, but only on their ratio. Thus, the solution in part a) would not change if the exponents were not 3 and 2, but, say, 15 and 10 or 0.3 and 0.2. The last circumstance is due to the fact that utility functions related by a monotonically increasing transformation represent the same system of preferences (ordinal concept of utility). Let x be a vector representing a set of goods, U^x) and U2(x) be utility functions, and U2(x) = φ(^1(x)), where φ is a monotonically increasing function. In this case, if ^1(x1) > U^x2), then U2(x^ > > U2(x2), i.e., the set evaluated by the function U as more preferable, is also evaluated by the function U2. Reduction to a positive degree is a monotonically increasing transformation, and the function x15y10 = (x3y2)5 describes the same system of preferences as the function in task a). For example, logarithm gives the same result:
U3(x) = 3 ln x + 2 ln y = 1п(х3у2).
In the tasks, the consumer was limited to two goods, but the conclusions remain valid for an arbitrary number of goods. Let x = (x1, x2, xn) and
We will use the notation for marginal utilities,
From this we obtain an expression for the limiting rates of substitution:
U.a. X. MRS.. = = , U. a.) x.
The resulting expression allows, at given prices, to express the costs of all consumed goods through the costs of one, for example the first:
MRS;/ = P = - X, from where:
р x\% = -рл. (3)
Now the budget constraint can be represented as
so taking into account equality (2) p1x1 = a17, and equality (3) shows that similar expressions are valid for all goods: p.x. = aI. Thus, if the utility function has the form (1), then the shares of expenses for individual goods in the total amount do not depend on either the amount of income or prices. They are constant quantities proportional to the parameters ai, and if these parameters are normalized in accordance with equality (2), then the shares coincide with the parameters. The quantity demanded for each good is x. = a.I/p..
SOLUTION TO PROBLEM No. 5
It is easy to see that U1(x, y) = In U2(x, y). Logarithm is an increasing function. If the first consumer prefers the set (x1, y1) to the set (x2, y2), i.e. if U1(x1, y1) > U1(x2, y2), then U2(x1, y1) > U2(x2, y2) , which means that the second consumer also prefers the first set to the second. Under ordinal utility theory, consumer preferences are indistinguishable.
Comments.
From the formulaic notation of utility functions, it is not always easy to guess that one of them is a function of the other. But this can always be clarified by comparing the marginal rates of substitution: if the marginal rates of substitution coincide for any combination of goods, then they express the same system of preferences of individuals. When solving Problem 2, the maximum replacement rate for the first individual was determined:
MRS1^ (x, y) = [Уj .
For the second person
dU2 = 1 1 y dU2 = 1 1 x
dx x x + y x(x + y) dy y x + y y(x + y) so
MRSxy (x, y) = dU/dx = (yT. xyK У" dU2/ dy ^ x J
Thus, for any combinations (x, y), the marginal rates of substitution for both individuals coincide, and therefore their preferences also coincide.
The concept of ordinal utility serves as the basis for the theory of consumer choice in the absence of risk. It turns out to be insufficient for a theoretical description of consumer behavior in a risky situation. The theory of choice under risk states the existence of such a utility function, the mathematical expectation of which the consumer strives to maximize (von Neumann-Morgenstern utility function). In this regard, the preferences of individuals in a given problem are different if the conditions are given by the von Neumann-Morgenstern utility functions. Let us assume that in the examples under consideration the prices of the products are numerically equal, so that, as can be easily verified, in the sets chosen by both consumers x = y. Let us also assume that the consumer is asked to indicate a set of goods that is as useful as a lottery from the sets (1, 1) and (5, 5) with equal probabilities. Since x x/(x + x) = x/2, the first consumer will specify a set (x, x) satisfying the condition:
0.5 .1 + 0.5 . 5 = x,
from where x = 3, so it will indicate the set (3, 3). The corresponding condition for the second consumer is:
0.5. ln1 + 0.5 . ln5 = lnx,
whence x = \PxPy" Py +4 PxPy
c) The last equalities show that with a fixed amount of income, the volume of demand for each good decreases
both with an increase in the price of this and with an increase in the price of another good.
