no incomplete answers\

a. Using the aggregate production function, show that the output/income per capita at time t is a function of the private physical capital per capita at time t and the public physical capital per capita at time t. (5 points)

b. From your answer to a, show that the consumption per capita at time t and the equilibrium private investment per capita at time t are functions of the private physical capital per capita at time t and the public physical capital per capita at time t. (10 points)

c

d. In equilibrium, show that the change in the private physical capital per capita from time t to time t+1 is a function of the private physical capital per capita at time t and the public physical capital per capita at time t. (5 points)

e. Derive the steady-state formulas for the private physical capital per capita and for the public physical capital in per capita. (20 points)

f. Derive the steady-state formulas for the output per capita and the consumption per capita. (10 points)

g. Using your answer to f., derive the tax rate that maximizes the steady-state consumption per capita. (10 points)

The Solow-Swan Model Consider an economy with the general Cobb-Douglas production function: Y, = AK, "Let-a. Answer the following questions, assuming that labour grows at the rate n = 0 and adopting the assumptions made in lecture. The equation describing capital dynamics is: KE+1= 66+16-dk1 where d is a constant parameter. a) Obtain the steady state levels of the capital stock (K), output (X), capital per worker (K), output per worker () and real interest rates (r). Make sure to show all your work. b) Using the help of the Solow-Swan diagram developed in lectures and tutorials, explain the effect of a decrease in the savings rate on the standard of living of the economy. Make sure to include: i) The economic explanation of why the economy experienced a change in the steady state level of capital per worker. ) A diagram illustrating what happened to the relevant curves. ili) Another diagram illustrating the dynamics of the stock of capital in the economy (before, at and after the change in savings). iv) What must have happened to the growth rate of output per worker in the transition between the initial equilibrium and the final one?

1. The Size of the Closed City

Since each square block contains 15,000 square feet of housing and each apartment has 1500 square feet, each square block of the city has 10 households living on it. As a result, a city with a radius of x* blocks can accommodate 10x*2 households (x*2 is the area of the city in square blocks). Suppose the city has a population of 250,000 households. How big must its radius be in order to fit this population? Use a calculator and round off to the nearest block.

2. Housing Prices at the Periphery of the Closed City

Recall that the zoning law says that each developed block must contain 15,000 square feet of floor space. Suppose that the annualized cost of the building materials needed to construct this much housing is $75,000. [Note: $75,000 = iK, where K is the capital needed to build 15,000 square feet of floor space. If i= 5%, then K = $75,000/.05 = $1,500,000.] The annual profit per square block for the housing developer is equal to 15000p-75000-r, where r is land rent per square block. In equilibrium profit for the developer is just equal to zero everywhere. In the absence of any other activities, land rent at x* (call it r*) is just equal to zero. What is the value of p at x*? Call this p*.

3. The Closed City's Corn Consumption Level

Using your results from above, suppose that income per household equals y=$25,000 per year. Next suppose that the commuting cost parameter t equals $10 per block. This means that a person living ten bocks from the CBD will spend 10*10=$100 per year getting to work. The consumers' utility functions are all U(c, q), where c is the quantity of bread (at $1 per loaf) consumed by the household and q is the quantity of housing per household, i.e. 1500 sq. ft. The consumers' budget constraints are all c+pq=y-tx. Under our special assumptions this reduces to c+1500p=25000-10x. Given that a household at x* faces a budget constraint, c+1500p*=25000-10x*, what is the value of c at x*? Call this c*. Explain why in equilibrium every household, regardless of location, must be consuming c*. Notice that since q is the same throughout the city, to have an equilibrium in which U is the same throughout the city, c must be the same throughout the city and everywhere equal to c*

4. The Closed City's Housing Price Function

Substituting the value of c* in place of c into the budget constraint c+1500p=25000-10x solve for p in terms of x. The solution tells what the price per square foot must be at a given location for the household to be able to afford exactly c* worth of bread. This is the city's housing price curve. How does p vary with location?

5. The Closed City's Land Rent Function

Substitute the function for p you just calculated in (4) into the zero-profit constraint for developers: 0=15000p-75000-r. Now solve for r in terms of x. What is the rent per square block at the CBD (x=0)? Plot the bid rent function for land in the city.

Scenario You own a bakery caf which has been successful ior the last 10 years. You have opened 3 branches in the Sydney metropolitan area. As your business gained a great reputation in the market, there were several approaches for franchise inquiries. However, you are concerned that the quality of products and services may be affected; therefore, you have decided not to consider the franchise option. Considering your business objective (to increase sales) and intense competition in the industry, you are looking for collaborative alliances with potential collaborators. As the current production capacity can meet up to 7 times greater than the current sales volume, you will need to find an appropriate collaborator to boost up your sales. The business has the following objectives for business expansion or sales increase:

The quality of the products must be managed and produced under the main branch supervision.

The products are in very high demand as the customers are satisfied with the quality.

The product know-how must be treated as intellectual property and is not an option to share.

