Mathematical formula of utility function

I. Explanation of Nouns

1. economies of scale and scope.

2. Transaction cost and settlement cost (Sunkcost)

3. Vertical merger and horizontal merger.

4. Pustule formation and asymmetric information.

5. Gini coefficient and price index.

6. Laffer curve

7. Difference between gross national product and gross domestic product

8. Neo-classical macroeconomics "policy non-interaction theory"

9. The "menu cost" of inflation

10. Intertemporal substitution of leisure

Second, short answer questions

1. Which of the following production functions belongs to increasing, constant and decreasing returns to scale?

( 1)F(K，L)=K×K×L

(2)F(K，L)=K+2×L

(3)F(bK，bL)=sqrt(b)×F(K，L)

2.( 1) Assuming that orange juice and apple juice are perfect substitutes, draw the appropriate price-consumption (the price change of orange juice).

Curve and income-consumption curve; (2) The left shoe and the right shoe are completely complementary, and a suitable price-consumption curve is drawn.

And income-consumption curve.

3. When economists observe that the price of a product rises, but the demand increases, what explanation will they give?

4. The consumer's utility function is V=X4Y3 (note: the fourth power of X, the third power of Y), so his expenditure on Y goods accounts for.

What is the proportion of the total expenditure? What is the relationship between the demand for Y and the price of X?

5. How do external influences lead to market failure? What measures should the government take to correct market failure?

6. At present, some domestic industries implement self-discipline pricing, giving support and opposition.

7. Give an example of a prisoner's dilemma you encountered in real life.

8. It is proved that the optimal price of the products of a complete monopoly manufacturer is higher than the marginal cost and negatively related to the elasticity of demand.

9. Please explain the possible impact of money illusion on consumer spending.

10. Briefly describe the real money balance effect of consumption (haberler-Pigou effect).

1 1. Try to explain the meaning of Baum-Tobin's "square root formula".

12. Explain the J curve phenomenon.

13. Please explain (with the help of graphs) how to determine the position of the IS curve in the R-Y space.

Third, the calculation problem

1. For consumer economy, assume that the direct utility function of consumers with income of m is u(x) and the indirect utility function is

Five o'clock in the afternoon. Let's assume that P is the production price. If T is the corresponding tax vector, then the consumer's price is p+t, so

Indirectly expressed by v(p+t, M), the government gains from consumers are:

(1) the government's behavior is to maximize the whole social welfare under the condition of its own income, assuming that the government income is R.

Write the government's best questions.

(2) Deduce the law of tax inverse elasticity.

2. The demand curve faced by two oligopolistic enterprises is P=a-bQ, where Q=Q 1+Q2, and the cost function is Ci=ai+bi, and i= 1.

2, a, b, ai and bi are constants.

(1) What is the maximum output when two oligarchs are united? How much output does each oligarch have to produce separately to unite?

(2) If two oligarchs adopt a non-cooperative strategy, and the oligarch 1 is in the leading position, find out their respective equilibrium output and profit.

Market price, and equipped with appropriate graphics.

(3) Oligopoly 1 How much are you willing to pay to merge with Oligopoly 2?

3. Suppose that consumers live in two periods. In the first period, consumers worked and earned income to meet their consumption and savings.

Consumers don't work in the second period, and use the savings in the first period to meet the consumption in this period. Suppose the consumer's consumption in the first period

For C 1, savings is s and labor income is w; The consumption in the second period is C2, assuming that the market interest rate is R and the discount factor is 0.

& lt 1。 Let the utility function of consumers be: (where q is a normal number)

u(C)= C 1-Q- 1/ 1-Q

(1) Write about the utility maximization of consumers;

(2) Find out the consumer's saving function and discuss the relationship between interest rate change and saving;

(3) Analyze the relationship between interest rate decline and savings by combining the above conclusions with the current situation in China.

If consumers are divided into three categories according to their preference for public TV service, they will get the following benefits from public TV service.

The marginal benefits are as follows:

Mr 1 = A-aT； MR2 = B-bT； MR3=C-cT

Where t is the broadcast time of public TV, and A, B, C, A, B and C are all constants, assuming that the public TV service is pure public TV.

The marginal cost of products and public goods is equal to a constant, that is, m yuan per hour. Q:

(1) What is the effective broadcast time of TV?