This means that the benefits are mutually complementary.
SOLUTION TO PROBLEM No. 8
a, b) Reasoning by analogy with the solution to the previous problem, we find:
MRSxy =l \% = ^ =
Hence y = x (p /pY)2 = 4x. From the equality of income and expenses, 10x + 5 4x = 60, we find x = 2, y = 8.
The dependence of demand volumes on prices and income is described by the equalities
px ■ (1 + px / py) py ■ (1 + py / px)
c) The last equalities show that for a fixed amount of income, the volume of demand for each good decreases
decreases as the price of this good increases, but increases as the price of another good increases. This means that goods are mutually substitutable.
SOLUTION TO PROBLEM No. 9
Let's find expressions for marginal utilities: dU = J__ dU_ = 1
Since at the consumer optimum point the marginal rate of substitution is equal to the price ratio, the equality
where is the volume of consumption of good X directly found: x = E = Ё5 = 1.25
The demand for good X, as we see, does not depend on income (in the future we will have to clarify this statement). The demand for Y clearly depends on income. At I = 70 we have:
70 -у/16 25 „
It is clear that the volume of consumption cannot be negative. But, according to the obtained expression for y, the condition y > 0 is satisfied for I > 4~PxPy = 20 and is violated otherwise. It is natural to assume that if this condition is violated, the household completely abandons the good Y, so that y = 0. But in this case, the expression for x becomes incorrect: since all income is spent on the good X, the volume of its consumption is equal to x = I/ PX.
To test this assumption, let us find out what values the marginal rate of substitution takes on the budget boundary, described by the equality pXx + pYy = I. From the condition y > 0 it follows that pXx< I и x < I /pX. Поэтому на бюджетной границе
MRSxy = x > Т2" 7^The equality MRSxy = pX /pY = 16/25 = 0.625 is the condition for the internal consumer optimum. At I = 70, the marginal rate of substitution at the budget boundary is not less than 162/702 ~ 0.052, and at some point (a namely x = 1.25, y = = 2) it is equal to 0.625. This is the internal optimum found above. At I = 15 on the budget line MRSxy > > 162/152 ~ 1.138 and a value equal to 0.625 does not exist on the budget line. This means that the consumer optimum occupies a boundary, or, as it is more often called in economics, angular position. Thus,
Y = at / > ^рхРу;
METHODOLOGICAL EXPLANATIONS
Below are solution examples typical tasks and exercises
Topic 1. System of macroeconomic relationships
Problem 26. Based on the data given in the table, determine:
1) GNP by income;
2) GNP by expenses;
5) national income.
Solution
1 We determine GNP by income. It includes depreciation, indirect business taxes, employee wages, dividends, interest, income from individual investments, rent, retained earnings, corporate income taxes, rent:
GNP by income = 1,010 + 786 + 5,810 + 196 + 290 + 158 + 40 + 650 + 784 = $9,724 million.
2 Determine GNP based on expenses:
GNP by expenditure = C + Ig + Xn + G,
where C is personal consumer spending;
Ig – gross domestic private investment;
G – government procurement of goods and services;
Xn is net export.
GNP by expenses = 6,452 + 1,530 – 186 + 1,928 = 9,724 million dollars.
GNP in terms of expenses must be equal to GNP in terms of income.
3 Define GDP:
GDP = GNP – Хn = 9,724 + 186 = 9,910 million dollars.
4 Let’s determine the NNP:
NNP = GDP – A = 9,910 – 1,010 = 8,900 million dollars.
where A is depreciation.
5 Let’s determine national income:
ND = NNP - indirect taxes on business,
ND = 8,900 – 786 = 8,114 million dollars.
