### Identify the steady state of relationship commitment

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Homework #3

1. The total adult population in your country is constant and equal to 500,000 people. In your country there are initially 400,000 people in committed relationships (C) and 100,000 people who are not in committed relationships (F). In this example, "C" stands for committed and F for "free". There are pros and cons to being in either group and in your country there is substantial movement between these two categories.

a. What is the initial percentage of people in committed relationships in this country?

Now, suppose that each year the rate of commitment to committed relationships , m, is 10% of the previous year's free population while the rate of separation, s, is 20% of the previous year's committed population. Use this information to answer the following questions. Note: the adult population in your country is unchanging and should therefore always sum to 500,000.

b. Calculate the values for the end of the first year given the values of m and s. You can do this without Excel, but you will find it infinitely easier to do this problem using Excel.

 Committed Relationships (C) Free from Relationships (F) Total Adult Population Initial Levels at Beginning of Year Change due to committing to a relationship XXX Sub-total Change due to leaving committed relationship XXX Total

c. At the end of the first year, what is the percentage of people in committed relationships in this country?

d. Calculate the values for the end of the second year given the values of m and s, and the values you calculated in part (b). Use the following table to report your answers.

 Committed Relationships (C) Free from Relationships (F) Total Adult Population Initial Levels at Beginning of Year Change due to committing to a relationship XXX Sub-total Change due to leaving committed relationship XXX Total

e. At the end of the second year, what is the percentage of people in committed relationships in this country?

f. Calculate the values for the end of the third year given the values of m and s, and the values you calculated in part (d). Use the following table to report your answers.

 Committed Relationships (C) Free from Relationships (F) Total Adult Population Initial Levels at Beginning of Year Change due to committing to a relationship XXX Sub-total Change due to leaving committed relationship XXX Total

g. At the end of the third year, what is the percentage of people in committed relationships in this country?

h. Do you see any pattern emerging from this exercise? Briefly describe the pattern you see and its implications.

i. The rate of commitment to relationships will neither rise nor fall when rise when sC = mF. An alternative way to way to convey this idea is to say that the steady rate of relationship commitment (C/adult population)SS, equals m/(m+s). In this example, what is the steady state of relationship commitment?

j. If this economy is at its steady state of relationship commitment, what do C and F equal?

k. Using the figures you gave in part (j), verify that your answers really do identify the steady state of relationship commitment by filling in the table below.

 Committed Relationships (C) Free from Relationships (F) Total Adult Population Initial Levels at Beginning of Year Change due to committing to a relationship XXX Sub-total Change due to leaving committed relationship XXX Total

l. Suppose as a policymaker you determine that the rate of commitment to relationship is too low and leads to instability in your economy. You decide to offer a variety of programs and incentives to encourage a greater level of commitment to relationships. Suppose you decide that ideally 60% of the adult population in your economy should be in committed relationships. Furthermore, suppose that you cannot alter the rate of separation through public policy. To reach your policy goal of 60%, what would the rate of commitment to relationships need to be in your economy?

m. Using the information you calculated in part (l), verify that your answers really do identify the steady state of relationship commitment by filling in the table below.

 Committed Relationships (C) Free from Relationships (F) Total Adult Population Initial Levels at Beginning of Year Change due to committing to a relationship XXX Sub-total Change due to leaving committed relationship XXX Total

2. You are given the following information about a small, open economy.

Y = F(K, L) = A Ka L1-a
C = C(Y - T) = 20 + .6(Y - T)
I = I(r) = 100 - 1000r
Y = C + I + G + NX
Y = C + SP + T
G = 20
T = 20
K = 100
L = 100
a = .25
A = 2
(M/P)S = 1000
(M/P)D = 2000 - 20000r

a. Find the level of aggregate output, Y , for this economy.

b. Find the level of consumption spending in this economy.

c. Find the level of private saving, SP , for this economy.

d. Find the level of government saving, SG , for this economy.

e. Find the level of national saving, S, for this economy.

f. Find the level of investment, I, for this economy.

g. Assume this economy is in equilibrium where the level of production equals the level of spending. What is the relationship between S, I, and NX in this economy?

Now, suppose the government in this small open economy increases government spending by 10, holding everything else constant. Calculate the rest of this problem based on this change in government spending.

h. Find the level of aggregate output, Y , for this economy.

i. Find the level of consumption spending in this economy.

j. Find the level of private saving, SP , for this economy.

k. Find the level of government saving, SG , for this economy.

l. Find the level of national saving, S, for this economy.

m. Find the level of investment, I, for this economy.

n. Assume this economy is in equilibrium where the level of production equals the level of spending. What is the relationship between S, I, and NX in this economy?

