Lecture 4 - The Malthusian Hypothesis

Question 1

Definition

What is the Malthusian epoch characterized by

Also what is endogneous in the Malthusian model

Answer 1

(1) a positive association between income and the size of population (y ↑ ⇒ L ↑) and (2) diminishing marginal returns to labour due to the land constraint ( L ⇒ y ↓)

income and population

Question 2

Proposition

What is income per capita in the Malthusian epoch

Answer 2

stagnant and fluctuating around the subsistence level

Question 3

What does the technological progress do during the Malthusian epoch overall

Answer 3

• increases income per capita above subsistence in S.R
• population increases as long as income remains above subsistence
• average labour productivity declines and income per capita ultimately returns to its long-run level

Draw Graph that represents this

because income increases, people will start having more children as they can sustain more people with the same fixed amount of land, however, as population increases marginal benefits start to diminish

Question 4

What is the output equation (production function) in Malthusian Epoch

Answer 4

Y_t = (AX)^αL_t^1-α

technological progress is exogenous

Question 5

What is output per worker in the Malthusian Epoch

Answer 5

y_t = Y_t/L_t = [AX/L_t]^α

Question 6

what is the utility equation for the preferences of adults at time t

Answer 6

u_t = (n_t)^γ(c_t)^1-γ

n_t = number of children and c_t = consumption

homothetic preferences: if you double consumption and number of children, then you will double utility

Question 7

What is the budget constraint

Answer 7

ρn_t + c_t ≤ y_t
ρn_t/ y_t + c_t/ y_t = y_t / y_t = 1

ρ = cost of raising a child

where ρn_t/ y_t = γ is the proportion of income dedicated to raising children and c_t/ y_t = 1- γ is the proportion of income allocated to consumption

Question 8

Proposition

What is the optimal expenditure on consumption and children a direct proportion of

Answer 8

share of income determind by preferences
(1) ρn_t^* = γy_t
(2)c_t^* = (1- γ)y_t

Draw Graphs to represent this

Proof:

Question 9

Proposition

In the Short run what is the size of the population

Answer 9

(1) directly proportional to income in the previous period (2) a positive and concave function φ(·) of past population, for a given technology
L_t+1 = γ/ρY_t = γ/ρ(AX)^αL_t^1-α = φ(L_t: A)

Question 10

Proof

What is the proof for the size of the population in the short run

Answer 10

1. L_t+1 = n_tL_t and n^*_t = (γ/
   ρ)y_t
2. then L_t+1 = γ/ρy_t L_t
3. Knowing that y_t = Y_t/L_t then we can substitute again for L_t+1 = γ/ρY_t/L_t L_t
4. cancelling out L_t leaves L_t+1 = γ/ρY_t
5. substituting in the production function gives L_t+1 = γ/ρ(AX)^αL_t^1-α

SHORT RUN

Question 11

evoultion of the size of the population graphs

Answer 11

Concave function so will grow exponentially meaning it will increase less and less over time

increase is going to get smaller and smaller until it reaches the steady state L_t+1 = L_t

Question 12

Proposition

what does the concavity of the steady state level of population ensure

(long - run)

Answer 12

unique steady state where population is directly proportional to technology
L_t+1 = L_t = L bar and L bar = [ γ/ρ]^1/αAX = L bar(A)

Question 13

Proof

Proof for the Steady-state level of population

Answer 13

1. if L_t+1 = L_t = L bar then L bar = γ/ρ (AX)^αL bar^1-α
2. the dividing L bar by L bar^1-α gives L bar ^α on the left hand side
3. dividing through by ^α gives L bar =(γ/ρ)^1/α AX

LONG - RUN

The only difference to the short run equation is there is no L_t on the right hand side
This shows that population in the given period does not impact the steady state level of population

Question 14

Proposition

In the short - run what is income per capita

Answer 14

(1) a positive function of land and technology and a negative function of past fertility and population (2) a positive and concave function ψ(·) of current income per capita
y_t+1 = [AX/L_t+1]^α = [AX/n^*_tL_t]^α = y_t/(n^*_t)^α = [ρ/γ]^α y_t^1-α ≡ ψ(y_t)

Technology has a lasting impact on income per capita (increase in income in one period and on the next)

Question 15

What is the proof for the income per capita in the short run

Answer 15

1. substitute in L_t+1 = n_tL_t into y_t+1 = [AX/L_t+1]^α to get [AX/n_tL_t]^α
2. substitute AX/L_t for y_t from the production output equation to get [y_t/n_t^*]
3. if n_t^* = (γ/ρ)y_t
4. y_t/ (γ/ρ)y_t^α = (ρ/γ)^α y_t^1-α

Question 16

Proposition

Steady- state level of income per capita

long - run

Answer 16

The concavity of ψ(y_t) ensures a unique steady-state where income per capita is independent of technology: y_t+1 = y_t = y bar and y bar = ρ/γ

Question 17

Proof of steady state of income per capita

Answer 17

1. y_t+1 = y _t = y bar
2. y bar = (ρ/γ)^α y bar^1-α
3. Dividing y bar by y bar^1-α gives y bar ^α on the left hand side
4. y bar ^α = (ρ/γ)^α
5. cancelling the ^α we have y bar = (ρ/γ)

It is not a function of anything else- there is no technology in the equation
tells you that in the L.R. technology will not impact income per capita
only thing that will impact inome per capita is one of the paramters in the model i.e. cost of raising children and consumption costs

Question 18

Steady-state level of income per capita graphs

Answer 18

If you have a shock to technology, in the S.R. there will be small increase in income but in L.R. because population increases as well, you will return still along line to same steady state level

Question 19

Overall what is the effect of technological progress and better land quality

Answer 19

they lead to higher population density but have no effect on income per capita in the long run

thre is tech progress in S.R. and its effect will temporarily inceease income per capita but then as population increases income per capita is going to return to its inital level near subsistence level