Problem 27. GNP is equal to 9,000 den. units, consumer expenses – 5,200 den. units, government spending – 1,900 den. units, and net exports – 180 den. units Calculate:
1) the amount of gross investment;
2) NNP, if the amount of depreciation is 850 den. units;
If net exports are positive in this example, can it be negative? In which case?
Solution
1 GNP by race = C + Ig + G + Xn,
where C is consumer spending;
Ig - gross investment;
G - government spending;
Xn is net export.
Ig = GNP – C – G – Xn,
Ig = 9000 – 5200 – 1900 – 180 = 1820;
2 NNP = GNP – A,
NNP = 9000 – 850 = 8150.
3 If imports are greater than exports.
Topic 2. Consumption, saving, investment
Problem 14. The economy is characterized by the following data:
a) consumption function C = Ca + MPC Y;
b) autonomous investments of Ia units;
c) government procurement of G units;
d) marginal tax rate t;
e) transfer payments TR.
Existing production capacities make it possible to increase national income by 1.25 times. How should the government change its purchasing to ensure full capacity utilization while balancing the government budget? What could be the change in transfer payments?
Solution
Let us determine the equilibrium level of income:
m =1/(1– mpc (1– t)) = 1/(1 – 0.55· (1 – 0.1)) = 1.98;
A = Ca + G + Ia +TR mpc,
A = 50 + 100 + 400 + 200 · 0.55 = 660;
Y = 1.98 660 = 1,306.8.
Let us determine the change in the equilibrium volume of production:
Y2 = 1.25 · 1,306.8 = 1,633.5.
Let’s determine how much government procurement should change:
DY = 1,633.5 – 1,306.8 = 326.7;
DG = 326.7/1.98 = 165.
Let us determine by what amount transfer payments should change:
DTR mpc = 165;
DTR = 165/0.55 = 300.
Problem 16. If the saving function is described by the formula S = –30+0.1Y, and autonomous investment is 125, then what will be the equilibrium level of national income?
Solution
At the equilibrium level of national income, the amount of autonomous investment is equal to saving.
Then I = S,
125 = – 30 + 0.1 y,
Y equals = 1,550.
Answer: equilibrium level of national income 1,550.
Problem 17. The consumption function is given by the formula C = 100 + 0.2 Y.
1) build a consumption schedule;
2) build a savings schedule;
3) determine the equilibrium volume of national income;
4) determine the value of the expense multiplier, provided that income is 0; 200; 400; 600; 800.
Solution
Let's build a table:
1 Using the table data, we build a consumption schedule:
Let us calculate the equilibrium income analytically.
Let us equate Y = C, then Y = 100 + 0.2Y.
2 Using the table data, we build a savings schedule:
 |
The marginal propensity to save is:
MPS = 1 – MPC = 1– 0.2 = 0.8.
Then the value of the multiplier μ is equal to:
.
Subject. Methods of linear algebra in economic analysis.
Target. Solving economic problems with modeling elements based on the basic framework of linear algebra.
1. Reference material.
The concept of a matrix is often used in practical activities, for example, data on the output of several types of products in each quarter of the year or the cost rates of several types of resources for the production of several types of products, etc. It is convenient to write it in matrix form.
Task 1. In some industry, m factories produce n types of products. The matrix sets the production volumes at each plant in the first quarter, the matrix - accordingly in the second; (a ij, in ij) - volumes of products of the j-th type at the i-th plant in the 1st and 2nd quarters, respectively:
a) production volumes;
b) an increase in production volumes in the second quarter compared to the first by types of products and plants;
c) the value expression of manufactured products for six months (in dollars), if l is the dollar exchange rate against the ruble.
Solution:
a) The production volumes for the half-year are determined by the sum of the matrices, i.e. C=A+B=, where c ij is the volume of products of the j-th type produced by the i-th plant over the six months.
b) The increase in the second quarter compared to the first is determined by the difference in matrices, i.e.
