midterm

I have a midterm is on Feb12th 2:40-3:45pm

you must make sure you can get 90+!

Practice Problem for Midterm 2
Econ

1

00B, Winter 2021

1. State whether the following production functions exhibit increasing, constant, or decreasing

returns to scale in K and L.

(a) Y = K1/3L1/2

(b) Y = K2/3L

(c) Y = K1/2L1/2

2. Write down the firm’s profit maximizing problem. Be sure to identify the variables the firm

can choose and which it takes as given. What should the firm facing the following scenarios

do?

ˆ If the marginal product of capital is greater than the rental price of capital.

ˆ If the marginal product of labor is less than the wage.

3. Show the transition dynamics in the Solow model if s̄Yt < d̄Kt.

4. Given a production function Yt = ĀK
1/3
t L̄

2/3, if Ā = 2, L̄ = 4, s̄ = 0.2, and d̄ = 0.05.

(a) Calculate the steady-state level of capital.

(b) Does the above production function exhibit constant returns to scale? Or does it exhibit

diminishing marginal returns? Explain, including defining the difference between these

two concepts.

5. Consider the following Romer model of economic growth:

Yt = AtLyt

∆At+1 = z̄AtLat

Lat + Lyt = L̄

Lat = l̄L̄

(a) If A0 = 100, l̄ = 0.1, z̄ = 1/3000 and L̄ = 1000, what is the growth rate of knowledge

in this economy?

(b) What is the growth rate of per capita output in this economy?

(c) Using the information from year 1, what is the level of per capita output in this economy

in year 5?

1

Chapter 4: A Model of Production

Hikaru Saijo

University of California, Santa Cruz

4.2 A Model of Production

• Vast oversimplifications of the real world in a model can
still allow it to provide important insights.

• Consider the following model

• Single, closed economy

• One consumption good

Setting Up the

Model

• Inputs

• Labor (L)
• Capital (K )

• Production function
• Shows how much output (Y ) can be produced given any

number of inputs

• Production function:

Y = F (K, L) = ĀK
1
3L

2
3

• Output growth corresponds to changes in Y .

• There are three ways that Y can change:

• Capital stock (K ) changes.

• Labor force (L) changes.

• Ability to produce goods with given resources (K , L)
changes.

• Technological advances occur (changes in A).
• TFP is assumed to be exogenous in the Solow model.

• The Cobb-Douglas production function is the particular
production function that takes the form of

Y = KaL1−a

Assumed to be a = 1/3. Explained later.

• A production function exhibits constant returns to scale if
doubling each input exactly doubles output.

• Standard replication argument

• A firm can build an identical factory, hire identical
workers, double production stocks, and can exactly
double production.

If the sum of exponents in the inputs. . .

• sum to more than 1 −→ returns to scale

• sum to 1 −→ returns to scale

• sum to less than 1 −→ returns to scale

Returns to Scale Example

Are these production functions IRS, CRS, or DRS?

1 Y = K
1
3 L

1
3

2 Y = L1.1

3 Y = K1.5L0.9

Allocating Resources

max
K,L

Π = F (K, L) − rK −wL

• The rental rate and wage rate are taken as given under
perfect competition.

• For simplicity, the price of the output is normalized to
one.

• The marginal product of labor (MPL)
• The additional output that is produced when one unit of

labor is added, holding all other inputs constant.

• The marginal product of capital (MPK)
• The additional output that is produced when one unit of

capital is added, holding all other inputs constant.

MPL =
∂Y

∂L
=

MPK =
∂Y

∂K
=

If the production function has constant returns to scale in
capital and labor, it will exhibit decreasing returns to scale in
capital alone.

max
K,L
Π = F (K, L) − rK −wL

• The solution is to use the following hiring rules:

• Hire capital until the MPK = r

• Hire labor until MPL = w

Solving the Model

: General Equilibrium

• The model has five endogenous variables:

• Output (Y )
• the amount of capital (K )
• the amount of labor (L)
• the wage (w)
• the rental price of capital (r)

• The model has five equations:

• The production function
• The rule for hiring capital
• The rule for hiring labor
• Supply equals the demand for capital
• Supply equals the demand for labor

• The parameters in the model:

• The productivity parameter
• The exogenous supplies of capital and labor

• Unknowns/endogenous variables:

