Detailed explanation of Jones (1965)
1 Outline
2 Notation
3 Basic relations
4 Equations of change (comparative statics)
5 An alternative derivation of the equations of change.
6 Factor intensities
7 Optimal use of inputs
8 Substitution between factors.
9 Endogenous technology
10 Magnification effects:
10.1 Rybzcynski Theorem
10.2 Stolper Samuelson Theorem
11 Demand
2 Notation
L – labor endowment
T – land endowment
M – manufacturing output
F – food output
w – wage
r – rent
pF, pW – competitive market prices
- labor’s share in the value of manufacturing output: and
is the matrix of factor shares in the value of output
- share of factor i used in the production of good j.
- share of labor used in manufacturing: and
is the matrix of sector shares in the factor endowments
- infinitesimal percentage change in x (rate of change in x)
, - elasticities of substitution between land and labor in manufacturing and food.
3 Basic relations
Technology is defined in terms of the amounts of land and labor used to produce one unit of each good:
- quantity of factor i required to produce a unit of commodity j. So, is the amount of land required to produce one unit of Food. The technology matrix is defined as
The dimensions of the technology matrix is (number of factors)X(number of goods). The technology matrix ties together the vector of endowments and the vector of outputs:
or
The equation above corresponds to the following two input market clearing conditions
1)
The two equations above just say that the labor and land are completely used up in production.
Perfect competition will guarantee that the prices equal costs:
or
2)
The input coefficients are a function of the input prices
A micro reminder: unit cost minimization is dual to the profit maximization
or
4 Equations of change (comparative statics)
In what follows we denote percentage changes in variables (also called rates of change) rather than changes in levels. - infinitesimal percentage change in x (rate of change in x).
Using percentage changes allows us for example to make judgments about the real changes in incomes. If the percentage increase in wage is larger than percentage increases in prices of all goods then the real wage unambiguously increases.
Rates of change are also a natural way to describe a magnification effect, when one variable changes more than proportionately to the change of another variable. For example, output of manufacturing changes more than proportionately to the changes in labor endowment. Magnification would just mean that the rate of change in one variable, manufacturing output, is larger than the rate of change in another variable, input endowment. Difference in rates of change of prices is a change in relative price.
The derivation of equations [1.1] through [4.1] follows the same steps. We will perform one derivation in detail leaving the rest to the homework. Let’s derive the first equation of change (equation [1.1] in Jones, 1965) by starting with the first input market clearing condition (equation [1] in Jones,1965) which states that all labor is allocated between production of manufactures, M, or food, F:
3)
Total differentiation of the above equation gives us
4)
A gentle calculus reminder about total differentiation of a sum:
Endowment of labor (L), outputs (M and F), and the input requirements (and) can all change. So the terms in round brackets can be expanded as
5)
A gentle calculus reminder about total differentiation of a product:
Substituting expanded terms from back into gives us:
Now we need to transform this expression from changes to the rates of change. First, divide both sides by L to get the rate of change in labor endowment,, on the right hand side:
To complete the transition from changes (such as and ) to rates of changes (such as and respectively) we divide and multiply every term in the above equation by the necessary variable. We also drop the brackets at this time.
Rearranging multipliers in the above equation produces a slightly better looking expression
6)
Every term on the left hand side consists of two parts: (a) share of industry in the labor endowment and (b) rate of change. Take the first term
Consider each of the two parts of this term:
(a) The first part of this term is . The numerator, , is the amount of labor used in the production of manufactures:
The denominator is the amount of labor, so that the ratio is the share of labor used in the production of food. Denote it as . More generally
- share of factor i used in the production of good j.
Note that the relative endowment is bounded by the ratio of inputs in both industries
(b) The second part of the term is . This is the rate of change in . Denote rates of change in a variable with a hat over that variable: , etc.
Now equation can be rewritten as
rearranging the above equation produces equation (1.1) from Jones (1965)
Following a similar procedure we can derive the rest of the equations.
5. An alternative derivation of the equations of change
There is a more elegant alternative to derive the rate of change which makes use of some properties of total differentiation in terms of rate of change. These properties are outlined in the “gentle calculus reminders” below. Start from the same labor market clearing condition
Using the rule below totally differentiate the equation in terms of rate of change
7)
A gentle calculus reminder:. A useful interpretation of this relation is that the partial elasticity (holding y constant) of the sum, which is z, with respect to one of the summands, say x, is the share of x in z. Two intuitive examples: (1) if every part of some total increases by the same percentage the total will increase by the same percentage; (2) if one percent of a total doubles, increases by 100%, the total will increase only by 1%. Convince yourself, take 2+3=5, 3 is 60% of the total, a 20% increase in 3 should increase the total by 20%x60%=12%. Does it?
The rates of changes on the left hand side can be further decomposed as
8)
A gentle calculus reminder: . A percentage change in a multiplier changes the product the same percentage. Convince yourself, take 20x5=100, if 5 increases by 20% to 6 the product will increase by 20% as well: 20x5x1.2=120
Proof.
1) Total differentiation yields:. Divide both sides by z: . Since , we get .