D=B-A= . Negative elements show that the production volume at this plant has decreased, positive elements have increased, and zero elements have not changed.
c) The product lC = l(A+B) gives an expression for the cost of production volumes for the quarter in dollars for each plant and each enterprise.
Task 2. An enterprise produces n types of products using m types of resources. The cost rates of the resource of the i-th product for the production of a unit of product of the j-th type are specified by the cost matrix. Let the enterprise produce the quantity of each type of product recorded in the matrix over a certain period of time.
Determine S - the matrix of the total costs of resources of each type for the production of all products for a given period of time, if
, . Solution. The matrix of total resource costs S is defined as a product of matrices, i.e. S=AX.
That is, over a given period of time, 930 units will be consumed. resource of the 1st type, 960 units. resource of the 2nd type, 450 units. resource of the 3rd type, 630 units. resource of the 4th type.
Task 3. The plant produces engines that may either immediately require additional adjustment (in 40% of cases) or can be used immediately (in 60% of cases). As statistical studies show, those engines that initially required adjustment will require additional adjustment after a month in 65% of cases, and in 35% of cases they will work well after a month. The same engines that did not require initial adjustment will require it after a month in 20% of cases and will continue to work well in 80% of cases. What is the percentage of engines that will work well or require tuning 2 months after release? In 3 months?
Solution.
At the moment after release, the share of good engines is 0.6, and the share of those requiring adjustment is 0.4. In a month, the share of good ones will be: 0.6. 0.8+0.4. 0.35=0.62. Proportion requiring adjustment: 0.6. 0.2+0.4. 0.65=0.38. enter the status line X t at moment t; X t =(x 1t; x 2t), where x 1t is the share of good engines, x 2t is the share of engines that require adjustment at time t.
Transition matrix, where is the proportion of engines that are currently in good condition (1- “good”, 2- “needs adjustment”), and after a month - in good condition.
Obviously, for the transition matrix, the sum of the elements of each row is equal to 1, all elements are non-negative.
Obviously =(0.6 0.4), .
Then in a month,
2 months later; in 3 months.
Let's find the matrices;
Note that if is a transition matrix, then is also a transition matrix for any natural t. Now
Obviously, .
Task 3. The company consists of two branches, the total profit of which last year amounted to 12 million conventional units. units This year it is planned to increase the profit of the first branch by 70%, the second - by 40%. As a result, the total profit should increase by 1.5 times. What is the amount of profit of each of the departments: a) last year; b) this year?
Solution.
Let it be the profits of the first and second branches last year. then the condition of the problem can be written in the form of a system: Having solved the system, we obtain Investigator, a) profit in the past year of the first department -4 million conventional units. units, and the second - 8 million conventional units. units; b) the profit this year of the first department is 1.7. 4=6.8 million conventional units units, second 1.4. 8=11.2 million conventional units units
2.1. Three factories produce four types of products. It is necessary: a) find the matrix of product output for the quarter, if the matrices of monthly outputs A 1, A 2, A 3 are given; b) find the growth matrices of output for each month B 1 and B 2 and analyze the results:
2.2. The company produces three types of furniture and sells it in four regions. The matrix specifies the selling price of a unit of furniture of the i-th type in the j-th region. Determine the enterprise's revenue in each region if furniture sales for the month are given by the matrix.
2.3. According to the conditions of task 2, determine: 1) the total costs of resources of 3 types for the production of monthly products, if the cost rates are specified by the matrix and the output volume of each of the two types of products;
2) the cost of all expended resources, if the cost of units of each resource is given.
2.4 . The repair shop receives telephones, 70% of which require minor repairs, 20% - medium repairs, 10% - complex repairs. It has been statistically established that 10% of devices that have undergone minor repairs require minor repairs after a year, 60% require medium repairs, and 30% require complex repairs. Of the devices that have undergone average repairs, 20% require minor repairs after a year, 50% require medium repairs, and 30% require complex repairs. Of the devices that have undergone complex repairs, after a year, 60% require minor repairs, 40% require medium repairs. Find the share of the devices repaired at the beginning of the year that will require repairs of one kind or another: after 1 year; 2 years; 3 years.