1 Production function:

2 Rule for hiring capital:

3 Rule for hiring labor:

4 Demand = supply for capital:

5 Demand = supply for labor:

• Parameters/exogenous variables:

• A solution to the model

• A new set of equations that express the five unknowns in
terms of the parameters and exogenous variables

• Called an equilibrium

• General equilibrium

• Solution to the model when more than a single market
clears

Solving the Model

1 Find a market clearing (equilibrium) capital and labor.

2 Plug in the equilibrium capital and labor to the
production function and get output.

3 Given equilibrium capital and labor, solve for prices.

Solving the Model

Solving the Model

Interpreting the Solution

• If an economy is endowed with more machines or people,
it will produce more.

• The equilibrium wage is proportional to output per
worker.
• Output per worker = Y /L

• The equilibrium rental rate is proportional to output per
capital.
• Output per capital = Y /K

In the United States, empirical evidence shows:

• Two-thirds of production is paid to labor.
• One-third of production is paid to capital.
• The factor shares of the payments are equal to the

exponents on the inputs in the Cobb-Douglas function.

Y ∗ = F (K̄, L̄) = ĀK̄1/3L̄2/3

w∗L∗

Y ∗
=

r∗K∗

Y ∗
=

All income is paid to capital or labor.

• Results in zero profit in the economy.
• This verifies the assumption of perfect competition.
• Also verifies that production equals spending equals

income.

w∗L∗ + r∗K∗ =

4.3 Analyzing the Production Model

• Per capita = per person
• Per worker = per member of the labor force

• In this model, the two are equal.
• We can perform a change of variables to define output

per capita (y) and capital per person (k).

• Output per person equals the productivity parameter
times capital per person raised to the one-third power.

y∗ ≡
Y ∗

L∗
=

• What makes a country rich or poor?

• Output per person is higher if the productivity parameter
is higher or if the amount of capital per person is higher.

Draw graph for the per capita versions of the production
function.
Y = ĀK0.7L0.3 and Y = ĀK0.5L0.5

Draw graph for the per capita versions of the production
function.
Y = K − 3ĀL and Y = K + 3ĀL

Comparing Models with Data

• The model is a simplification of reality, so we must verify
whether it models the data correctly.

• The best models:

• Are insightful about how the world works

• Predict accurately

The Empirical Fit of the Production Model

Development accounting:

• The use of a model to explain differences in incomes
across countries

y∗ = Āk̄1/3

• Setting the productivity parameter = 1

y∗ = k̄1/3

• Diminishing returns to capital implies that:

• Countries with low K will have a high MPK

• Countries with a lot of K will have a low MPK , and
cannot raise GDP per capita by much through more
capital accumulation

• If the productivity parameter is 1, the model overpredicts
GDP per capita.

Productivity Differences: Improving the Fit of the

Model

• The productivity parameter measures how efficiently
countries are using their factor inputs.

• Often called total factor productivity (TFP)

• If TFP is no longer equal to 1, we can obtain a better fit
of the model.

• However, data on TFP is not collected.

• It can be calculated because we have data on output
and capital per person.

• TFP is referred to as the “residual.”

• A lower level of TFP

• Implies that workers produce less output for any given
level of capital per person

4.4 Understanding TFP Differences

• Output differences between the richest and poorest
countries?

• Differences in capital per person explain about one-third
of the difference.

• TFP explains the remaining two-thirds.

• Thus, rich countries are rich because

• they have more capital per person.

• more importantly, they use labor and capital more
efficiently.

Why are some countries more efficient at using capital and
labor?

• Human capital

• Technology

• Institutions

• Misallocation

Chapter 5: The Solow Growth

Model

Hikaru Saijo

University of California, Santa Cruz

5.

2

Setting Up the Model: Production

•

Start with the previous production model
• Add an equation describing the accumulation of capital

over time.

• The production function:
• Cobb-Douglas
• Constant

returns

to scale in capital and labor
• Exponent of one-third on K

• Variables are time subscripted (t).

Yt = F (Kt , Lt) =

• Output can be used for consumption or investment.

= Yt

• This is called a resource constraint.
• Assumes no imports or exports

Capital Accumulation

• Goods invested for the future determines the
accumulation of

capital.