2) Another way to show the above rule of differentiation in terms of rate of change is to use log-differentiation. Taking natural logarithms of each side of gives us . Total log-differentiation gives. Remember that . So, . This will come in useful later when we derive elasticity of substitution between factors.
Substituting into , using our definition of ’s ( and ), and denoting rates of change with hats we get
,
which can be rearranged to obtain equation (1.1) in Jones (1965)
Homework:
1. Using the same procedure please derive equations (2.1) – (4.1) in Jones (1965).
2. Rewrite equations of change (1.1) – (4.1) in Jones (1965) for the case of constant, Leontieff, input coefficients.
The equations of change are therefore
9)
Where - labor’s share in the value of manufacturing output
To gain some intuition for the equations of change consider a graphical representation of change in endowment. First, consider the change in factor allocation. Note that the length of the factor usage vectors connecting the origin with the endowment point is proportional to the output. When endowment of labor increases (from E to E’) and the input coefficients do not change the output of M increases and the output of F decreases. This can be seen by reduced length of a the factor use vector (parallel to F through E’). The change in outputs does not have to be so dramatic if the input requirements change (both industries will use more labor). The length of the factor use vector sloped F’ that goes through E’ increases as the slope increases from F to F’.
Second, consider the zero profit condition.
As a reminder, full employment of labor and land implies
dividing each equation by the left hand side and using definition of ’s give us
10)
Zero profit conditions imply
dividing each equation by the left hand side and using definition of ’s give us
11)
6 Factor intensities
Factor intensities distinguish the two industries. Manufacturing is arbitrarily chosen to be labor intensive. It means that the amount of labor relative to the amount of land is greater in manufacturing:
12)
One implication of the factor intensity is that labor’s share in manufacturing is greater than labor’s share in food
13)
Proof.
Using the definitions of and inequality can be rewritten as
Using the fact that under perfect competition the price equals cost the above inequality becomes
Divide each side by the numerator
Invert the inequality changing the inequality sign
Subtract 1 from each side, divide both sides by , and invert both side of the resulting inequality
This is the definition of factor intensity given in , Q.E.D.
Another implication of the definition of the factor intensity is that the percentage of labor force used in manufacturing must exceed the percentage of total land that is used in manufacturing.
14)
Proof. Using the definitions of ’s the above inequality becomes
Using factor market clearing conditions in the inequality becomes
Divide each side by the numerator, invert the obtained inequality, subtract 1 from both sides, divide both sides by , and multiply both sides by
; ; ; ;
The last equation is the definition of factor intensity in , Q.E.D.
This result looks very intuitive on the graph
The implications of the factors intensity can be also summarized in the matrix form. Define two matrices
and
The determinants of both matrices is given by
and
Applying restrictions imposed by factor clearing and perfect competition in equations and . The determinants can be simplified to
15)
and
16)
Both determinants are positive by the inequalities and implied by factor intensity.
7 Optimal use of inputs
Producers take prices of inputs and prices of the outputs as given and chooses inputs in such combination that the unit cost is minimized (the first derivative is equal to zero).
Zero profit condition guarantees that the denominator of each equation equals the price. The two equations above can be further transformed (see the previous gentle calculus reminders)
Using our definitions of ’s and noting that the rate of change in is equal to the rate of change in (because w is taken as given) we get two equations that describe optimal input requirements
17)
The two equations above correspond to equations (6) and (7) in Jones (1965). Alternatively, equations could have been obtained from the equations of change in by noting that for a price taking producer it is always the case that
8 Substitution between factors.
The elasticity of substitution between factors is a measure of responsiveness of the amounts of factors used in the production to the changes in input prices. Elasticity of 2 means that if the price of labor increases by 2% the labor to land ratio will decrease by 4%. The elasticity of substitution between land and labor can be defined in one of two equivalent ways (the definition for Food is obviously the same):
or
These expressions can be now expressed in rates of change.
A gentle detour: A rate of change in a fraction is the difference between rates of change in the numerator and the rate of change in the denominator
Proof. There are two ways to see it.
1)
2) The second method uses log differentiation (recall )
The elasticity of substitution between inputs in both industries are given by
18) and
Intuitively, these elasticities determine the shape of the isoquants
Homework:
1. What is the elasticity of substitution for a Cobb-Douglass production function given by where subscript M denotes the amount of factor used in the production of manufacturing output?
2. What is this elasticity for a Constant Elasticity of Substitution (CES) production function given by the following expression?
where
- to guarantee convex isoquants
- because is a share parameter
A useful reminder: notice that Leontief (inputs are perfect complements), Cobb-Douglas (input cost chares are constant), and linear (inputs are perfect substitutes) production functions are all special cases of the CES production function:
If then
If then (this could be shown using L’Hopital’s rule, see below)
If then (this also can be shown applying L’Hopital’s rule)
These properties of the CES production (and similarly utility) function often come in useful in trade models.
L’Hopital’s Rule is useful to calculate limits of indeterminate expressions. If or then .
Application #1:
Let’s apply this rule to the CES production function to show that Cobb-Douglas is a special case. First take natural logarithms of both sides.
Notice that both the numerator and the denominator converge to 0. Applying L’Hopital’s rule we get