The business considers collaborative alliances to expand its business. The partner business must meet the following selection criteria determined by the business:

- The partner business must have a good market reputation.

- The partner business has an annual turnover of at least $2 million.

- The partner business must be able to accelerate your business sales.

- The partner business must be able to support training for collaborative works.

- The partner business must have appropriate communication channels for collaborative alliances.

1. (15 Minutes - 20 Points) Answer each of the following subquestions BRIEFLY. (a) (5 points) In my second lecture I defined a solution concept called "pure strategy iterated strict dominance." In my fifth lecture I defined a more powerful version of iterated strict dominance. What was the difference betweeen them and why was the game below a useful example? L C R U M D 10, 9 10, 6 10, 10 -5, 9 15, 10 11, 12 -35, 10 10, 7 15, 5 (b) (5 points) Find all pure strategy Nash equilibria of the game below. X Y Z A B C 4, 3 5, 2 5, 1 1, 7 6, 6 4, 3 2, 3 7, 3 5, 3 (c) (5 points) In class I discussed the general discrete choice model of price competition between N firms: the firms choose prices p1, . . . , pn and each consumer i decides to purchase from the firm j for which vpj +Eij is largest. Describe briefly what happens to equilibrium prices as the number of firms N goes to infinity both for uniformly distributed Eij and under general distributions. (d) (5 points) Describe precisely an example of a game that has no pure or mixed strategy Nash equilibrium. Describe as well as you can a theorem that provides conditions under which a game with an infinite number of pure stategies must have a Nash equilibrium. What conditions of your theorem are violated in your example? 1 2. (25 Minutes - 30 Points) When my daughter Anna was 3 years old, she liked to play Rock-Paper-Scissors. However, she faced a difficulty - three year olds find it hard to make "scissors" with their fingers. Suppose that we capture this problem by treating her playing Rock-Paper-Scissors against her older sister using the asymmetric 3 3 game shown below (with Anna as player 1). R P S R P S 0, 0 -1, 1 1, -1 1, -1 0, 0 -1, 1 1 c, 1 1 c, -1 c, 0 (a) (13 Points) Consider first the version of this game where c > 1. (You can think of this as a model for the extreme situation where Anna is physically incapable of playing scissors.) Find a mixed-strategy Nash Equilibrium of this game. (b) (3 points) What is Anna's expected payoff in the equilibrium you found in part (a)? (c) (14 points) Consider now the version of this game with 0 < c < 1. Find a mixed strategy Nash equilibrium of this game in which both players play every strategy with positive probability. 2 3. (15 Minutes - 22 Points) Two students are deciding how long to spend studying for 14.12 on the night before the exam. Let ei be the fraction of the available time student i devotes to studying with 0 ei 1. Assume that the students' utilities are u1(e1, e2) = log(1 + 3e1 e2) e1 u2(e1, e2) = log(1 + 3e2 e1) e2 (A story for this would be that the first term reflects the benefits they get from learning and getting a good grade, whereas the second reflects the opportunity cost of time. The negative effect of e2 on student 1's utility could reflect that student 1 will get a lower grade if student 2 studies more and does better on the exam.) (a) (5 points) What is player 1's best response to a choice of e2 by player 2. (b) (13 points) Find a pure strategy Nash equilibrium of the game where players 1 and 2 choose e1 and e2 simultaneously. (c) (4 points) Is this game solvable by iterated strict dominance? How do you know this? 3 4. (25 Minutes - 28 Points) Suppose Prof. Ellison decides to run a classroom experiment to teach about mixed strategy equilibrium (and make some money). He chooses two students from the class. Each student is required to write down an integer from 1 to 100 inclusive. The rules of the game are that the student who writes down the smaller number must pay Prof. Ellison that number of dollars. The student who writes down the larger number pays nothing. If both students write down the same number assume that both pay. Assume that both students are risk-neutral and self-interested so that this game can be represented as S1 = S2 = {1, 2, . . . , 100} with s if s s u1(s1, s2) = 1 1 2 0 if s1 > s2 s2 if s2 s1 u2(s1, s2) = 0 if s2 > s1 (a) (3 points) Are any strategies in this game strictly dominated? (b) (7 points) This game has two pure strategy Nash equilibria. What are they? (c) (3 points) Discuss briefly why you should expect given what I told you in part (b) that this game would also have a mixed strategy Nash equilibrium. (d) (15 points) Find a symmetric mixed strategy Nash equilibrium of this game

a. Using the aggregate production function, show that the output/income per capita at time t is a function of the private physical capital per capita at time t and the public physical capital per capita at time t. (5 points)

b. From your answer to a, show that the consumption per capita at time t and the equilibrium private investment per capita at time t are functions of the private physical capital per capita at time t and the public physical capital per capita at time t. (10 points)

c. Show that the change in the public physical capital per capita from time t to time t+1 is a function of the private physical capital per capita at time t and the public physical capital per capita at time t. (5 points)

d. In equilibrium, show that the change in the private physical capital per capita from time t to time t+1 is a function of the private physical capital per capita at time t and the public physical capital per capita at time t. (5 points)

e. Derive the steady-state formulas for the private physical capital per capita and for the public physical capital in per capita. (20 points)

f. Derive the steady-state formulas for the output per capita and the consumption per capita. (10 points)

g. Using your answer to f., derive the tax rate that maximizes the steady-state consumption per capita. (10 points)

