economic essay

SESSION 6 23RD JANUARY 20

INTRODUCTION TO

ECONOMIC GROWTH

SOMASRI MUKHOPADHYAY

Assumptions Throughout

Some Common Choices to Make • How much to consume today and

how much to save • How much time to spend in to

accumulate skills and how much time to spend working in the labour

market

Summarise the Problems of Optimisation

Summarise the results of optimisation with Elementary Rules Individuals save a constant fraction of

their income And

Spend a constant fraction of their time accumulating skills

Two Equations

Production Function Y = f(K,L)

Capital Accumulation ∆K = sY-βK

The Solow Model (1956) The Two Foundation Pillars

• Y = f (K,L) – Where – Y= Output – K = Capital – L = Labour

• A Cobb-Douglass Production Function – Exhibiting Constant Returns to Scale – Y = KαL 1- α

• Where α ranges between 0 and 1

The Production Function in Solow Model Cobb–Douglas Production Function Functional form of Production Function, representing the technological relationship between the amounts of two or more inputs, particularly physical capital and labor, and the amount of output that can be produced by those inputs.

Constant Rule • f(x) = c then df(x)/dx = 0

Constant Multiple Rule • g(x) = c f(x) then df(x)/dx = c · f0(x)

Power rule • f(x) = xn then df(x)/dx = nx n−1

Sum and difference • h(x) = f(x)±g(x) then dh(x)/dx = df(x)/±dg(x)/dx

Product Rule • h(x) = f(x)g(x) then dh(x)/dx = g(x)df(x)/dx + f(x)dg(x)/dx

Quotient Rule • h(x) = f(x)/g(x), then dh(x)/dx =[ g(x)df(x)/dx − f(x)dg(x)/dx] g(x)2

Chain Rule • h(x) = f(g(x)) then dh(x)/dx = df(g(x)dx)dg(x)/dx

A Brief Look at the Basics (1) Derivatives: The Rules

Finding the Derivative of a Function is the known Differentiation

Derivatives

Slope Or

Rate of change

Optimisation Tools (1)

• For a Firm – the Optimisation Problems Are – Output Maximisation Subject to Cost Constraint

• Produce the Maximum Output at a given Cost – Cost Minimisation subject to Output Constraint

• Minimise cost to Produce a Particular Level of Output

A Brief Look at the Basics (2) Optimisation Behaviour of Producers

• Output = Y • Total Cost Required/Incurred for

Producing the Output = C • Factors of Production = Labour (L) and

Capital (K) • Prices of Factors of Production =

w(Wages to Labour and r (Rent to Capital)

Cost Function C = rK + wLOutput Function

Y = f(K,L)

Capital

La bo

ur

La bo

ur

The Isocost LinesThe Isoquants

Optimisation Tools (2)

A Brief Look at the Basics (3) Optimisation Behaviour of Producers

Cost Minimisation Min C = rK + wL Subject to f(K,L)= Constant

Output Maximisation Max f(K,L) Subject to C = rK + wL = Constant

Capital

La bo

ur La

bo ur

Capital

E

E’

• Slope of Isoquant = -(MPk/MPL) • Slope of Isocost Line = -(r/w) • -(MPk/MPL) =-(r/w)

Widening the Gap Between Output Earning and Factor Spending

Widening the Gap Between Revenue and cost

Maximising Profit

Optimisation Tools (3)

• Maximum Profit Implies a level of Output where if output is further improved there will not be any increase in Profit

• dπ/dy = 0 – The First Order Condition for Optimisation • dπ/dy = dR/dy - dC/dy = 0 • MR = MC

A Brief Look at the Basics (4) Profit Maximisation Behaviour of Producers

• Profit = Revenue Earned – Cost Incurred

• π = TR – TC – Where

• Π = Profit • TR = total Revenue • TC= Total Cost

• Both are functions of the Output • π = R(Y) – C(Y)

Optimisation Tools (4)

• Y = f(K,L) • C = wL+rk • So, Following the Profit Maximisation Condition: �δf/δL – w = 0

�MPL = w �δf/δK – r = 0

�MPK = r

A Brief Look at the Basics (4) Profit Maximisation – Two Inputs Detailed

Note: • Output Changes if Input are Changed. • With Price remaining constant,

Revenue will change if Output Changes.

• Output will change if inputs are changed.

