ECN101: Intermediate Macroeconomic Theory TA Sectionjaewookjung.weebly.com/uploads/2/1/5/2/21526362/2014fq_ecn101_ta... · Announcement Review: Chapter 5 Homework Guide for HW #2

Announcement Review: Chapter 5 Homework Guide for HW #2 Review: Chapter 6 Homework Guide for HW #3 Take-Home Message

ECN101: Intermediate Macroeconomic TheoryTA Section

Jae-Wook Jung([email protected])

Department of Economics, UC Davis

October 27, 2014Slides revised: October 27, 2014

ECN101 Section Jae-Wook Jung


Outline1 Announcement

2 Review: Chapter 5

3 Homework Guide for HW #2Problem 2Problem 4Problem 5Problem 6

4 Review: Chapter 6

5 Homework Guide for HW #3Problem 1Problem 2Problem 3

6 Take-Home Message



Review Sessions for Midterm

• Midterm (35%): Next Thursday, Nov. 6, in class• Review sessions for midterm next week

• Mon.(Nov. 3), 6:10-8:00 p.m., Wellman 119: led by TA SeungduckLee

• Tue.(Nov. 4), 6:10-8:00 p.m., Chemistry 179: led by TA Jae-WookJung

• FYI, my office hour: Tue. this week 3:10 - 5:10 p.m. at 139 SSH• next week office hour: Mon. 3 - 5 p.m. for your HW #3• Other office hour:

• TA Seungduck Lee: Wed. 1-3 p.m. at 117 SSH• Professor Geromichalos: Tue. 10:40-1:30 at 1102 SSH

• Piazza online Q&A forum is always available.



Solow Model

The Steady State Solution of the 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, capital stock stays at constant,Kt = Kt+1 = Kt+2 = · · · .

• Solow model does not explain long-run economic growth.• (∆Kt+1 = It − dKt = sYt − dKt = 0)• I∗ = sY ∗ = dK ∗



Solow Model


• Write K as a function of parameters/exogenous variables.• From sY ∗ = dK ∗,

=⇒ K ∗ =( s

d

)Y ∗ =

( sd

)A(K ∗)1/3(L∗)2/3

•

=⇒ (K ∗)2/3 =( sA

d

)(L∗)2/3

•

=⇒ K ∗ =( sA

d

)3/2L∗ =

( sAd

)3/2L



Solow Model


• 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



Solow Model

Transition Dynamics

• What happens if the parameters change?• Compare new steady state K ∗∗ to original steady state K ∗.

• Keep in mind the steady state solution of K ∗ =(

sAd

)3/2L

• What if s goes down (saving policy), Yt , Ct , It , and Kt?• What if d goes down (smaller depreciation),• What if A goes up (technological progress),• What if L goes up (immigration),• What if K0 goes down (destruction),

• Solow model helps to explain some differences across countries.



Solow Model

Transition Dynamics (s ↓)

Figure : The effect of decreasing sECN101 Section Jae-Wook Jung


Solow Model

Transition Dynamics (s ↓)

Figure : The effect of decreasing sECN101 Section Jae-Wook Jung


Solow Model

Transition Dynamics (d ↓)

Figure : The effect of decreasing dECN101 Section Jae-Wook Jung


Solow Model

Transition Dynamics (d ↓)

Figure : The effect of decreasing dECN101 Section Jae-Wook Jung


Solow Model

Transition Dynamics (A ↑)

Figure : The effect of increasing AECN101 Section Jae-Wook Jung


Solow Model

Transition Dynamics (A ↑)

Figure : The effect of increasing AECN101 Section Jae-Wook Jung


Solow Model

Transition Dynamics (L ↑)

Figure : The effect of increasing LECN101 Section Jae-Wook Jung


Solow Model

Transition Dynamics (L ↑)

Figure : The effect of increasing LECN101 Section Jae-Wook Jung


Solow Model

Transition Dynamics (Destruction of K )

Figure : The effect of destroying Kt



Solow Model

Transition Dynamics (Destruction of K )

Figure : The effect of destroying Kt



Problem 2

HW #2 Problem 2

• Consider the production model that we studied in class (Chapter4) and assume that the production function is now given byY = AK 3/4L1/4. Everything else in the model remainsunchanged.

a) Reproduce the analogue of Table 4.1 (5 equations and 5unknowns) for this new economy. That is, state carefully what areyour endogenous variables, parameters, and what equations youhave in order to solve for equilibrium.

b) Fully describe the solution of the model (in other words provideformulas that relate each endogenous variable of the model withthe known parameters).



