L3 - Model of Production

Question 1

What are the benefits of economic growth?

Answer 1

1) Increased GDP per capita - more money/wealth per person
2) Greater life expectancy - people live longer

Question 2

Has steady growth always occurred?

Answer 2

No, steady growth is a recent phenomenon (since the 1870s - after the Industrial Revolution)

Question 3

How would you calculate the growth rate between t and t + 1 (discrete time) and the newer/present GDP?

Answer 3

g = (Yt+1 - Yt) / Yt

Note t and t+1 is in the subscript of Y

In words the above equation means that the growth rate is equal to the GDP now minus the GDP in the past over the GDP in the past - essentially change of rate formula (new minus original over original)

Rearranging the formula above for Yt+1:

gYt + Yt = Yt+1

By factorising this becomes:

Yt+1 = Yt (1 + g)

Question 4

We have encountered exponential growth in a previous lecture - what was that?

Answer 4

Yt = Y0*e^gt

Question 5

What is the difference between the exponential growth we encountered previously and in this lecture?

Answer 5

This weeks equation, Yt+1 = Yt (1 + g), is for discrete time when you have 2 time periods, Yt+1 and Yt to mark the newer/present time period and past time period respectively.

Last weeks equation, Yt = Y0*e^gt, is for continuous time.

Question 6

How would you derive the annual growth in discrete time formula from, Yt+1 = Yt(1+g)?

Answer 6

Y1 = Y0(1+g)
… by substituting Y1 into next formula:
Y2 = Y1(1+g) = Y0(1+g)(1+g) = Y0(1+g)^2

If you continued this for Y3 you would get:
Y3 = Y0(1+g)^3

This eventually gives the general equation for calculating the annual growth in discrete time:
Yt = Y0(1+g)^t

Question 7

What is the main equation for discrete time annual growth?

Answer 7

Yt = Y0(1+g)^t

Question 8

Why is the equation, Yt+1 = Yt (1 + g), still important?

Answer 8

To help derive the discrete time equation:
Yt = Y0(1+g)^t

Question 9

What else can you do with the annual growth in discrete time equation, Yt = Y0(1+g)^t?

Answer 9

You can use to derive the continuous growth
equation learnt in the previous lecture:
Yt = Y0*e^gt

This can be done when Yt grows continuously (here n tends to infinity) - I don’t think you need to know how exactly to do this as long as you know it can be done

Question 10

State the continuous time and the discrete time exponential growth formulas?

Answer 10

Continuous time:
Yt = Y0*e^gt

Discrete time:
Yt = Y0(1+g)^t

Question 11

When drawing graphs using the continuous and discrete time exponential formulas respectively, what will be on the y-axis and the x-axis?

Answer 11

Yt on the y-axis and t on the x-axis

Question 12

What will the graphs for the discrete and continuous time formulas look like?

Answer 12

For a small growth rate, g, both formulas will produce very close results - both graphs will be upward sloping (increase exponentially) and are essentially mapped onto each other

Question 13

State the useful growth rate arithmetic

Answer 13

Assume that Xt grows at rate gx and Yt grows at rate gy

If Zt = Xt * Yt then gz = gx +gy
In words this means that if Zt is equal to Xt multiplied by Yt, then the growth rate of Z is approximately (squiggly equals) equal to the growth rate of X plus the growth rate of Y

If Zt = Xt/Yt then gz = gx - gy
In words this means that if Zt is equal to Xt divided by Yt, then the growth rate of Z is approximately (squiggly equals) equal to the growth rate of X minus the growth rate of Y

If Zt = Yt/Xt then gz = gy - gx
In words this means that if Zt is equal to Yt divided by Xt, then the growth rate of Z is approximately (squiggly equals) equal to the growth rate of Y minus the growth rate of X

If Zt = Xt^alpha then gz = alpha * gx
In words this means that if Zt is equal to Xt to the power of alpha, then the growth rate of Z is approximately (squiggly equals) equal to alpha multiplied by the growth rate of X

If Zt = Yt^alpha then gz = alpha * gy
In words this means that if Zt is equal to Yt to the power of alpha, then the growth rate of Z is approximately (squiggly equals) equal to alpha multiplied by the growth rate of Y

Question 14

How can the useful growth rate arithmetic be seen in practice?

