Imagine an economy consisting of 1000 members. There is one produced good, corn, which all like to consume. Corn is produced from inputs of labor and seed corn. All members of this society are equally skilled and productive, and all have knowledge of the technologies that exist for producing corn. Each person needs to consume 1 unit of corn per week. Beyond that we assume that people care only about leisure -- that is after consuming one unit of corn they would like to work as few days as possible and use as many days of the week in leisure activities as possible. In this economy there are two ways of producing corn, or two techniques of production: a labor intensive technique, LIT, and a capital intensive technique, CIT.
Labor Intensive Technique:
4 days of labor + 0 units of seed corn; yield 1 unit of corn
Capital Intensive Technique:
1 day of labor + 1 unit of seed corn; yield 2 units of corn
In either case it the corn produced does not appear until Sunday. That is, if I work on Monday through Thursday using the LIT I will get a "yield" of one unit of corn on Sunday. If I work with a unit of seed corn on Monday using the CIT I will get a "yield" of 2 units of corn on Sunday. There is no need to replace seed corn used in the LIT since none is used. On the other hand, if we are to get back to where we started after using the CIT, we would need to use 1 of the 2 units of corn produced to replace the unit of seed corn we used up. Another way of saying this is that the CIT produces 2 gross units of corn but only 1 net unit of corn. So we can think of each technique as producing one net unit of corn, available for consumption at the end of the week. We measure the degree of inequality in the economy as the difference between the maximum number days anyone works and the minimum number of days anyone works. And we measure the efficiency of the economy as the average number of days worked to produce a unit of net corn.
Situation I: There is a total of 500 units of seed corn and 50 people each have 10 units of seed corn and the other 950 have no seed corn at all.
(1) Under autarky, i.e., if we outlaw labor and credit markets:
(a) What will the seedless do?
(b) What will the seedy do?
(c) What will be the degree of inequality in the economy?
(d) What will be the efficiency of the economy?
(2) If we legalize a labor market and each employer hires as many employees as she can:
(a) What will the wage rate be?
(b) What will each seedless person do?
(c) What will each seedy person do?
(d) What will be the degree of inequality in the economy?
(e) What will be the efficiency of the economy?
(3) If we legalize a credit market and each lender lends as much corn as she can:
(a) What will the interest rate be?
(b) What will each seedless person do?
(c) What will each seedy person do?
(d) What will be the degree of inequality in the economy?
(e) What will be the efficiency of the economy?
(4) Based on your answers in Situation I, how does opening a labor market affect:
(a) the efficiency of the economy?
(b) the degree of inequality in the economy?
(c) Explain the intuition behind your answers to (a) and (b).
Situation II: There are 500 units of seed corn and each person has 1/2 unit of seed corn.
(5) Under autarky, i.e. if we outlaw labor and credit markets:
(a) What will each person do?
(b) What will be the degree of inequality in the economy?
(c) What will be the efficiency of the economy, i.e., how many days on average will it take to produce a unit of corn to eat?
(6) If we legalize a labor market and each employer hires as many employees as she can:
(a) What will the wage rate be?
(b) What will each employee do?
(c) What will each employer do?
(d) What will be the degree of inequality in the economy?
(e) What will be the efficiency of the economy?
(7) If we legalize a credit market and each lender lends as much corn as she can:
(a) What will the rate of interest be?
(b) What will each borrower do?
(c) What will each lender do?
(d) What will be the degree of inequality in the economy?
(e) What will be the efficiency of the economy?
(8) In Situation II, in the case of a labor market, how would outcomes be affected if people, on average, preferred to work for themselves rather than for someone else, i.e., if on average people preferred self-managed to alienated labor? What if people, on average, preferred working in the CIT than in the LIT?
Now suppose there is a labor saving technical change in the LIT so it only takes 3 days of labor to make a unit of corn on Sunday, while the CIT technology remains as before. So now we have:
Labor Intensive Technique:
3 days of labor + 0 units of seed corn; yield 1 unit of corn
Capital Intensive Technique:
1 day of labor + 1 unit of seed corn; yield 2 units of corn
Situation III: Suppose the inegalitarian distribution of seed corn now becomes somewhat less unequal. Instead of 50 people starting with 10 units of seed corn each (and 950 starting with none), now 100 people begin with 5 units of seed corn each (leaving only 900 to start with none.)