• So, Partial Derivatives ……..to understand the change in output for change in one input keeping the other constant

Profit Maximisation: • Max [f(K,L) – wL – rK] • dY = δf/δL + δf/δK • When Capital is Left

Unchanged • δf/δK = 0

• When Labour is Left Unchanged

• δf/δL = 0

Optimisation Tools (5)

• Apply the Techniques discussed in the Refreshing Slides to – Optimise Profit

• Using a – Cobb-Douglas Production Function

» revealing Constant Returns to Scale » using two inputs Labour and Capital

• and the – Linear Cost Function,

» w being the wage of labour and » r being the rent for capital

Optimisation Tools (6)

Cobb-Douglas Production Function

Q(L,K) = A Lβ Kα

Where:

- Q is the quantity of products.

- L is the quantity of labor.

- K is the quantity of capital.

- A is a positive constant.

- β and α are constants between 0 and 1.

Marginal product is the change in total

production, with per unit change in an

input.

∂Q/∂L- Marginal Product of Labour

In the case of the Cobb-Douglas

production function:

∂Q/∂L = Aβ L(β-1) Kα

If L or K increases, the total output will

increase, that is, the marginal product for

factors is positive.

Cobb-Douglas Production Function and Returns to scale Returns to scale – Change in output level w.r.t change in factors of production by same proportion. If the output increases more than proportionally - increasing returns to scale. If the output increases less than proportionally - decreasing returns to scale. In Cobb-Douglas production function, increasing all factors by a certain proportion C, we get, . Y’ = A (cL)β (cK)α = A cβ Lβ cα Lα = cβ cα A Lβ Kα = c(β+α) Y If all inputs change by a factor of c, output increases by c(β+α). β+α=1 , the production function has constant returns to scale. β+α > 1 , the production function has increasing returns to scale. β+α < 1 , the production function has decreasing returns to scale.

What are these ß and α The are the Output Elasticities of Labour and Capital. Output elasticity measures the responsiveness of output to a change in levels of either labor or capital used in production, ceteris paribus.

• Firms Maximise Profit – Max f(K,L) – (rK+wL)

• Going by Optimisation Problem – Firms will hire labour till their wage per unit is

equal to the contribution in output for an additional unit of labour employed.

– Firms will rent capital to the point where the per unit rent is equal to the contribution in output for an additional unit of capital employed.

Back to Solow Model The Production Function

Reference • Y = f (K,L)

– Where – Y= Output – K = Capital – L = Labour

• A Cobb-Douglass Production Function – Exhibiting Constant Returns to Scale

– Y = KαL 1- α • Where α ranges between 0 and 1

Some More Assumptions • Workers are paid wages – w • Capital is paid rent - r • Perfect Competition Prevails

– Firms are Price Takers

Note: Refer to the “Optimisation Tools” Slides

• Firms Maximise Profit – Max f(K,L) – (rK+wL)

• Going by Optimisation Problem – Firms will hire labour till their

wage per unit is equal to the contribution in output for an additional unit of labour employed.

– Firms will rent capital to the point where the per unit rent is equal to the contribution in output for an additional unit of capital employed.

The Production Function Continued

Reference • Y = f (K,L)

– Where – Y= Output – K = Capital – L = Labour

• A Cobb-Douglass Production Function – Exhibiting Constant Returns to Scale

– Y = KαL 1- α

• Where α ranges between 0 and 1

Some More Assumptions • Workers are paid wages – w • Capital is paid rent - r • Perfect Competition Prevails

– Firms are Price Takers • Let us Derive the

Mathematical Equations

In the given Scenario wL + rK = Y

Max f(L,K) –rK-wL For Maximisation……. MPL = w MPK = r

Output per capita y = Y/L Capital per Worker k = K/L

Refer back to the Profit Equation we looked at.

Note: Refer to the “Optimisation Tools” Slides

• w = δF/ δL ¾ = δ(kαL1-α)/δL ¾ = kα(1-α)L-α

¾ = [(1-α) kα L1-α L-α]/ L1-α

¾ = [(1-α) kα L1-α]/L ¾ = (1-α) Y/L

The Profit Maximisation ……..

Max f(L,K) –rK-wL For Maximisation……. • MPL = w • MPK = r

• Similarly……. • r = δF/ δK ¾ = αY/K

Thus • w = (1-α) Y/L • r = αY/K

Thus • (wL)/Y = (1-α) • (rK)/Y = α

Thus The Factor Shares are Constant over time

Fact 5 of chapter 1 • Real Life Data for

USA for - rK/Y and wL/Y shows no Upward Trend*

*Note: The stylised Facts discussed in Chapter 1 of Text Book shall be discussed in the

Course Gradually.

