Lecture 14' Inputoutput analysis II'

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Lecture 14. Input-output analysis II.

• Learning objectives. By the end of this lecture
  you should
• Know more about input output analysis
• Understand how to check for the productiveness of
  an input-output system.
• Introduction Inputs and outputs.
• In the last lecture we learnt the basics of
  input-output analysis.
• A key question is whether a given input
  coefficients matrix makes sense, meaning
• Given the input coefficient matrix and provided
  there are sufficient primary inputs, can any
  pattern of final demands be produced?
• Example. OK Computers PLC buys PCs from 3
  suppliers. From the first it keeps the keyboard
  and throws everything else away. From the second
  it keeps the monitor, throwing everything else
  away and from the third it throws away the
  monitor and keyboard. So from 3 computers it
  produces 1 new one.

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2. Example

• A I-A
• So (I-A)-1
• If then x (I-A)-1d
• In other words, to meet final demand for 1 unit
  of the first good, 1.33 units of that good must
  be produced along with 0.46 units of good 2 and
  0.53 units of good 3.

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3. Meeting demand

• Another way to think about this
• To meet final demand d we require
• d the final demand
• Ad the direct intermediate inputs
• A(Ad) the inputs required for the direct
  intermediate inputs
• A(A2d) the inputs required
• Or x d Ad A2d A3d . (IAA2A3)d
• Question on feasibility
• Given any final demand vector d, is there an
  input vector x that will produce d?
• Obviously to be feasible no element of x can be
  negative
• And no element can be infinite

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3. Meeting demand

• Given that,
• x d Ad A2d A3d . (IAA2A3)d and
• All the elements of A are non-negative (so that
  all the elements of An must be positive), then
• Its clear that x wont be negative. But can it
  be infinite?
• Define S (IAA2A3)
• Define Sn (IAA2A3An)
• In other words S is the limit of Sn as n ?8.
• Note that ASn AA2A3An1
• So Sn ASn (IAA2A3An) (AA2A3An1)
• I - An1
• Or
• (I-A)Sn I - An1
• So if (I-A)S I or S (I-A)-1 and x exists.

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4. Hawkins-Simons conditions

• The Hawkins-Simons conditions are conditions on A
  which guarantee that given any final demand
  vector, d, there is an input vector, x, which
  will produce d.
• There are different, but equivalent statements of
  the conditions. We shall state 3 and consider the
  first 2
• The principal minors of (I-A) are all positive.
  
• The dominant eigenvalue is less than 1.
• (dont know what an eigenvalue is? Given a matrix
  A its a solution, ? to the equation . The
  dominant eigenvalue is the one with the largest
  absolute value)
• we are going to define this term on the next
  slide
• If the conditions are satisfied then the
  input-output system is said to be productive.

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4. Hawkins-Simons conditions

• This is the condition we derived earlier. Its
  simple, but may be hard to calculate.
• The principal minors of (I-A) are all positive.
  
• Given an nxn matrix, A, a principal matrix is
  found from A by deleting k rows (e.g. rows 2, 5
  and 7) and the same k columns (so columns 2, 5
  and 7). 0 k n-1. k is called the order of the
  principal matrix.
• The principal minor is the determinant of the
  relevant principal matrix.
• So for a 3x3 matrix, there are
• 1 0-order principal matrix (A itself)
• 3 first order principal matrices
• 3 second order principal matrices.
• So you would need to check 7 determinants.

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4. Hawkins-Simons conditions -example

• Suppose A Are the conditions satisfied?
• I-A is
• The Principal matrices are I-A itself and,
• The relevant determinants are
• 0.352, 0.52, 0.59, 0.54, 0.8, 0.7, 0.8, so the
  conditions are satisfied.
• Note how when we eliminate n rows and columns we
  end up with the diagonal elements of A.

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4. Hawkins-Simons conditions -exercise

• Suppose A
• Find the principal matrices (no need to calculate
  their determinants).
• Note principal minors will resurface in the next
  topic.

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4. Hawkins-Simons conditions OK computers example

• Suppose two goods, computers and software. To
  produce the computer 0.1 unit of software is
  required and to produce 1 unit of software, 0.3
  units of computer and 0.2 units of software are
  needed.
• A
• I-A
• Principal minors are, -2, 0.8 and -1.63.
  Obviously not all positive.
• Note the basic point here if it takes more than
  one unit of a good to produce that good, then the
  Hawkins-Simons conditions can never be satisfied.

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5. Employment multipliers

• Often the number of jobs apparently lost as the
  result of a business closing or the number of
  jobs created as the result of new business
  opening seems far in excess of the number of jobs
  actually with the specific company.
• Input output analysis can be used to calculate
  the knock on effects of changes in employment.

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5. Employment multipliers

• Suppose the only primary input is labour. Recall
  that,
• Note first that this labour demand is in value
  terms
• Consider a change in final demand of ?d, then the
  change in the value of labour demand will simply
  be ?d (because of the circular flow of payment
  within the economy and the fact that we assumed
  that there is only one input).
• It is still useful to break this down into
  sectors using

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How we calculated l (previous lecture)

• Recall final inputs receive the final payments
  the requirement for primary inputs is given by 1
  sum of the column entries in the input matrix
• In the example, 1 of IT goods needs 0.25 input
  of primary inputs.
• The primary inputs coefficient vector, l, is the
  vector of these values

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5. Employment multipliers

• Example. Suppose d1 falls by one unit, what
  happens to the value of employment?

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5. Employment multipliers

• Now l1 0.25, so from a 1 unit drop in demand
  we get
• a 0.25 direct drop in the value of employment in
  sector 1
• a further 0.25 indirect drop in the value of
  employment in sector 1
• a 0.27 indirect drop in sector 2
• a 0.23 indirect drop in sector 3.
• The employment multiplier is the ratio of the
  total change in employment to the direct drop.
• The multiplier is therefore
• I.e. for every one job lost due to the immediate
  effect of the drop in demand, there are 3 jobs
  lost indirectly.

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Conclusion.

• Exercise.
• identify the direct and indirect effects on
  employment if ?d2 2
• what is the employment multiplier for d2?

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Conclusion.

• 7. Summary
• 4 definitions learnt
• Hawkins-Simons conditions
• Principal minors
• 4 skills you should be able to do
• Identify the Principal minors of a square matrix
• Check that the Hawkins Simons are satisfied
• Calculate employment multipliers.