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## 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 the 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 utility maximization problem as;

Maximize U= *f* (X_{1}, X_{2})

Subject to B= P_{1}X_{1}+P_{2}X_{2}

Where X_{1} and X_{2} are consumption goods. ‘B’ is money income or budget constraint to the consumer and P_{1} and P_{2} are the per-unit prices of the goods.

Now, Setting the Lagrangian expression as;

L= X_{1}X_{2} + λ(B-P_{1}X_{1}-P_{2}X_{2}) ………………(i)

Where λ is an unknown Lagrange multiplier. The first-order conditions for a maximization are;

∂L/∂X_{1}= 0 Or, ∂ {X_{1}X_{2} + λ(B-P_{1}X_{1}-P_{2}X_{2})}/∂X_{1} = 0 Or, ∂L/∂X_{1}=X_{2}-λP_{1}=0

∂L/∂X_{2}= 0 Or, ∂ {X_{1}X_{2} + λ(B-P_{1}X_{1}-P_{2}X_{2})}/∂X_{2} = 0 Or, ∂L/∂X_{2}=X_{1}-λP_{2}=0

∂L/∂ λ = 0 Or, ∂ {X_{1}X_{2} + λ(B-P_{1}X_{1}-P_{2}X_{2})}/∂λ = 0 Or, B=P_{1}X_{1+}P_{2}X_{2}

The above three equations must be solved simultaneously for finding the values of X_{1}, X_{2,} and λ. Solving the above equations we will get, X_{1}=B/2P_{1} and X_{2}= B/2P_{2}.

Here, X_{1} and X_{2} are the Marshallian demand functions. Thus, the indirect utility function is V=B/2P_{1}* B/2P_{2}=B^{2}/4P_{1}P_{2}=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=P_{1}X_{1}+P_{2}X_{2}

Subject to the constraint U*= f (X_{1}, X_{2})

Setting up the Lagrangian expression as

L^{D}= [P_{1}X_{1}+P_{2}X_{2} + λ(U*-X_{1}X_{2})]

Where *D *denotes the dual concept

The following first-order conditions are needed for a minimization problem

∂L^{D}/∂X_{1}=0 Or, ∂ [P_{1}X_{1}+P_{2}X_{2} + λ(U*-X_{1}X_{2})]/ ∂X_{1}=0 Or, P_{1}-λX_{2}=0

∂L^{D}/∂X_{2}=0 Or, ∂ [P_{1}X_{1}+P_{2}X_{2} + λ(U*-X_{1}X_{2})]/ ∂X_{2}=0 Or, P_{2}-λX_{1}=0

∂L^{D}/∂λ=0 Or, ∂ [P_{1}X_{1}+P_{2}X_{2} + λ(U*-X_{1}X_{2})]/λ=0 Or, U*=X_{1}X_{2}

Solving the above equations, we get X_{1}={U*P_{2}/P_{1}}^{1/2} and X_{2}={U*P_{1}/P_{2}}^{1/2}

If we substitute the value of indirect utility function U* from the primal solution to the value of X_{1} and X_{2} in dual solution, we get;

X_{1}=B/2P_{1} and X_{2}=B/2P_{2}. 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 help us to derive similar results from both the systems. The duel 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 indirect utility function showing the dependence of utility on price and income and consumer expenditure function showing the minimum cost of attaining a given utility level for a given set of prices.