Problem 2

HW #2 Problem 2

c) Let A = 1, K = 100, L = 1000. What are the equilibrium values ofthe wage, the rental rate of capital, and total output?

d) How does the equilibrium wage that you reported in (c) change ifL = 1500 and everything else remains the same? Why?

e) Again consider general values for A, K , L. What is the equilibriumvalue of output per person?



Problem 2

HW #2 Problem 2

• We have an alternative production function Y = AK 3/4L1/4.• The exponent of K means the contribution of capital on the

production output, and this is the same as the portion of outputdistributed to capital or capital income share.

• From the profit maximization problem of a representative firm,•

maxK ,L

Π = F (K ,L)− rK − wL = AK 3/4L1/4 − rK − wL



Problem 2

HW #2 Problem 2

• The necessary conditions for maximization problem, F.O.C.’s:•

K : A34

K 3/4−1L1/4 − r = A34

(KL

)−1/4− r =

34

YK− r = 0

L : A14

K 3/4L1/4−1 − r = A14

(KL

)3/4− w =

14

YL− w = 0

• From these two conditions, we get the “rule for hiring capital” andthe “rule for hiring labor”

34

YK

= r ,14

YL

= w



Problem 2

HW #2 Problem 2

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

K = K , L = L

• From the production function,

Y = AK 3/4L1/4

• 5 unknowns(endo. var.), Y , K , L, r , w and 5 equations.• Parameters/exog. var.: A, K , L



Problem 2

HW #2 Problem 2

• How to solve the model• In the solution, or called equilibrium, these 5 equations should be

satisfied by the solutions, the set of the optimal chosen prices,r∗, w∗ and the optimal quantities Y ∗, K ∗, L∗

• The solution must be described with only parameters and exog.variables, here A, K , L



Problem 4

HW #2 Problem 4

• Suppose a country enacts a tax policy that discouragesinvestment. As a result, the value of the parameter s now goes toa smaller value s′.

a) Assuming that the economy starts at its initial steady state, use theSolow model to explain what happens to the economy (after thechange of s) over time and in the long run.

b) Draw a graph showing how output evolves over time (put Yt on thevertical axis and time on the horizontal axis). What happens toeconomic growth over time?



Problem 5

HW #2 Problem 5

• Suppose the level of TFP in an economy rises permanently fromA to A′.

a) Assume that the economy starts at the initial steady state. Use theSolow model to explain what happens to this economy (after thechange of TFP) over time and in the long run.

b) Draw a graph showing how output evolves over time, and explainwhat happens to per capita income.



Problem 5

HW #2 Problem 5

c) How is the response of the economy to the increase in TFPdifferent from the economy’s response to an increase in theinvestment rate (like the one of Problem 4 above)? (Hint: Thinkabout what happens to consumption)



Problem 6

HW #2 Problem 6• In class we pointed out that, in the baseline Solow model, the

variables K ∗, Y ∗ remain constant in the long-run (that is, there isno long run growth since the economy settles at thesteady-state).

a) Verify that the variables y∗, C∗, c∗ are also constant in thelong-run.

b) Now suppose that everything in the Solow model remains the sameexcept one assumption: labor supply is not constant over time anymore, but assume that it grows at a constant rate per period.Intuitively (no need to write down any equations), will the variablesK ∗, Y ∗, y∗, C∗, c∗ still reach a long run steady state?

• Can we define the steady state in aggregate variables?• Yes, aggregate variables grow at the constant rate of gL since per

capita variables stay constant and only labor force Lt grows at theconstant rate of gL. This supports a balanced growth path.



Romer Model

Romer Model: Basic Difference from Solow Model

• Data: Most countries show continuous economic growth atpositive growth rates.

• Solow model: exogenous growth model but no sustained longrun growth

• Why no sustained growth in the long run?• Due to diminishing MPK or constant returns to scale property of

the Cobb-Douglass production function

• How can we solve this?



