This document discusses constrained multiplier analysis by relaxing the assumption of unlimited factor resources. It introduces the concept of constraining some sectors' production levels to model resource constraints in agriculture, mining, and government services. The constrained multiplier formula is derived, distinguishing between supply-unconstrained and constrained sectors. A matrix format is used to represent the formula, with the constrained multiplier calculated as the inverse of the identity matrix minus an adjusted coefficient matrix, multiplied by the exogenous components matrix. Readers are directed to a worksheet exercise to calculate constrained multipliers using the mathematical equations and Excel functions.

1. Emerta A. Aragie
November 2023
Addis Ababa, Ethiopia
Constrained Multiplier Analysis

2. Dropping a key assumption
• The three assumptions
• Unconstrained multipliers assume that factor resources are unlimited or
unconstrained
• such that any increase in demand is matched by increased supply.
• We now drop this assumption by preventing/constraining changes in some
sectors’ production levels.
• This is true in the case of resource constrained activities such as agric, mining
and govt services.
• For the sectors with constrained supply, it is net exports that now decline
when demand increases
• (i.e., imports increase to replace any shortfall in domestic production).

3. Constrained multiplier formula (1)
• Earlier, we derived the following demand equations
• We now distinguish between supply unconstrained (Z1), and those with
supply constraints (Z2).
• Previously exogenous components of demand are now treated as endogenous
(E2) (i.e., net exports will now be able to change if domestic production
cannot).
• As with the unconstrained multiplier formula, we group exogenous
components onto the right-hand side…
( ) ( )
11 1 1 1 1 1 12 2 1 2 2 2 1
1-a b -c v b Z -a b -c v b Z =E
+
( ) ( )
21 1 2 1 1 1 22 2 2 2 2 2 2
-a b -c v b Z 1-a b -c v b Z E
+ =
( ) ( )
11 1 1 1 1 1 1 12 2 1 2 2 2
1-a b -c v b Z =E a b c v b Z
+ +
( ) ( )
21 1 2 1 1 1 2 22 2 2 2 2 2
-a b -c v b Z E 1-a b -c v b Z
- = -

4. Constrained multiplier formula (2)
• The equation above can be expressed in matrix format…
• The first term on the left-hand side is the identity matrix (I) minus an
adjusted coefficient matrix (M*). We will call the first term on the right-
hand side the B-matrix.
• Finally, by rearranging terms, we arrive at the constrained multiplier
formula.…

5. Constrained Multiplier Model → Exercise 5 (Task 5)
Complete “Task 5 Worksheet.xls” in the
“Exercises” folder. The solution file is in the
“Solutions” sub-folder.
Task 5:
Constructing a Constrained Multiplier
Model
• Activity: Calculate constrained multipliers
• Translate the mathematical equations presented above into Excel
• Use the MINVERSE and MMULT Excel functions
• Answer the questions in red at the bottom of the worksheet
• Interpret the multipliers