Production Function How Does it Look Like y

y = Kα

k

A Typical Cobb-Douglas Production Function

Output per worker = Y/L= y Capital per worker = K/L = k

• Y = kα L1-α ¾ Y/L = (kα L1-α )/L

= kα/Lα

= (K/L)α

• Per-capita Output As A Function Of Capital Labour Ratio

• With More Capital Per Worker Firms Produce More Output Per Worker

Thus y = (k)α

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

0

1000

2000

3000

4000

5000

6000

1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970

A nn

ua l P

er ce

nt ag

e C

ha ng

e in

G D

P P

er C

ap ita

G D

P P

er C

ap ita

Year

GDP Per Capita (Current USD)

GDP Per Capita in the USA (1960-1970)

19.50

20.00

20.50

21.00

21.50

22.00

22.50

23.00

23.50

24.00

24.50

0

1000

2000

3000

4000

5000

6000

1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970

C ap

ita l F

or m

at io

n as

a p

er ce

nt ag

e of

G D

P

O ut

pu t P

er C

ap ita

(U S

D )

Year

Chart Title

GDP Per Capita Capital Fromation as a percentage of GDP

USA’s GDP Per Capita (Current USD) and Capital Formation as a Percentage of GDP

C ap

ita l F

or m

at io

n as

a P

er ce

nt ag

e of

G D

P

GDP Per Capita

Relating USA’s GDP Per Capita (Current USD) and Capital Formation as a Percentage of GDP

• dK/dt = sY - ΩK Lets Explain the Equation for Capital Accumulation • dK/dt = change in the Capital stock over time

¾ Capital Accumulation

• sY is the Gross Savings of the Economy • ΩK is the Depreciation Occurring during the

Production Process

Capital Accumulation Savings of the Economy • Consider a Closed Economy • Y = C + S (Consumption and Savings) • Y = C(Y) +S(Y) (Both are functions of

Income/Output Y) • Y = sY + cY • Total Savings S = sY • sY = I (Investment I rented out by

consumers to earn Rent r)

Depreciation of Capital Stock • Depreciation takes out a share of the Gross

Investment • Takes out by a Fixed Portion Ω….Assume

.05. • That is, 5 percent annual depreciation.

Capital Accumulation – Another angle • dK/dt = sY - ΩK Lets Explain the Equation for Capital Accumulation • dK/dt = change in the Capital stock over time

¾ Capital Accumulation

• sY is the Gross Savings of the Economy • ΩK is the Depreciation Occurring during the Production Process

Workers and Consumers Save a certain / Constant Fraction, s, of their Combined Wage and Rental Income Y = wL + rK • Closed Economy • Savings = Investment • The Only Use of this Investment is Capital Accumulation • Consumers Rent out this Capital to firms for Production

• Re-writing the Capital Accumulation Equation in terms of Capital Per Person ¾ The Production function will tell the amount of Output Per

Person produced for the Economy’s Capital Stock Per Person

Evolution of Output Per Person in the Economy…..

• Y = C + S ¾ = cY + sY

¾ c = (1-s)Y ¾ S = I ¾ I = sY

Bringing in Population in the Model

When Population Increases, say by a percentage n • Decreases the Capital

Per Worker.

• Increase in Population by n • Depreciation of Capital by Ω Erodes away the Capita Per worker, which is technically the Capital Labour Ratio

Refer back to the Production function of Solow • L/Y = α K/L ¾Y = αkAny Indication about an

Economy?

Assumption Labour force Participation Rate is constant • If Population Growth Rate is n ¾ Labour force Growth Rate is also n ¾ If n = .01 …..

¾Population and Labour force Growing at one per cent annually.

¾ L(t) = L0ent

Population in the Model

• k = K/L • log (k) = log K - log L • Differentiating…we get • (1/k)(dk/dt) = (1/K)(dK/dt) - (1/L)(dL/dt)

The Mathematical Derivation

• y = αk • log y = α log k • Differentiating we get • (1/y) dy/dt = α(1/k)dk/dt)

• L(t) = L0ent • Using log and differentiating

• (1/L)dL/dt = n

• (1/k)(dk/dt) = (1/K)(dK/dt) - (1/L)(dL/dt) » = [(sY - ΩK)/K] – (n) » = (sY/K) - Ω - n » = (sY/L)/(K/L) – (Ω + n) » = sy/k – (Ω + n)

• dk/dt = sy – (Ω + n)k

The Capital Accumulation Equation

Solving the Solow Model