(2) If the public television service is provided by a competitive private market, how long should it be provided?

Only Wang Laohan and Zhang Laohan sell tomatoes in the free market. The market demand function of tomatoes is q.

=3200- 1600P, Q=Qw+Qz, Qw and Qz, which are the number of tomatoes sold by Wang Laohan and Zhang Laohan respectively.

The marginal cost of growing tomatoes is 0.5 yuan.

(1) The two old people decide the number of tomatoes planted every spring, and they both know the market demand for tomatoes.

How many tomatoes did the other party sell last year? Now suppose that both old people think that the quantity sold by each other this year will be the same as last year.

If Zhang Laohan sells a quantity in t- 1 and Wang Laohan decides to plant it in the spring of T, then he must count the tomatoes.

What is the market price? What is his marginal income?

(2) In order to maximize profits, how many tomatoes should Wang Laohan plant in the spring of T year?

(3) What are the response functions of Wang Laohan and Zhang Laohan respectively? When the equilibrium is reached, the output of Wang Laohan and Zhang Laohan are respectively

How much/how much? What are the profits of both parties? What is the market price? What kind of equilibrium is this?

(4) If Zhang Laohan sowed two days earlier and Wang Laohan secretly learned that Zhang Laohan's output in that year was Q, what would Wang Laohan do?

What kind of decision? If Zhang Laohan had anticipated Wang Laohan's behavior and knew Wang Laohan's decision-making method, what would Zhang Laohan do?

What kind of decision? What is the output of Zhang Laohan and Wang Laohan when they reach equilibrium? What are the respective profits of both parties? city

What is the market price? What kind of equilibrium is this? If Zhang Laohan delays planting time, will it be beneficial to him?

(5) If two old people decide the total output together, and then distribute their respective outputs according to the marginal cost, the market price is

Power grid, their respective output, their respective profits? What kind of equilibrium is this?

6. Assume that the government's objective function is:

u(π，y)= cπ2(y-k)，c & gt0，k & gt 1

Where π is the real inflation rate, Y is the real output, and Y is the equilibrium output under the natural unemployment rate. A letter explaining the government's objectives

The meaning of numbers.

Now suppose that the relationship between output and inflation rate is given by the following Phillips curve with expected inflation:

y=？ +β(π-p)，β& gt； 0。

Where p is the expected inflation rate. Explain the output function.

(1) How does the government's optimal inflation rate depend on the public's expected inflation rate?

(2) Assuming that the public has rational expectations, what is a balanced inflation level? At the level of balancing inflation, the government's

What is the level of utility?

(3) Will the public believe if the government announces a zero inflation policy? Why?

7. Assume that the utility function of A is U=x3y, where X represents the number of times he provides services to others, and each unit needs 2 hours. Y table.

Each unit needs 1 hour to show the number of times he provides services for himself; He asks for leave every day 12 hours.

(1) How many hours will he work for others and himself?

(2) What are his total utility and marginal utility respectively? What is the marginal utility of time? Test Ghosn's second theorem

Established?

(3) If he helps others and is a complete altruist (that is, he spends all his time working for others), give A.

Three possible utility functions (modifying the original utility function).

(4) If the utility function of a certain B is U=xy3, which one do you prefer to cooperate with? Why?

(Peking University 2002)

8. Given the CES production function Q=(KP+LP) 1/P, Q is the output, and K and L are the input of assets and labor respectively.

(1) proves that the scale income of the enterprise remains unchanged.

(2) What are the marginal products of capital and labor?

(3) What is the marginal rate of labor's technological substitution for capital?

(4) Prove that the sum of the output elasticity of capital and labor is equal to 1.

(5) Divide the enterprise into two identical enterprises. What is the change between the total output after separation and the output of the original enterprise?

Write out the calculation process in detail.

9. A man has a wealth of 6,543,800 yuan, and there is a 25% chance of losing a car worth 200,000 yuan next year. Suppose he

The utility function is V(W)= 1nW, and w is his total wealth. Please answer the following questions:

(1) If he doesn't participate in the insurance next year, what is his expected effect?

(2) If the management fee of the insurance company is zero, how much insurance premium does he have to pay to participate in complete fair insurance? now

What is the expected cost? Has the situation improved?

(3) If he participates in insurance, how much premium is he willing to pay at most?