• Capital accumulation equation:

Kt+1

=

• Depreciation rate

• The amount of capital that wears out each period
• Mathematically must be between 0 and 1 in this setting
• Often viewed as approximately 10 percent

d̄ = 0.10

• Change in capital stock defined as

∆Kt+1 ≡ Kt+1 − Kt

• Thus:

∆Kt+1 =

• The change in the stock of capital is investment
subtracted by the capital that depreciates in production.

Labor

• To keep things simple, labor demand and supply not
included

• The amount of labor in the economy is given exogenously
at a constant level.

Lt = L̄

Investment

• Farmers eat a fraction of output and invest the rest.

It = s̄Yt

• Therefore:

Ct =

• Consumption is the share of output we don’t invest.

• Unknowns/endogenous variables:

1 Production function:

2 Capital accumulation:
∆Kt+1 =

3 Labor force: Lt = L̄

4 Resource constraint:

Yt =

5 Allocation of resources:
It =

• Parameters/exogenous variables: Ā, s̄, d̄ , L̄, K̄0

5.3 Prices and the Real Interest Rate

• If we added equations for the wage and rental price, the
following would occur:
• The MPL and the MPK would pin them.
• Omitting them changes nothing.

• The real interest rate
• The amount a person can earn by saving one unit of

output for a year
• Or, the amount a person must pay to borrow one unit of

output for a year
• Measured in constant dollars, not in nominal dollars

• Saving
• The difference between income and consumption
• Is equal to investment

Yt − Ct︸ ︷︷ ︸
saving

= It︸︷︷︸
investment

• A unit of investment becomes a unit of capital
• The return on saving must equal the rental price of

capital.

• Thus:
• The real interest rate equals the rental price of capital

which equals the MPK .

5.4 Solving the Solow Model

• The model needs to be solved at every point in time,
which cannot be done algebraically.

• Two ways to make progress
• Show a graphical solution
• Solve the model in the long run

• We can start by combining equations to go as far as we
can with algebra.

• Combine the investment allocation and capital
accumulation equation.

∆Kt+1 =

• Substitute the fixed amount of labor into the production
function.

Yt =

• We have reduced the system into two equations and two
unknowns (Yt , Kt).

The Solow Diagram

• Plots the two terms that govern the change in the capital
stock

s̄Y d̄K

• New investment looks like the production functions
previously graphed but scaled down by the investment
rate.

s̄Y = s̄ ĀK 1/3L̄2/3

Drawing the Solow diagram

Using the Solow Diagram

• If the amount of is greater than
the amount of :
• The capital stock will increase until investment equals

depreciation.
• here, the change in capital is equal to 0
• the capital stock will stay at this value of capital forever
• this is called the steady state

• If is greater than
, the economy converges to the

same steady state as above.

Notes about the dynamics of the model:

• When not in the steady state, the economy exhibits a
movement of capital toward the steady state.

• At the rest point of the economy, all endogenous variables
are steady.

• Transition dynamics take the economy from its initial
level of capital to the steady state.

Output and Consumption in the Solow Diagram

• As capital moves to its steady state by transition
dynamics, output will also move to its steady state.

• Consumption can also be seen in the diagram since it is
the difference between output and investment.

Drawing the Solow diagram with output.

Solving Mathematically for the Steady State

• In the steady state, investment equals depreciation.

s̄Y ∗

= d̄K ∗

• Sub in the production function

= d̄

K ∗

Solving for K ∗

The steady-state level of capital is

• Positively related with

•

•
•

• Negatively related with

•

Solving for Y ∗

The steady-state level of output is

• Positively related with
•
•
•
• Negatively related with
•

• Finally, divide both sides of the last equation by labor to
get output per person (y) in the steady state.

y ∗ ≡ Y
∗

L∗
=

• Note the exponent on productivity is different here (3/2)
than in the production model (1).
• Higher productivity has additional effects in the Solow

model by leading the economy to accumulate more
capital.

By what proportion does per capita output change in the long
run in response to the following changes?

1 Saving rate decreases by 10%.

2 The productivity level falls by 20%.

3 The capital stock increases by 50% as a result of foreign
investment.

5.5 Looking at Data through the Lens of the Solow

Model: The Capital-Output Ratio

• Recall the steady state.

s̄Y ∗ = d̄K ∗

• The capital to output ratio is the ratio of the investment
rate to the depreciation rate:

K ∗

Y ∗
=

s̄

d̄

• Investment rates vary across countries.

