ECN101: Intermediate Macroeconomic Theory TA Section

Homework #1 Return Review Suggestions for Homework #2 Take-Home Message Appendix

ECN101: Intermediate Macroeconomic TheoryTA Section

Jae-Wook Jung([email protected])

Department of Economics, UC Davis

October 20, 2014Slides revised: October 20, 2014

ECN101 Section Jae-Wook Jung


Outline

1 Homework #1 ReturnBrief Review of Homework #1

2 ReviewChapter 4Chapter 5

3 Suggestions for Homework #2Problem 3

4 Take-Home Message



Homework Policy

(Revisited) Homework Policy

• 6-7 Homework assignments (15%)• No late submission is accepted.• Homework is due in class on the due day posted. If you can’t

make it to class,• you can drop it in my mailbox at 1111 SSH• or email me before class on the due day.

• Each assignment will be graded by one of TAs for consistency. Itwill be graded on a “V+ (4 pts)”, “V (3 pts)”, and “V- (2 pts)” scalebased on your attempt including the correctness of answers.

• Anyone who earns 15 points or more from assignments gets fullcredit for homework.



Homework Policy

(Revisited) Homework Policy

• Homework assignments will be returned in section next week ofthe due. You have until one week after homework grades areposted on smartsite to notify your TA of a missing grade.

• The University’s Honor Code• You can work together on assignments but EACH individual

student must write up their OWN answers to the questions aswell as the NAMES of ALL your study group members.



Brief Review of Homework #1

Brief Review of Homework #1 – Problem 1

• By how much does GDP rise in each of the following scenarios?Explain.

• What is counted in GDP?• : GDP counts only goods and services “newly” produced “in the

domestic country” and “within a specified quarter or year”• GDP = the value of “final goods”• GDP = sum of the “value-added” at each stage of production• National income identity: Y = C + I + G + NX



Brief Review of Homework #1

Brief Review of Homework #1 – Problem 1

• By how much does GDP rise in each of the following scenarios?Explain.(a) You buy a used car from a friend for $2,500.(e) A computer company buys parts from a local distributor for $1

million, assembles the parts, and sells the computers for $2 million.(f) A real estate agent sells a house for $200,000 that the previous

owners had bought 10 years earlier for $100,000. The agent earnsa commission of $6,000.

(h) A US airline purchases and imports $50 million worth of airplanesfrom the European company Airbus.



Chapter 4

Review: Chapter 4 – A Model of Production

• (Revisited) Example:• Production function: Y = AK 1/3L2/3

• The supply of capital and labor are given exogenously by K = 100and L = 1000.

• Firms maximize their profits taking the prices of all factors as givenby w and r .

• If A = 10, find the equilibrium quantities, Y∗, K∗, and L∗.• Calculate the equilibrium values of prices of inputs, w∗ and r∗.• Why do we assume that the labor is raised to the power of 2/3?



Chapter 4

(Revisited) Review: Chapter 4 – Solving the Model

• First, set up 5 equations as shown in Table 4.1.• From a firm’s profit (= revenue - costs) max. problem,

maxK ,L

Π = AK 1/3L2/3 − rK − wL

• Take partial derivatives with respect to K and L each, then setthem equal to zero.

•

K : A13

K 1/3−1L2/3 − r = A13

(KL

)−2/3− r =

13

YK− r = 0

L : A23

K 1/3L2/3−1 − w = A23

(KL

)1/3− w =

23

YL− w = 0



Chapter 4

(Revisited) Solving the Production Model• These two rules can be written as follow. See 3.5 in my handout

on Cobb-Douglas production function for the detail.

13

YK

= r ,23

YL

= w

• Can you interpret these two rules?

• From the “market clearing condition” which means “(Demand) =(Supply)”

K = K , L = L

• From the production function,

Y = AK 1/3L2/3

• Now we have 5 equations for 5 unknowns (or called,endogenous variables) Y , L, K , w , and r .



Chapter 4

(Revisited) Solving the Production Model1 From the market clearing condition (supply = demand),

K ∗ = K = 100 and L∗ = L = 1000.2 Put these solution into the production function, then

Y ∗ = AK 1/3L2/3 = 10× (100)1/3 × (1000)2/3 = 4641.59.3 From the solution level Y ∗, K ∗ L∗, the equilibrium rental rate of

capital is calculated by

r∗ =13

Y ∗

K ∗=

13

AK 1/3L2/3

K=

13

A( L

K

)2/3= 15.47.

4 Similarly, the optimal wage is given by

w∗ =23

Y ∗

L∗=

23

AK 1/3L2/3

L=

23

A( K

L

)1/3= 3.09.

• We have got the solution for 5 unknowns, K ∗, L∗, Y ∗, r∗, w∗,described with numbers and given values, A, K , L.



Chapter 4

Converting to Per-Capita Output• “Per Capita” means “per person.”• Divide variables with L!• From the Cobb-Douglas production function, Y = AKαL1−α,• Production per capita is give by•

YL≡ y =

AKαL1−α

L

=AKα L1−α

L=AKαL−α

=A(K

L)α

=Akα



Chapter 4

Converting to Per-Capita Output

• With the previous example,

Y = AK 1/3L2/3

•

y =YL

=AK 1/3L2/3

L= AK 1/3L2/3−1 = A

(KL)1/3

= Ak1/3

• Problem 1 in Homework #2 asks this!• Higher k?• Higher A?