Romer Model

Romer Model: Basic Difference from Solow Model

• The Romer model introduces nonrivalrous idea stock.• Ideas (A) : infinitely accessible without competition =⇒

nonrivalrous• Objects (K , L) : limited and finite =⇒ rivalrous• Fixed research cost would never be recovered in P = MC

economy.• Problems with pure competition!• =⇒ Introducing increasing returns to scale production function in

ideas



Romer Model

Romer Model: Basic Structure

• Unknowns/Endog. var’s: Yt , At , Lyt , and Lat

• Parameters/Exog. var’s: z, L, l , and A0

• Equations:

Output prod. func. : Yt = AtLyt (1)Flow of knowledge : ∆At+1 = zAtLat (2)

Resource constraint: Lyt + Lat = L (3)

Allocation of labor: Lat = l L (4)



Romer Model

Solving the Romer Model

• What are we solving for?• To show sustained long run growth or positive growth rates in the

long run

• First, note that (3) and (4) make Lyt = (1− l)L.• Then, combine with (1), Yt = At (1− l)L• Now, we can consider the growth rate of output according to the

rule of growth rates.•

g(Yt ) = g(At ) + g(1− l) + g(L)

= g(At )



Romer Model

Solving the Romer Model

• Recall g(Xt ) ≡ ∆Xt+1Xt

= Xt+1−XtXt

.• From (2) and (4),•

g(At ) =∆At+1

At= zLat = z l L > 0

• Thus, g(Yt ) = g(At ) = z l L. i.e., The output and the idea stocksgrow at the same positive constant rate of z l L.

• Balanced Growth (Path)! - All endogenous variables grow at aconstant rate.

• If plugging A0 into (2), we get series of At ’s. Apply these to (1),then we also get series of Yt ’s.



Problem 1

HW #3 Problem 1

• Suppose the economy is on a balanced growth path in theRomer model, and then, in the year 2030, research productivityrises permanently to z ′ > z

a) Solve for the new growth rate of knowledge and yt .b) Make a graph of yt over time (on a ratio scale).

• On a ratio scale, the slope of yt = its (constant) growth rate.c) Why might the parameter z increase in an economy?

• Build your own example. It could be anything to promote technologyin the research sector.



Problem 2

HW #3 Problem 2

• Consider Romer’s growth model of Chapter 6 and let A0 = 100,l = 0.06, z = 1/3000, and L = 1000.

a) What is the growth rate of output per person in this economy?b) What is the initial level of output per person? What is the output per

person in 100 years?• Use yt = A0(1 − l)(1 + g)t , equation (6.9)

c) How do your answers in parts a, b change if the follwoing changesoccur? A doubling of A0, a doubling of l , a doubling of thepopulation L, and a doubling of z.(Note: Explain these changes one at a time).

d) If you could advocate for one of the changes above, which onewould you choose? Why?



Problem 3

HW #3 Problem 3• Consider the following simple variation of the Romer model.

Yt = A1/2t Lyt

At+1 − At = zAtLat

Lyt + Lat = L

Lat = l L

• In short, the only difference is the exponent of At in the (finalgood) production function, so that now there are diminishingreturns to ideas in that sector.

a) Provide an economic interpretation for the first equation.• Now idea exhibits diminishing returns. What does it mean?

b) What is the growth rate of knowledge in this economy?c) What is the growth rate of output per person in this economy?

• First, get an equation similar to (6.9)• Use useful tips for growth rates in Chapter 3.



Take-Home Message

• The rule of thumb for transition dynamics of the Solow model:• Compare old and new steady state K ∗. Which one is bigger than

other?• Change rapidly after shock, then slow down gradually.

• The Solow model can explain the difference of incomes acrosscountries

• It cannot explain the long-run growth.• The Romer model allows the growth of A• All variables grow at the same rate, z l L in the long-run: Balanced

Growth Path.


Documents

ECN101: Intermediate Macroeconomic Theory TA Sectionjaewookjung.weebly.com/uploads/2/1/5/2/21526362/2014fq_ecn101_ta... · Announcement Review: Chapter 5 Homework Guide for HW #2