Practical lesson.
Subject. Methods of mathematical analysis for constructing SEP models.
Target. Solving economic problems with modeling elements using mathematical analysis methods.
1. Reference material.
The functions are widely used in economic theory and practice. The range of functions used in economics is very wide: from the simplest linear ones to functions obtained according to a certain algorithm using recurrent relations that connect the states of the objects under study in different periods of time.
The most commonly used functions in economics are the following:
1. Utility function (preference function) - the dependence of the result, the effect of some action on the level (intensity) of this action.
2. Production function - the dependence of the result of production activity on the factors that determined it.
3. Output function - the dependence of production volume on the availability or consumption of resources.
4. Cost function - the dependence of production costs on the volume of production.
5. Functions of demand, consumption and supply - the dependence of the volume of demand, consumption or supply for individual goods or services on various factors (for example, price, income, etc.).
Considering that economic phenomena and processes are determined by the action of various factors, functions of several variables are widely used to study them. Among these functions, multiplicative functions are distinguished, which make it possible to represent the dependent variable as a product of factor variables, turning it to zero in the absence of the action of at least one factor.
Separable functions are also used, which make it possible to isolate the influence of various variable factors on the dependent variable, and in particular, additive functions that represent the same dependent variable both under the total but separate influence of several factors, and under their simultaneous influence.
In addition to the geometric and mechanical meaning, there is also an economic meaning of the derivative. Firstly, the derivative of the volume of production with respect to time is labor productivity at the moment. Secondly, there is another concept that characterizes the economic meaning of a derivative. If production costs y considered as a function of the quantity of output x, - increase in production, - increase in production costs, and - average increase in production costs per unit of production, then the derivative equal expresses marginal cost production and approximately characterizes the additional costs of producing a unit of additional products.
Marginal costs depend on the level of production (quantity of output) x and are determined not by constant production costs, but only by variable ones (for raw materials, fuel, etc.). In a similar way, marginal revenue, marginal income, marginal product, marginal utility and other marginal values can be determined.
Limit values characterize not a state, but a process, that is, a change in an economic object. Thus, the derivative acts as the rate of change of some economic object (process) over time or relative to another factor under study. It should be taken into account that economics does not always allow the use of limit values due to the indivisibility of many objects of economic calculations and the discontinuity (discreteness) of economic indicators over time (for example, annual, quarterly, monthly, etc.). At the same time, in a number of cases it is possible to ignore the discreteness of indicators and effectively limit values.
To study economic processes and solve applied problems, the concept of elasticity of a function is often used.
The elasticity of a function is the limit of the ratio of the relative increment of a function y to the relative increment of the variable x at:
The elasticity of a function shows approximately how many percent the function will change y=f(x) when the independent variable changes x by 1%. This is a measure of the response of one variable to a change in another.
Let us note the elasticity properties of the function.
1. The elasticity of a function is equal to the product of the independent variable x on the rate of change of the function, i.e. .
2. The elasticity of the product (quotient) of two functions is equal to the sum (difference) of the elasticities of these functions: , .
Elasticity of functions is used in the analysis of demand and consumption. For example, elasticity of demand y regarding price x- coefficient determined by formula (1) and approximately showing by what percentage demand (volume of consumption) will change when price (or income) changes by 1%.
If the elasticity of demand (in absolute value), then demand is considered elastic, if - neutral, if - inelastic with respect to price (or income).
In practical activities, one often encounters problems that can be rationally solved by methods of mathematical analysis. These are problems of finding the volume of production with a known profit value, determining the level of consumption of goods with a known income, determining the point in time of production profitability, determining the size of the contribution with known initial investments, etc.