3. The following equations describe a small open economy using a Classical (long run) Model.

Y = F(K, L) = A Ka L1-a
C = C(Y - T) = 20 + .8(Y - T)
I = I(r) = 100 - 10r (where r is expressed as the percentage, e.g. if r = 5%, then in the equation r would have a value of 5)
Y = C + I + G + NX
Y = C + SP + T
G = constant at 200
T = constant at 100
K = 100
L = 400
a = .5

A = a measure of technology, exogenously given as 2

a. What is aggregate output, Y, for this economy?

b. If the world real interest rate is 5%, or r = 5, then calculate C and I.

c. Given the information you have, what is the value of net exports for this economy?

d. Fill in the following table for this economy.

 Disposable Income Private Saving Government Saving National Saving Net Capital Outflows

e. Suppose government spending increases by 50. Fill in the following table.

 Y C I G NX Private Saving Government Saving National Saving Net Capital Outflows

f. Did the increase in government spending crowd out investment as it did with a closed economy using a Classical Model? Explain your answer.

g. What is the relationship between spending and domestic production in this economy?

h. What happens to this country's real exchange rate when government spending increases, holding everything else constant? Explain your answer.

4.

a. Fill in the missing values in the table below.

 Sector Expenditures on Goods and Services Produced Domestically Expenditures on Goods and Services Produced Internationally Total Expenditures Domestic Households Cd = 2000 Cf = 1000 C = Domestic Businesses Id = If = 200 I = 500 Domestic Government Gd = 400 Gf = G = 500 TOTAL

b. Using the above information and assuming exports equal 500 complete the following table.

 Y C I G EX IM

c. Using the above data and information, fill in the following table.

 Y Cd Id Gd EX

d. Is the value of Y the same in the second and third tables?

5. Complete the following table.

 Year Y C I G NX Private Saving Govt. Saving Nat'l Saving Trade Surplus Net Capital Outflow T A 2000 1200 400 200 200 B 2000 1200 400 200 200 C 2000 1200 400 100 100

6. Suppose you are given the production function

Y = F(K, L) = AK1/2L1/2

Where Y is real GDP, K is capital, L is labor, and A is a parameter expressing the level of technology available in this economy.

a. Suppose K and L are both tripled in this economy. Does this production function exhibit constant returns to scale? Provide a proof of your answer.

b. Rewrite the above production function in terms of output per worker, y. That is, normalize the function so that it can be written as y = f(k) where k is capital per worker. Provide a verbal interpretation of your mathematical result.

c. Using the expression you found in part (b) and assuming labor is equal to 25 units, fill in the following table. Recall that k, the capital to labor ratio (K/L), measures capital intensity: that is, it provides a measure of how much capital each unit of labor has to work with in the economy. Assume that A equals 2 in this problem. Hint: this might be a good time to pull out your Excel spreadsheet program.

 L K k =K/L y Y MPk 25 0 XXX 25 100 25 200 25 300 25 400 25 500 25 600 25 700 25 800 25 900

d. Use your Excel spreadsheet program to graph k and y, placing k on the horizontal axis and y on the vertical axis.

e. Let's define s as the saving rate per worker, i as the amount of investment per worker, and c as consumption per worker. Suppose that you are told that the saving rate in this economy is constant and equal to .2. Use this saving rate and the values you found in part (c) to fill in the following table. Note that i equals sy and y = c + i.

 L K k = K/L y s i c 25 100 25 200 25 300 25 400 25 500 25 600 25 700 25 800 25 900

f. Suppose you are also told that depreciation of the capital stock, δ, is equal to 10% each year. Using this information, plus the information you have previously been given or have calculated compute the change in the capital intensity, Δk, in this economy. Find Δk for each level of K in the economy by filling in the table below. Recall that the change in the capital intensity is equal to the difference between the level of investment per worker and the amount of capital per worker that depreciates during the given time period.

 L K k = K/L y s i c δk Δk 25 100 25 200 25 300 25 400 25 500 25 600 25 700 25 800 25 900

g. Now, graph the amount of depreciation, δk; the level of investment per worker, i (which is also equal to sf(k)); and the capital intensity, k. Place k on the horizontal axis and δk and sf(k) on the vertical axis.

h. Suppose we define k* as the steady-state level of capital intensity or the level of capital per worker where the amount of capital depreciation per worker equals the level of investment per worker in each period. At k* the capital intensity is constant over time. Calculate the value of k* for your economy. Does your calculation agree with your findings in parts (f) and (g)?

i. What would happen to k* if s increased, holding everything else constant? Draw a sketch illustrating the initial situation and the effect of an increase in s. [Hint: why not run your Excel program again but this time with a higher value for s (I used .3)?]

j. What would happen to k* if δ decreased, holding everything else constant? Draw a sketch illustrating the initial situation and the effect of an increase in δ? [Hint: why not run your Excel program again but this time with a higher value for δ (I used .08)?]

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