Contents
Meaning of Duality
The term duality is habitually used to denote a dissimilarity between two correlated concepts, such as duel characteristics of developing economies. It means duality is related to looking for a particular thing in different two alternative ways. Thus, in consumer theory, duality is the alternative way of looking at the consumer’s utility maximization decision. Here we explain the concept and mechanism of duality in consumer theory.
In the microeconomic analysis, duality refers to the relationships between quantities and prices that rise as a consequence of the hypotheses of optimization and convexity. Relating to this, duality is the logical relation between utility and expenditure functions (and profit and production functions), primal and dual linear programs, shadow prices, and several other economic concepts. In the consumer theory, the duality between, say, utility, and expenditure function, arises from a sleight-of-hand/skillful trick with the first order conditions for optimization. These dual relationships, though, are not naturally a product of calculus.
While examining the consumer theory, we can examine the behavior in terms of maximizing utility subject to money income or budget constraint or minimizing the expenditure/cost subject to utility constraint. It means we can look up consumer behavior from both of these perspectives and that is known as consumption duality or the concept of duality in consumer theory.
The duality concept related to consumer behavior (the concept of duality in consumer theory) is based on the fact that preferences can be represented in two forms other than the utility function; these are the expenditure function and the indirect utility function. The practicality of the duality results from two facts. First is the Marshallian demand function. It can be measured from the indirect utility function by differentiation. The second is the Hicksian demand function. It can be computed from the expenditure function by differentiation. The Marshallian and Hicksian demand functions both are obtained only as implicit functions while deriving demand directly from the utility function by the conventional Lagrange method.
Thus, the concept of duality shows that there is a clear relationship between the problem of maximizing a function subject to constraint and the problem of assigning values to constraints. Any constrained maximization that focuses on the constraint in the original (primal) problem. Here its dual problem is to minimize the expenditure or cost required to attain the given utility. Or, a firm’s primal problem may be to minimize the total cost of inputs used to produce a given level of output, whereas the dual problem is to maximize the output for a given level of the total cost of inputs purchased.
Example of Duality
Primal Solution
For instance, we have given the primal problem as a utility maximization problem as;
Maximize U= f (X1, X2)
Subject to B= P1X1+P2X2
Where X1 and X2 are consumption goods. ‘B’ is money income or budget constraint to the consumer and P1 and P2 are the per-unit prices of the goods.
Now, Setting the Lagrangian expression as;
L= X1X2 + λ(B-P1X1-P2X2) ………………(i)
Where λ is an unknown Lagrange multiplier. The first-order conditions for maximization are;
∂L/∂X1= 0 Or, ∂ {X1X2 + λ(B-P1X1-P2X2)}/∂X1 = 0 Or, ∂L/∂X1=X2-λP1=0
∂L/∂X2= 0 Or, ∂ {X1X2 + λ(B-P1X1-P2X2)}/∂X2 = 0 Or, ∂L/∂X2=X1-λP2=0
∂L/∂ λ = 0 Or, ∂ {X1X2 + λ(B-P1X1-P2X2)}/∂λ = 0 Or, B=P1X1+P2X2
The above three equations must be solved simultaneously for finding the values of X1, X2, and λ. Solving the above equations we will get, X1=B/2P1 and X2= B/2P2.
Here, X1 and X2 are the Marshallian demand functions. Thus, the indirect utility function is V=B/2P1* B/2P2=B2/4P1P2=U*
Duality or Dual Solution
The dual of this constrained maximization problem is that for a given utility, the consumer wishes to minimize the cost required to get the given level of utility. Mathematically, the problem is to minimize
Cost E=P1X1+P2X2
Subject to the constraint U*= f (X1, X2)
Setting up the Lagrangian expression as
LD= [P1X1+P2X2 + λ(U*-X1X2)]
Where D denotes the dual concept
The following first-order conditions are needed for a minimization problem
∂LD/∂X1=0 Or, ∂ [P1X1+P2X2 + λ(U*-X1X2)]/ ∂X1=0 Or, P1-λX2=0
∂LD/∂X2=0 Or, ∂ [P1X1+P2X2 + λ(U*-X1X2)]/ ∂X2=0 Or, P2-λX1=0
∂LD/∂λ=0 Or, ∂ [P1X1+P2X2 + λ(U*-X1X2)]/λ=0 Or, U*=X1X2
Solving the above equations, we get X1={U*P2/P1}1/2 and X2={U*P1/P2}1/2
If we substitute the value of indirect utility function U* from the primal solution to the value of X1 and X2 in a dual solution, we get;
X1=B/2P1 and X2=B/2P2. These are known as Hicksian demand functions.
Conclusion
Duality is an important feature of several modern economic theories. Consumer theory is one of them. Duality, in general, is defined as the existence of two logical systems characterized by certain interrelationships. The essence of a dual system is a correspondence or similarity between concepts in one system and concepts in the other which helps us to derive similar results from both systems. The dual approach in studying economic phenomena is very helpful in applied econometrics research as well as in the development of economic theories. In the consumer theory, the concept of duality (the concept of duality in consumer theory) goes through an indirect utility function showing the dependence of utility on price and income and the consumer expenditure function showing the minimum cost of attaining a given utility level for a given set of prices.
The two concept were linked together (duality and consumer theory) , for example when marshallian and hecksian demand functions derived to come up with same results (duality)
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