Task 1. Costs y (in rubles) for the production of a batch of parts are determined by the formula, where is the volume of the batch. For the first variant of the technological process. For the second option, it is known that (rub.) at (det.) and (rub.) at (det.). Evaluate two technological process options and find the cost of production for both options at (details)
Solution.
For the second option, we determine the parameters and from the system of equations:
from where and, i.e. .
The point (x 0 ,y 0) of intersection of two lines is found from the system of their equations:
from where, .Obviously, with the volume of the batch, the second option of the technological process is more profitable, with the first option. The cost of production (rubles) for the first option is, and for the second - .
Task 2. Fixed costs amount to 125 thousand rubles. per month, and variable costs - 700 rubles. for each unit of production. Unit price 1200 rub. Find the volume of production at which profit is equal to: a) zero (break-even point); b) 105 thousand rubles. per month.
Solution:
a) The production costs of units of production will be: (thousand rubles). Total income (revenue) from the sale of these products, and profit (thousand rubles). The break-even point at which is equal to (units).
b) Profit (thousand rubles), i.e. at (units).
Task 3. The execution duration (min.) for repeated operations is related to the number of these operations by a dependency. Calculate how many minutes the work takes for 50 operations, if it is known that at, and at.
Solution. Let's find the parameters and, taking into account that, . We get the system: solving which we find, .
So, at, (min.)
Task 4. The volume of production u produced by a team of workers can be described by the equation (units), where t- working time in hours. Calculate labor productivity, the speed and rate of its change an hour after the start of work and an hour before its end.
Solution. Labor productivity is expressed as a derivative (units/hour), and the speed and rate of change in productivity are the derivative and logarithmic derivative, respectively: (units/hour 2),
At given moments of time and, accordingly, we have: z(t)=112.5 (units/hour), z"(t)=-20(units/hour 2), T z (7)=-0.24 ( units/hour).
So, by the end of work, labor productivity decreases significantly; at the same time, a change in the sign of z"(t) and T z (t) from plus to minus indicates that the increase in labor productivity in the first hours of the working day is replaced by its decrease in the last hours.
Task 5. The functions of supply and demand have been established empirically, where q And s- the quantity of goods respectively purchased and offered for sale per unit of time, p- the price of the product.
Find: a) the equilibrium price, i.e. the price at which demand is equal to supply;
b) elasticity of supply and demand for this price;
c) change in income when price increases by 5% from the equilibrium price.
Solution. a) The equilibrium price is found from the condition q=s, then where from p= 2, i.e. the equilibrium price is 2 monetary units.
b) Let’s find the elasticity of supply and demand using formula (1)
; . For equilibrium price p=2 we have; . Since the obtained values of elasticity in absolute value are less than 1, then both the demand and supply of this product at the equilibrium (market) price are inelastic relative to the price. This means that a change in price will not lead to a sharp change in supply and demand. So, with an increase in price p by 1%, demand will decrease by 0.3%, and supply will increase by 0.8%.
c) When the price increases p by 5% of the equilibrium demand will decrease by 5. 0.3=1.5%, therefore, income will increase by 3.5%.
Task 6. Relationship between production costs y and volume of products x expressed by a function (denominated units). Determine the average and marginal costs for a production volume of 10 units.
Solution. The average cost function is expressed by the ratio; at x= 10 average costs (per unit of production) are equal to (den. units). The marginal cost function is expressed by its derivative; at x= 10 marginal costs will be (monetary units). So, if the average cost of producing a unit of output is 45 monetary units, then the marginal cost, i.e. additional costs for the production of an additional unit of production at a given level of production (volume of output 10 units) amount to 35 monetary units.
Task 7. Find out what the marginal and average total costs of an enterprise are if the elasticity of total costs is equal to 1?
Solution. Let the total costs of the enterprise y are expressed by a function, where x- volume of products produced. Then the average cost y 1 per unit of production. The elasticity of the quotient of two functions is equal to the difference in their elasticities, i.e. .
