L3 - Model of Production Flashcards
What are the benefits of economic growth?
1) Increased GDP per capita - more money/wealth per person
2) Greater life expectancy - people live longer
Has steady growth always occurred?
No, steady growth is a recent phenomenon (since the 1870s - after the Industrial Revolution)
How would you calculate the growth rate between t and t + 1 (discrete time) and the newer/present GDP?
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)
We have encountered exponential growth in a previous lecture - what was that?
Yt = Y0*e^gt
What is the difference between the exponential growth we encountered previously and in this lecture?
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.
How would you derive the annual growth in discrete time formula from, Yt+1 = Yt(1+g)?
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
What is the main equation for discrete time annual growth?
Yt = Y0(1+g)^t
Why is the equation, Yt+1 = Yt (1 + g), still important?
To help derive the discrete time equation:
Yt = Y0(1+g)^t
What else can you do with the annual growth in discrete time equation, Yt = Y0(1+g)^t?
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
State the continuous time and the discrete time exponential growth formulas?
Continuous time:
Yt = Y0*e^gt
Discrete time:
Yt = Y0(1+g)^t
When drawing graphs using the continuous and discrete time exponential formulas respectively, what will be on the y-axis and the x-axis?
Yt on the y-axis and t on the x-axis
What will the graphs for the discrete and continuous time formulas look like?
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
State the useful growth rate arithmetic
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
How can the useful growth rate arithmetic be seen in practice?
It can be seen using the continuous and discrete time formulas
How can you see the useful growth arithmetic with continuous time formulas?
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
How can you see the useful growth arithmetic with discrete time formulas?
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)
What is the purpose of the useful growth rate arithmetic?
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
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?
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
In the last 100+ years, in most developed countries what has the GDP per capita growth been?
GDP per capita (income divided by population so income per person) has grown at a roughly constant rate of 1-3%
In the last 100+ years, in most developed countries what has the capital per worker growth been?
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)
In the last 100+ years, in most developed countries what has the capital per output ratio been?
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
In the last 100+ years, in most developed countries what has labour’s share of income been?
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
In the last 100+ years, in most developed countries what has the rate of return on capital been?
The rate of return on capital has remained roughly constant
In the last 100+ years, in most developed countries what has the real wages growth been?
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
What is long run growth determined by?
Economists believe that long run growth is determined by our increasing (produce more and more) ability to produce goods and services (supply side)
What does an economy’s output (GDP) depend on?
1) quantities of inputs (FOP)
2) ability to turn inputs into output (production function)
Define factors of production
Inputs used to produce goods and services
Give examples of factors of production and define them
2 most important ones:
1) Capital (K) - tools and machinery used by workers
2) Labour (L) - time people spend working
What are some assumptions that can be made about K and L?
That they are in fixed supply
That they are fully utilised
How can you model the relationship between inputs and output?
Using a production function
What is a basic production function?
Y = F(K,L)
Income/output Y is equal to some function/combination of capital K and labour L
What is the most functional form of a production function - state the name and the general formula?
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
If we change the inputs in a production function, like the Cobb-Douglas production function, what happens to output?
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
What is the replication argument?
That we should be able to double our production if we replicate existing factories (I.e find similar land and hire identical workers)
How would you show constant returns to scale?
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
State the equation for the firms profit
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
What do firms aim to do and how do they do this?
Maximise their profit
They can do this by increasing an input - so either the time worked by labour or the amount of capital hired
How do firms know how much to increase their inputs by?
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
What is the marginal product of capital?
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)
How can MPK be calculated?
It is the derivative (partial derivative) of the production function, F(K,L) with respect to K (so keep L as constant)
What can be experienced as you increase capital whilst keeping labour constant?
Diminishing marginal product of labour
What is the diminishing marginal product of capital?
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
How can you show the diminishing marginal product of capital?
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