The Solow-Swan Model Consider an economy with the general Cobb-Douglas production function: Y, = AK, "Let-a. Answer the following questions, assuming that labour grows at the rate n = 0 and adopting the assumptions made in lecture. The equation describing capital dynamics is: KE+1= 66+16-dk1 where d is a constant parameter. a) Obtain the steady state levels of the capital stock (K), output (X), capital per worker (K), output per worker () and real interest rates (r). Make sure to show all your work. b) Using the help of the Solow-Swan diagram developed in lectures and tutorials, explain the effect of a decrease in the savings rate on the standard of living of the economy. Make sure to include: i) The economic explanation of why the economy experienced a change in the steady state level of capital per worker. ) A diagram illustrating what happened to the relevant curves. ili) Another diagram illustrating the dynamics of the stock of capital in the economy (before, at and after the change in savings). iv) What must have happened to the growth rate of output per worker in the transition between the initial equilibrium and the final one?

1. The Size of the Closed City

Since each square block contains 15,000 square feet of housing and each apartment has 1500 square feet, each square block of the city has 10 households living on it. As a result, a city with a radius of x* blocks can accommodate 10x*2 households (x*2 is the area of the city in square blocks). Suppose the city has a population of 250,000 households. How big must its radius be in order to fit this population? Use a calculator and round off to the nearest block.

2. Housing Prices at the Periphery of the Closed City

Recall that the zoning law says that each developed block must contain 15,000 square feet of floor space. Suppose that the annualized cost of the building materials needed to construct this much housing is $75,000. [Note: $75,000 = iK, where K is the capital needed to build 15,000 square feet of floor space. If i= 5%, then K = $75,000/.05 = $1,500,000.] The annual profit per square block for the housing developer is equal to 15000p-75000-r, where r is land rent per square block. In equilibrium profit for the developer is just equal to zero everywhere. In the absence of any other activities, land rent at x* (call it r*) is just equal to zero. What is the value of p at x*? Call this p*.

3. The Closed City's Corn Consumption Level

Using your results from above, suppose that income per household equals y=$25,000 per year. Next suppose that the commuting cost parameter t equals $10 per block. This means that a person living ten bocks from the CBD will spend 10*10=$100 per year getting to work. The consumers' utility functions are all U(c, q), where c is the quantity of bread (at $1 per loaf) consumed by the household and q is the quantity of housing per household, i.e. 1500 sq. ft. The consumers' budget constraints are all c+pq=y-tx. Under our special assumptions this reduces to c+1500p=25000-10x. Given that a household at x* faces a budget constraint, c+1500p*=25000-10x*, what is the value of c at x*? Call this c*. Explain why in equilibrium every household, regardless of location, must be consuming c*. Notice that since q is the same throughout the city, to have an equilibrium in which U is the same throughout the city, c must be the same throughout the city and everywhere equal to c*

4. The Closed City's Housing Price Function

Substituting the value of c* in place of c into the budget constraint c+1500p=25000-10x solve for p in terms of x. The solution tells what the price per square foot must be at a given location for the household to be able to afford exactly c* worth of bread. This is the city's housing price curve. How does p vary with location?

5. The Closed City's Land Rent Function

Substitute the function for p you just calculated in (4) into the zero-profit constraint for developers: 0=15000p-75000-r. Now solve for r in terms of x. What is the rent per square block at the CBD (x=0)? Plot the bid rent function for land in the city.

Scenario You own a bakery caf which has been successful ior the last 10 years. You have opened 3 branches in the Sydney metropolitan area. As your business gained a great reputation in the market, there were several approaches for franchise inquiries. However, you are concerned that the quality of products and services may be affected; therefore, you have decided not to consider the franchise option. Considering your business objective (to increase sales) and intense competition in the industry, you are looking for collaborative alliances with potential collaborators. As the current production capacity can meet up to 7 times greater than the current sales volume, you will need to find an appropriate collaborator to boost up your sales. The business has the following objectives for business expansion or sales increase:

The quality of the products must be managed and produced under the main branch supervision.

The products are in very high demand as the customers are satisfied with the quality.

The product know-how must be treated as intellectual property and is not an option to share.

The business considers collaborative alliances to expand its business. The partner business must meet the following selection criteria determined by the business:

- The partner business must have a good market reputation.

- The partner business has an annual turnover of at least $2 million.

- The partner business must be able to accelerate your business sales.

- The partner business must be able to support training for collaborative works.

- The partner business must have appropriate communication channels for collaborative alliances.