Answer 14

It can be seen using the continuous and discrete time formulas

Question 15

How can you see the useful growth arithmetic with continuous time formulas?

Answer 15

Remember that Yt = Y0e^gyt and … Xt = X0e^gxt

… if Zt = Xt * Yt then:

Zt = X0e^gxt Y0e^gyt = Z0*e^(gx + gy)t

… Zt = Z0*e^(gz)t

NOTE above that the x, y and z after g is to just make it clear which growth rate we’re talking about

Question 16

How can you see the useful growth arithmetic with discrete time formulas?

Answer 16

Remember that Yt = Y0(1+g)^t and Yt+1 = Yt (1 + gy) and Xt+1 = Xt (1 + gx)

… (using only the second equation - initial formula used to derive the first equation above) if
Zt+1/Zt = Xt+1/Xt * Yt+1/Yt

Then 1 + gz = (1 + gx)(1 + gy) = 1 + gy + gx + gxgy

If gx and gy are small then gx*gy is tiny and can be ignored … gz = gx + gy (gz is approximately equal, squiggly equals sign, to gx plus gy)

Question 17

What is the purpose of the useful growth rate arithmetic?

Answer 17

These useful rules allow you to calculate the growth rate without spending a lot of time doing the maths to work the growth rate out - for some it would take a lot of rearranging and time consuming maths - the example given for the continuous time formula is probably the simplest out of the 3

Question 18

Applying the growth rate arithmetic, what would the growth rate of GDP per capita be if GDP grows at 10% per year and population at 4% per year?

Answer 18

As GDP per capita is GDP growth / population growth - following the arithmetic rule that here you minus the growth rates (Zt = Xt/Yt hence gz = gx - gy) - you minus 4 from 10 to give 6% which is the growth rate of GDP per capita

Question 19

In the last 100+ years, in most developed countries what has the GDP per capita growth been?

Answer 19

GDP per capita (income divided by population so income per person) has grown at a roughly constant rate of 1-3%

Question 20

In the last 100+ years, in most developed countries what has the capital per worker growth been?

Answer 20

Capital (machinery) per worker (capital over labour) has grown again at a roughly constant rate of 1-3% (similar to that of GDP per capita)

Question 21

In the last 100+ years, in most developed countries what has the capital per output ratio been?

Answer 21

Capital to output ratio (capital over income) has remained at roughly a constant level of 3 which essentially means that capital (in terms of its value I presume) is 3 times greater than the income/output value

Question 22

In the last 100+ years, in most developed countries what has labour’s share of income been?

Answer 22

Labours share of income (wage rate * labour quantity / income/output) has remained roughly constant at 2/3 - which means that labour earns two thirds of the total income earned

Question 23

In the last 100+ years, in most developed countries what has the rate of return on capital been?

Answer 23

The rate of return on capital has remained roughly constant

Question 24

In the last 100+ years, in most developed countries what has the real wages growth been?

Answer 24

Real wages have grown at a rate similar to the growth rate of GDP per capita (1-3%) - makes sense as GDP per capita is quite literally the average wealth per person

Question 25

What is long run growth determined by?

Answer 25

Economists believe that long run growth is determined by our increasing (produce more and more) ability to produce goods and services (supply side)

Question 26

What does an economy’s output (GDP) depend on?

Answer 26

1) quantities of inputs (FOP)
2) ability to turn inputs into output (production function)

Question 27

Define factors of production

Answer 27

Inputs used to produce goods and services

Question 28

Give examples of factors of production and define them

Answer 28

2 most important ones:
1) Capital (K) - tools and machinery used by workers
2) Labour (L) - time people spend working

Question 29

What are some assumptions that can be made about K and L?

Answer 29

That they are in fixed supply
That they are fully utilised

Question 30

How can you model the relationship between inputs and output?