10. Suppose the demand curve of a commodity is Qd= 150-50P, and the supply function is QS=60+40P. Suppose the government is responsible for every order of the manufacturer.

A product is taxed in 0.5 yuan.

(1) Output Q and price P after tax.

(2) What is government tax?

(3) What is the net loss of welfare?

1 1. Given the income matrices of two brewing enterprises A and B, the following is the income matrix:

A company Chinese spirit beer

Enterprise B Liquor 700600900 1000

Beer 800900600800

The first number in each group represents the income of enterprise B, and the last number represents the income of enterprise A. Please answer the following questions:

(1) Is it dominant equilibrium or Nash equilibrium to find the equilibrium solution of this game problem?

(2) Is there Pareto improvement? If so, under what conditions can it be achieved? What is the increase in income?

(3) How to change the income of the above A and B enterprises and make the equilibrium become Nash equilibrium or dominant equilibrium? How to change the above a

Can the income of enterprise B make the game not equilibrium?

(4) Write the mathematical meaning of Nash equilibrium and explain its economic meaning concisely.

12. The cost function of duopoly enterprises is:

C 1=20Q 1, C2=2Q22, and the market demand curve is P=400-2Q, where q = q1+Q2;

(1) Calculate the output, price and profit under the Cournot equilibrium, and calculate the sum of their respective reaction functions.

Profit curve, and explain the equilibrium point;

(2) Calculate the output, price and profit under Stackelberg equilibrium, and draw a chart;

(3) Explain the reasons for the difference between the above two equilibrium results;

13. We use x 1 and X2 to represent the consumer's consumption quantity of commodities X 1 and x2, and now the utility function of consumers is given as U(x

1, x2)=x 1αx2β, the prices of two commodities are P 1 and P2 respectively, and the income of consumers is m, thus solving the following problems:

(1) What percentage of consumers' income will be spent on X 1 and X2 respectively?

(2) Find out the consumer's demand function for X 1 and X2.

(3) When consumers are balanced, what is the price elasticity of demand of two commodities?

14. A retired man with a fixed income needs to choose a city to live in between Beijing, Shanghai and Guangzhou.

Live, assuming that he only chooses according to the utility of consumption, regardless of regional, climate and cultural factors. His utility function is u=x 1x2.

, x 1x2∈R2+. It is known that Beijing price is (), Shanghai price is (), Guangzhou price is (), (here.

The superscript of p indicates the city and the subscript indicates the commodity). Known =,

Q: Which city will he choose to live in?

15. It is known that the cost function of an enterprise is C(q)= 100+5q+q2. When the market price is 20, the enterprise produces 5 units of products.

(1) At this time, what is the producer surplus of the enterprise?

(2) On the previous question, if an enterprise pays 50% of the environmental tax in one lump sum, how much will its producer surplus be reduced?

16. let the market demand cost of an industry be P= 13-x (let p be the price and x be the output), and there is only one monopolist, production.

The cost function is x+6.25, so is the cost function of new entrants. Q:

(1) What is the monopoly profit of the monopolist?

(2) Existing enterprises want to enter, and the original monopolist wants to make the profit of the entrants zero in order to restrain the entry of potential entrants. origin

How much output should the monopolist set to make the profit of new entrants zero?

(3) Can the output decision chosen by the original monopolist in question 2 scare off the entrants? Why?

(4) In the long run, what will be the output, price and profit of new entrants and original monopolists?

17. Some people spend all their income on X and Y consumer goods. When Px= 10 yuan and Py=5 yuan, the purchase quantity is X.

=5, y= 10. Now Px=8 yuan, Py=6 yuan. I want to ask: Has the living standard of this consumer increased or decreased after the price change?

Come down? Why?

18. Let the production function of a team (* * * has I members) be x=f(e 1, E2, e3…, eI=f(e), where e 1, e2, e.

3…, eI respectively represent the efforts of team members, 1, 2, 3…I, and let Si(x) represent the number that member I can get from X.

Well ... the following formula is called the budget balance rule, and the cost of individual effort is Ci(ei)=ei, and each member is risk neutral.

Yes

What is the Nash equilibrium between team individual behaviors?

Please prove that there is no Nash equilibrium about individual efforts in the team under the above conditions.

19. A monopoly enterprise consists of two factories. The production function of factory one is, and the production function of factory two is. Where X 1

And X2 are the inputs of two elements, and α and β are constants. If the factor market is completely competitive, r 1 and r2 are two.