Chapter 5

Review: Chapter 5 – Why do we need the Solow Model?

Figure : k is not enough!



Chapter 5

Review: Chapter 5 – Why do we need the Solow Model?

• The Production model can tell us which country is richer thanothers and why.

• Rich countries have higher capital stock (k ) and higher TFP (A).• But it cannot tell us how the country becomes rich over time.

• How does capital accumulate over time?



Chapter 5

Review: Chapter 5 – Ingredients of Solow Model

• production function : Yt = AKαt L1−α

t

• capital accumulation: Kt+1 = Kt + It − dKt

• or ∆Kt+1 = It − dKt

• labor force is exogenously given: Lt = L• resource constraint : Yt = Ct + It , no G and closed economy• allocation of resources or saving rule : It = sYt

• Ct =? ×YtSolution of Solow Model



Chapter 5

Review: Chapter 5 – Exercise

• Yt = AK 1/3t L2/3

t

• A = 1, Lt = L = 10, K0 = 10, s = 0.2, d = 0.1.• Draw a Solow diagram and explain what will happen to the

capital stock of this economy in the short run and in the long run,having as starting point period t = 0.

• Draw a Solow diagram like Fig 5.2. Make sure labeling axes andcurves correctly.

• steady-state K ∗ =(

sAd

)3/2L = 28.28 > K0 =⇒ Kt increases in the

short run because K0 < K ∗. It increases a lot at first, then graduallychanges.

• In the long run, it will converge to K ∗ = 28.28. The way to find thenumber is introduced in Appendix.



Chapter 5

Review: Chapter 5 – Exercise• What is the long run value of per person output, y∗, and per

person consumption c∗?• y∗ = Ak∗1/3 = 1.4142 and c∗ = (1− s)y∗ = 1.1314.

• What if s suddenly increases to s′ = 0.3 in period t = 100?Assume in t = 100 the economy reaches the long-runequilibrium.

• new steady state K ∗∗ =(

0.3Ad

)3/2L = 51.96 > K ∗

• Describe the value of capital in the periods t = 100, 101, 102,and the growth rate of capital between the periods 100− 101,and 101− 102?

• K100 = K ∗ = 28.28,• K101 = K100 + sY100 − dK100 = 29.6944• K102 = K101 + sY101 − dK101 = 31.0370• gK ,100−101 = K101−K100

K100= 0.0500, gK ,101−102 = K102−K101

K101= 0.0452

• Can you tell about the change in C?• gC = gY . Why? Ct = (1− s)Yt . gY = 1

3 gK from Yt = AK 1/3t L2/3

t .Note gL = 0.



Problem 3

HW #2 Problem 3 - Difference between Production Model andSolow Model

• The (steady state) equilibrium value of output per person in theSolow growth model (Chapter 5) is given by

y∗ = A3/2(s/d)1/2 (1)

• The equilibrium value of output per person in the productionmodel (Chapter 4) is given by

y∗ = Ak∗1/3 (2)

a) In (2), y∗ depends on equilibrium capital per person (i.e. k∗). Isthat true in (1) as well? (do not just say no because you do not seek∗ appearing in (1). Think first).

• Why does not k∗ show in (1)?



Problem 3

HW #2 Problem 3 - Difference between Production Model andSolow Model

b) Why do you think the technological parameter A enters with adifferent exponent in the equations? Or equivalently, what is thedifference in the two models that leads to this different result?

• Think about the role of A on output. What are two different roles of Aon output (or on capital)?



Take-Home Message

• The Solow model endogenize capital stock K to explain how Kaccumulate.

• Higher s raises K ∗

• Higher A raises K ∗

• Higher d reduces K ∗

• You need to know• how to get 5 equations and solutions• how to interpret the solution• how to compare Solow model with Production model.

• Homework #2 due by next Tuesday in class.



Solving Solow Model

• The steady state means the time period which all the variableseventually approach constant levels of variables no matter wherethey begin.

• At the steady state(∆Kt+1 = It − dKt = sYt − dKt = 0),I∗ = sY ∗ = dK ∗

•=⇒ K ∗ =

( sd

)Y ∗ =

( sd

)A(K ∗)1/3(L∗)2/3

•

=⇒ (K ∗)2/3 =( sA

d

)(L∗)2/3



Solving Solow Model

•

=⇒ K ∗ =( sA

d

)3/2L∗ =

( sAd

)3/2L

• From the value of K ∗, we can get

Y ∗ =ds

K ∗ =ds

( sAd

)3/2L = A3/2

( sd

)1/2L

C∗ = (1− s)Y ∗ = (1− s)A3/2( s

d

)1/2L

I∗ = sY ∗ = (sA)3/2d−1/2L

L∗ = L

Back to Review


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ECN101: Intermediate Macroeconomic Theory TA Section