(9) If we outlaw labor and credit markets:
(a)What will the seedless do?
(b)What will the seedy do?
(c)What will be the degree of inequality in the economy?
(d)What will be the efficiency of the economy?
(10) If we legalize a labor market:
(a) What will the wage rate be?
(b)What will each seedless person do?
(c)What will each seedy person do?
(d) What will be the degree of inequality in the economy?
(e) What will be the efficiency of the economy?
(11) If instead we legalize a credit market:
(a) What will the interest rate be?
(b) What will each seedless person do?
(c) What will each seedy person do?
(d) What will be the degree of inequality in the economy?
(e) What will be the efficiency of the economy?
(12) Based on your answers in Situation III, how does opening a credit market as compared to autarky affect:
(a) the efficiency of the economy?
(b) the degree of inequality in the economy?
(c) Explain the intuition behind your answers to (a) and (b).
(13) With a credit market, if there is another technical change that decreases the amount of labor it takes to produce a unit of corn in the LIT:
(a) Will the interest rate go up, down, or stay the same? Why?
(b) Will inequality in the economy increase, decrease, or stay the same? Why?
(c) Will efficiency increase, decrease, or stay the same? Why?
(14) With a credit market, if there is a technical change that decreases the amount of labor it takes to produce a unit of corn in the CIT:
(a) Will the interest rate go up, down, or stay the same? Why?
(b) Will inequality in the economy increase, decrease, or stay the same? Why?
(c) Will efficiency increase, decrease, or stay the same? Why?
Situation IV: Now suppose the 500 units of seed corn are distributed equally so each of the 1000 people have 1/2 unit of seed corn to begin with.
(15) Under autarky, i.e. if we outlaw labor and credit markets:
(a)What will each person do?
(b)What will be the degree of inequality in the economy?
(c)What will be the efficiency of the economy?
(16) If we legalize a labor market employers hires as many employees as they can:
(a)What will the wage rate be?
(b)What will each employee do?
(c)What will each employer do?
(d) What will be the degree of inequality in the economy?
(e) What will be the efficiency of the economy?
(17) If we legalize a credit market and each lender lends as much corn as she can:
(a)What will the rate of interest be?
(b)What will each borrower do?
(c) What will each lender do?
(d) What will be the degree of inequality in the economy?
(e) What will be the efficiency of the economy?
(18) Based on your answers in Situation IV, how does opening a credit market as compared to autarky affect:
(a) the efficiency of the economy?
(b) the degree of inequality in the economy?
(c) Explain the intuition behind your answers to (a) and (b).
(19) In situation IV, assuming a labor market, how many days would someone have to work in week one in order never to have to work again?
Math Notation, Eigenvalues and Eigenvectors, and the Frobenius-Perron Theorem
(20) What does v(i)a(ij) represent?
(21) What is the difference between v(i) and l(i)?
(22) What is the difference between x(i) and f(i)?
(23) What is the difference between a(ij) and a(ij)*?
(24) What does b(i)l(j) represent?
(25) What does it mean to say that p is unique "up to a multiplicative constant?"
(26) If a(ij)' = a(ij) for i,j = 1.... 25, except a(7,19)' < a(7,19), what can you deduce about the relationship between a' and a?
(27) In theorem 4, if condition i holds, min(j){(zA)(j)/z(j)} = a = max(j){(zA)(j)/z(j)}, what is z?
(28) Is this matrix non-negative and indecomposable?
a(11) = 0.3 a(12) = 0.2
a(21) = 0.2 a(22) = 0.0
(29) Is this matrix non-negative and indecomposable?
a(11) = 0.3 a(12) = 0.0
a(21) = 0.2 a(22) = 0.2
(30) If Az ³ kz and y' > 0 how can we deduce: y'Az > y'kz ?