Answer 30

Using a production function

Question 31

What is a basic production function?

Answer 31

Y = F(K,L)

Income/output Y is equal to some function/combination of capital K and labour L

Question 32

What is the most functional form of a production function - state the name and the general formula?

Answer 32

Cobb-Douglas:

Y = AK^alpha*L^1-alpha

Y = output/total production (real value of all good produced in a year)

A - parameter which captures productivity of available technology

Alpha is usually between 0 and 1 and is another parameter

Question 33

If we change the inputs in a production function, like the Cobb-Douglas production function, what happens to output?

Answer 33

If we change inputs by a factor greater than 1, and output changes by the same factor (the factor can be denoted at lambda) then we have constant returns to scale

If output changes by more than the factor we changed the inputs by (greater than factor lambda) then we have increasing returns to scale

If output changes by less than the factor we changed the inputs by (less than factor lambda) then we have decreasing returns to scale

Question 34

What is the replication argument?

Answer 34

That we should be able to double our production if we replicate existing factories (I.e find similar land and hire identical workers)

Question 35

How would you show constant returns to scale?

Answer 35

Remember that if we change inputs by a factor greater than 1, and output changes by the same factor (the factor can be denoted at lambda) then we have constant returns to scale

Also remember that the general form of a production function is: Y = F(K,L) and the Cobb-Douglas general form is Y = AK^alpha*L^1-alpha

If we multiply both inputs, capital and labour, by factor lambda then:

F (lambdaK, lambdaL) = A(lambdaK)^alpha(lambdaL)^1-alpha

Multiply out the brackets so that we get:
A(lambda)^alpha(K)^alphalambda^1-alpha*L^1-alpha

Add the powers of lambda’s being multiplied together - alpha minus alpha plus 1 = 1 …

Lambda[A(K)^aplha*(L)^1-alpha]

Which becomes the following in the general production function form:
Lambda*F(K,L)

Thus, we have constant returns to scale

Question 36

State the equation for the firms profit

Answer 36

Profit = P*F(K,L) - wL - rK

This means that the profit is equal to the price multiplied by the output produced minus the wage rate multiplied by time worked by labour (labour costs) minus the rental price of capital multiplied by the quantity of capital hired (capital costs)

So essentially the profit is equal to revenue minus labour and capital costs

Question 37

What do firms aim to do and how do they do this?

Answer 37

Maximise their profit

They can do this by increasing an input - so either the time worked by labour or the amount of capital hired

Question 38

How do firms know how much to increase their inputs by?

Answer 38

If profit increases by increasing an inout then they should hire more of this input and if it falls then they should hire less

Also need to consider the cost of increasing an input and the relative increase in profit - if cost of increasing inputs greater than increase in output (profit) then not worth doing so

Question 39

What is the marginal product of capital?

Answer 39

The marginal product of capital (MPK) is the extra amount of output produced when one unit of capital is added holding all other inputs constant (ceteris paribus)

Question 40

How can MPK be calculated?

Answer 40

It is the derivative (partial derivative) of the production function, F(K,L) with respect to K (so keep L as constant)

Question 41

What can be experienced as you increase capital whilst keeping labour constant?

Answer 41

Diminishing marginal product of labour

Question 42

What is the diminishing marginal product of capital?

Answer 42

Each additional unit of capital becomes less useful (increases output and therefore profit by less and less) if we keep the amount of labour input constant

In other words, MPK (the marginal product of capital) decreases as we increase K and hold input L constant

Question 43

How can you show the diminishing marginal product of capital?

Answer 43

Use the Cobb-Douglas production function (Y = AK^alphaL^1-alpha)

As we know the MPK is the partial derivative of the production function …

MPK = partial derivative sign Y / partial derivative sign K = alphaAK^alpha-1*L^1-alpha

Put L over K as a fraction so that they both have the same power and can both be to the power of 1-alpha

…

MPK = alpha*A (L/K)^1-alpha

Here you can see that increasing K (capital) will decrease the marginal product of capital as L is held constant

ADD Y/K bit and also do for MPL