By condition, therefore, . This means that with a change in production volume, the average cost per unit of production does not change, i.e., where.
The marginal cost of an enterprise is determined by the derivative. So, that is, marginal costs are equal to average costs (the resulting statement is valid only for linear cost functions).
2. Assignments for independent work.
2.1. The costs of transportation by two modes of transport are expressed by the equations: and, where are distances in hundreds of kilometers, are transport costs. From what distance is the second mode of transport more economical?
2.2. Knowing that the change in production volume with a change in labor productivity occurs in a straight line, create its equation if at =3 =185, and at =5 =305. Determine the volume of production at =20.
2.3 . The company bought a car worth 150 thousand rubles. The annual depreciation rate is 9%. Assuming the dependence of the cost of the car on time is linear, find the cost of the car in 4.5 years.
2.4. The dependence of the level of consumption of a certain type of goods on the level of family income is expressed by the formula: . Find the level of consumption of goods at a family income level of 158 monetary units. It is known that when =50 =0; =74 =0.8; =326 =2.3.
2.5. The bank pays annually 5% per annum (compound interest). Determine: a) the amount of the deposit after 3 years, if the initial deposit was 10 thousand rubles; b) the amount of the initial deposit, at which after 4 years the deposit (together with interest money) will be 10,000 rubles.
Note. The size of the deposit after t years is determined by the formula, where p is the interest rate for the year, Q 0 -initial contribution.
2.6. The costs of production (thousand rubles) are expressed by the equation, where is the number of months. Income from product sales is expressed by the equation. From what month will production be profitable?
2.7. The relationship between the unit cost of production y(thousand rubles) and product output x(billion rubles) is expressed by the function. Find the cost elasticity for a production output equal to 60 billion rubles.
Practical lesson.
Subject. Limit analysis of economic processes.
Target. Consider the use of mathematical methods for finding limit values in optimization problems.
1. Reference material.
Cost function C(x) determines the costs required for production x units of this product. Profit where D(x)- income from production x units of product.
Average costs A(x) in production x units of product are. Marginal cost.
Optimal value release for the manufacturer is the value x units of product at which profit P(x) turns out to be the greatest.
Task 1. The cost function has the form. At the initial stage, the company organizes production in such a way as to minimize average costs A(x). Subsequently, the price for the product is set at 4 conventional units. for a unit. By how many units should the firm increase output?
Solution. Average costs take a minimum value when x=10. Marginal costs. At a stable price, the optimal value P(x) output is given by the profit maximization condition: , i.e. 4= M(x), where. Therefore, production should be increased by 10 units.
Task 2. Determine the optimal output value for the manufacturer x 0 , provided that all goods are sold at a fixed price per unit p=14 , if the type of cost function is known.
Solution. Using the profit formula we get, .
We find the derivative of profit by volume: , then X wholesale = 2.
Task 3. Find the maximum profit that a manufacturing company can receive, provided that all goods are sold at a fixed price per unit R=10.5 and the cost function has the form.
Solution. Find the value of profit.
The derivative of profit by volume has the form: . Then, . .
2. Assignments for independent work.
2.1 Determine the optimal output value x0 for the manufacturer, provided that all goods are sold at a fixed price per unit p=8 and the type of the cost function is known.
2.2 Find the maximum profit that a manufacturing company can receive, provided that all goods are sold at a fixed price per unit p=40 and the form of the cost function is known.
2.3 When produced by a monopoly x units of goods per unit. Determine the optimal output value for the monopoly x 0 (it is assumed that all produced goods are sold), if costs have the form.
2.4 The cost function has the form. The income from the sale of a unit of production is 50. Find the maximum profit value that the manufacturer can receive.
2.5 At the initial stage of production, the firm minimizes average costs, and the cost function has the form: Subsequently, the price per unit of goods is set equal to R=37. By how many units should the firm increase output? How much will average costs change?
Test assignments.