What is the cost function of this enterprise?

20. It is known that the production function of an enterprise is f (x 1, x2) = min {x 1, x2} 1/a, x 1 and x2 are the number of two input factors, α.

& gt0 is a constant. Find the demand function, supply function and profit function of profit maximization, and discuss that α will be full when profit is maximized.

Foot restraint.

2 1. The marginal production cost of monopoly manufacturers is fixed at 5 units, that is, MC=5. The market demand function faced by this manufacturer is q (

P)=53-P .

(a) Calculate the price, output and profit that maximize the profit of the manufacturer and the net welfare loss caused by monopoly.

Now suppose a second manufacturer joins this market, which has the same cost function as the first manufacturer. Suppose there are two factories.

Business is Cournot competition (Cournot competition).

(b) Write the best response function of each manufacturer.

Find out the production level of Cournot equilibrium.

(d) What is the equilibrium price of the market? Calculate the profit of each manufacturer.

(e) If two manufacturers compete in bertram, there are many equilibrium prices in the market.

Less?

22. Consider the utility function of a pure exchange economy with only two consumers A and B and two commodities X and Y.

Defined as follows:

UA(xA，yA)=3xA+5yA。 UB(xB，yB)=9xB+2yB .

The total endowment of this economy is XA+xB= 10, yA+yB= 10.

Please give the definition of perfect competitive equilibrium.

(b) Please give the definition of Pareto optimal allocation.

Please give all possible Pareto optimal configurations of this economy.

(d) If the initial wealth distribution is X and Y with 5 units for A and B, and the price ratio of X and Y is Px/Py, when the economy reaches the end,

Can this price ratio be greater than 1 when the whole competition is balanced? Why?

(e) Assuming the above conditions, can this price ratio be less than 1? Why?

23. There are two hunters who catch rabbits for a living in a public hunting ground. There are 1000 rabbits in Hunting Ground 1. Every hunter faces a choice.

Another method is to determine the rabbit capture rate ri(i= 1, 2). The net communication cost of hunter I depends on the rabbit's catch rate and catch rate.

Ri, i.e.

ui=4qi+50ri-ri2

Where qi= 1000ri/(r 1+r2).

(a) If two hunters can achieve the best capture rate, what is it?

(b) If each hunter made his own decision, what would be the capture rate they chose? Please briefly explain why every hunter

The result of self-determination will be different from that obtained in (a).

This text briefly explains what a public tragedy is. Please also list the solution to this problem and briefly explain your solution.

Conditions and reasons for playing a role.

[Calculation problem analysis]

Judging from the comprehensive examination questions of famous schools, except Wuhan University and Peking University, the calculation questions of other schools account for a small proportion, and the examination questions are also difficult.

Don't be too high, so you should master basic concepts and basic calculation formulas. The calculation problems of Wu Da and Peking University will generally involve

The content of advanced economics lies in game theory, consumer surplus and market structure. So, it is accurate.

When preparing for these two schools, you should review the middle and advanced textbooks. When solving problems, you should be clear-headed and step by step.

I. Explanation of Nouns

1. demand

2. Opportunity cost

3. Economic cycle

4. Marginal propensity to consume

Second, short answer questions

1. Briefly describe the indifference curve and its characteristics.

2. What is an automatic stabilizer (internal stabilizer)? What systems are used to develop the function of financial automatic stabilizer?

Swing?

3. Draw and explain what are the Keynesian zone, classical zone and middle zone of LM curve?

Third, the calculation problem

1. The short-term total cost function of a given enterprise is STC (q) = 0.04q3-0.8q2+10q+s, and the lowest level of the enterprise is calculated.

Average variable cost value.

Fourth, the essay questions

1. Draw a diagram to discuss the conditions for a perfectly competitive firm to achieve short-term equilibrium, and derive the short-term supply curve of a perfectly competitive firm.

Line.

2. On the basic viewpoints and policy propositions of macroeconomics of rational expectation school.

I. Explanation of Nouns

1. demand theorem

2. Coase theorem

3. Money demand function

4. Disposable personal income

5. Endogenous variables and exogenous variables

6. Substitution effect and income effect

7. Internal economy and endogenous non-economy

8.CPI and PPI

9. Natural unemployment rate and potential total output level