• It is assumed that the depreciation rate is relatively
constant.

Differences in Y /L

• The Solow model gives more weight to TFP in explaining
per capita output than the production model.

• We can use this formula to understand why some
countries are so much richer.

• Take the ratio of y ∗ for two countries and assume the
depreciation rate is the same:

y ∗rich
y ∗poor︸ ︷︷ ︸
64

=

(
Ā∗rich
Ā∗poor

)3/2
︸ ︷︷ ︸

32

×
(
s̄∗rich
s̄∗poor

)1/2
︸ ︷︷ ︸

2

5.6 Understanding the Steady State

• The economy reaches a steady state because investment
has diminishing returns.
• The rate at which production and investment rise is

smaller as the capital stock is larger.

• Also, a constant fraction of the capital stock depreciates
every period.
• Depreciation is not diminishing as capital increases.

• Eventually, net investment is

zero.

• The economy rests in steady state.

5.7 Economic Growth in the Solow Model

• Important result: there is no long-run economic growth in
the Solow model.

• In the steady state, growth stops, and all of the following
are constant:

• Output

• Capital

• Output per person

• Consumption per person

• Empirically, however, economies appear to continue to
grow over time.

• Thus, we see a drawback of the model.

• According to the model:

• Capital accumulation is not the engine of long-run
economic growth.

• After we reach the steady state, there is no long-run
growth in output.

• Saving and investment

• are beneficial in the short-run
• do not sustain long-run growth due to diminishing

returns

5.8 Some Economic Experiments

• The Solow model:

• Does not explain long-run economic growth
• Does help to explain some differences across countries

• Economists can experiment with the model by changing
parameter values.

An Increase in the Investment Rate

• Suppose the investment rate increases permanently for
exogenous reasons.

s̄ −→ s̄ ′

• What happens to the economy (capital and output) over
time and in the long run?

• Use Solow diagram to answer the question.

A Rise in the Depreciation Rate

• Suppose the depreciation rate is exogenously shocked to a
permanently higher rate.

d̄ −→ d̄ ′

• What happens to the economy (capital and output) over
time and in the long run?
• Use Solow diagram to answer the question.

A Rise in the Productivity

• Suppose the productivity is exogenously shocked to a
permanently higher rate.

Ā −→ Ā′

• What happens to the economy (capital and output) over
time and in the long run?
• Use Solow diagram to answer the question.

A Destruction of Capital Stock

• Suppose the capital stock is exogenously shocked to a
lower level (due to, for example, a natural disaster).

K −→ K ′

• What happens to the economy (capital and output) over
time and in the long run?
• Use Solow diagram to answer the question.

5.9 The Principle of Transition Dynamics

• If an economy is below
• It will grow.

• If an economy is above
• Its growth rate will be negative.

• When graphing this, a ratio scale is used.
• Allows us to see that output changes more rapidly if we

are further from the steady state.
• As the steady state is approached, growth shrinks to

zero.

• The principle of transition dynamics

• The further below its steady state an economy is, (in
percentage terms)
• the the economy will grow

• The further above its steady state
• the the economy will grow

• Allows us to understand why economies grow at different
rates

Understanding Differences in Growth Rates

• Empirically, for OECD countries, transition dynamics
holds:

• Countries that were poor in 1960 grew quickly.
• Countries that were relatively rich grew slower.

• For the world as a whole, on average, rich and poor
countries grow at the same rate.

• Two implications of this:
• Most countries (rich and poor) have already reached

their steady states.
• Countries are poor not because of a bad shock, but

because they have parameters that yield a lower steady
state (determinants of the steady state invest rates and
A).

5.10 Strengths and Weaknesses of the Solow

Model

The strengths of the Solow Model:

• It provides a theory that determines how rich a country is
in the long run.
• long run = steady state

• The principle of
• allows for an understanding of differences in growth

rates across countries
• a country further from the steady state will grow faster

The weaknesses of the Solow Model:

• It focuses on investment and capital
• the much more important factor of

is still unexplained

• It does not explain why different countries have different
investment and productivity rates
• a more complicated model could endogenize the

investment rate

• The model does not provide a theory of sustained
long-run economic growth