1.   (Dissimilarity Index)

a.   Provide an intuitive interpretation of the dissimilarity index. Why is it important in terms of inequality?

i.      Answer: The dissimilarity index is the percentage of individuals from one of two groups that need to move in order to achieve perfect integration, i.e. if a fraction  X of type A workers is in region 1, then perfect integration requires that a            fraction X of type B workers in region 1, too. Note that this does NOT mean that the same number of type and B workers in region 1! The dissimilarity index is     important for the study of inequality because it provides a measure of the amount of segregation in a given area based on the total number of individuals between   two groups.

b.   Suppose a city that has N = 2 neighborhoods with the following population distribution:

What is the dissimilarity index for this city? How segregated is it? Show your work.

i.      Answer:  = 100  ∑1 | − |

1.   Here  1  = 50,  2  = 450,  1  = 300,  2  = 200,   =  500,   =  500

2.    = 100  (| − |  +  | − |) = 50

3.   This would mean that either 50% of low income people would need to be moved from Neighborhood 2 to Neighborhood 1, or 50% of high income people would need to be moved from Neighborhood 1 to Neighborhood

2 in order to get rid of segregation, which is pretty high!

c.    Suppose there is a difference in the quality of public education between Neighborhood 1 and Neighborhood 2. What do you think that difference is, and why do you think the two neighborhoods differ in this public good?

i.      Answer:  Given that there are far fewer low income people in Neighborhood 1     relative to Neighborhood 2, it’s likely that public education in Neighborhood 1 is of better quality than in Neighborhood 2 on average since education is funded via property taxes and property tax revenue is higher in locations with bigger, more   valuable houses as those commonly owned by wealthier people. One possible      explanation is that poor people care more about high property taxes since             spending on housing constitutes a larger share of their income. An alternative

explanation is that rents are much higher in neighborhood 1 since high-income workers are bidding up land prices there since the schools are better.

2.   (Non-rival goods) Consider the Romer model. Explain what’s special about ideas relative to goods and why they are the center of economic growth.

a.   Answer: Goods are rival while ideas are non-rival. This means only one person can         consume a specific good, but an idea can be used by many people at the same time. This means creating a new idea can raise the output of every worker in the economy at the      same time, while a new good, e.g., a computer, would only raise the output of the worker using it. In this way, ideas can make the whole economy more productive and a               consistent flow of new ideas can lead to sustained growth.

3.   (Romer Model) Suppose there is a city with a firm with a production function  =    , where the change in the “stock of ideas” is expressed as  +1  −   = ̅  . There are a total of 100      workers in the economy, city GDP is $1000 and there are a total of 10 patents in the city at time t.

a.   What output per worker at time t? What does this imply about production in this context?

i.      Answer:   = /  =  

1.     = 1000/100 = 10

2.   Output per worker is 10, which also implies that the production is fully utilizing all of the “ideas” in the economy.

b.   Suppose half of the labor force works in research, and the research is known to be 50%    efficient (use 0.5 rather than 50). Find the stock of ideas at time t+1 based on the equation for long run growth. Show your work.

i.      Answer: Long run growth is calculated as   = ( +1  −  )/  = ̅ × ι ×

1.   The total number of ideas is 10 (which is also the total number of           patents). There are 50 research workers, which means there are 1/5 ideas per research worker. ̅ = 10/50 = 1/5

2.   ( +1  − 10)/10  =  0.2  × 0.5 × 100

3.   ( +1  − 10)/10  =  10

4.    +1  − 10  =  100

5.    +1   =  110

c.   Now suppose there is an additional city. City 2 has a population of 160 workers, with 20 patents at time t, but the same levels of efficiency, same GDP and same fraction of the   labor force in research. Assume the production function and the change in knowledge    functions are the same. What is the average growth rate across both cities? What if you  weighted by population of each city instead of taking the simple average? Show your     work.

i.      Answer: Weighted average growth is   =  , /2 where  = 1,2 represents city

1.    , 1  = 10 from above

2.    ,2  = 0.25 × 0.5 × 160  =  20

a.   ̅2  = 20/80 = 1/4

b.   ι2  = 0.5

c.     =  160

3.   The unweighted average growth rates is   =  = 15 and the     weighted average one is   = (100/260) ⋅ 10 + (160/260) ⋅ 20  = 210/13  ≈ 16. 1538

4.   (Location Choice) Suppose we have the following wages for non-research workers (“normal”) in Cities 1 and 2

and the following levels of productivity for investors in Cities 1 and 2.

Let β =  0.5,  = 5 and  = 10.

a.   If the utility of normals in either location is 5, what is the total number of normals in City

2 in spatial equilibrium? Show your work.

i.      Answer: The utility for normals in city 2 in spatial equilibrium is ̅̅ = β

1.   5 = 10 ⋅ 2−0. 5  ⋅ 10

2.   2  = () −1/0.5  = () −2   =  400

b.   Does City 1 or City 2 have more normals? Why is this the case? Describe in your own words.

i.      Answer: City 1 would have fewer normals if the wages are lower relative to City

2 and/or the amenities are worse. In this case, both are true.

1.   1  = () −1/0.5  = () −2   =  25 < 400

c.    Suppose the utility of inventors in either location is 500. Let γ =  0.75. What is the relative number of inventors between City 1 and City 2? Show your work.

i.      Answer: The utility for investors in city i in spatial equilibrium is ̅̅ =

 ̅−β             

1                                                 1

ii.       1  = ()γ−β     =  ()0. 75−0.25     =  100

1                                                    1

iii.       2  = ()γ−β     =  ()0. 75−0.25     =  625

iv.       = 100/625 = 4/25

d.   Why do you think you got your answer in (c)? Explain.

i.      Answer: Both productivity and amenities are better in City 2.