Exercise 523 verify the leontief utility function on

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Exercise 5.23. Verify the Leontief utility function on R 2 + is not differentiable, but the induced Walrasian demand function is differentiable. 5.2 Comparative statics Suppose φ : R n × R m R , which we will write as φ ( x ; q ). The decision maker gets to choose x to maximize φ , while q are given parameters over which the decision maker has no control. Sometimes x is referred to as the endogenous variable and q as the exogenous variable. The basic comparative statics question is how the decision maker will adjust her optimal choice x * ( q ) in response to changes in the parameters q . In general, there can be multiple maximizers x * ( q ), but we will make life easier here and assume that there exists a unique solution. 30
5.2.1 Comparative statics without constraints using the Implicit Function Theorem Suppose φ ( x ; q ) is a smooth objective function, where x refers to the n -dimensional vector of choice (endogenous) variables and q refers to the m -dimensional vector of exogenous parameters. Consider the simple maximization problem: x * ( q ) = arg max φ ( x ; q ) . Suppose φ is strictly concave and differentiable, so the first order condition defines the maximizer: f ( x ; q ) = D x φ ( x ; q ) = 0 > n . The Jacobian of f is D x f = D xx φ ( x ; q ) and is nonsingular since φ is strictly concave. The first part of the Implicit Function Theorem says we can treat x * ( q ) as continuously differentiable function, at least locally. Moreover, the second part of the Implicit Function Theorem tells us how to compute the local response of the x * ( q ) to small changes in q : D q x * q ) = - [ D x f ( x * q ); ¯ q )] - 1 D q f ( x * q ); ¯ q ) = - [ D xx φ ( x * q ); ¯ q )] - 1 | {z } n × n D qx φ ( x * q ); ¯ q ) | {z } n × m By the Chain Rule, the change of the value function, φ * ( q ) = φ ( x * ( q ); q ) in response to changes in q is: D q φ ( x * q ); ¯ q ) = D q φ ( x, q ) | x = x * q ) ,q q + =0 z }| { D x φ ( x, q ) | x = x * q ) ,q q D q x * ( q ) = D q φ ( x, q ) | x = x * ( q ) In other words, for small changes, the “second order effect” of how the maximizer x * responds to q is irrelevant; simply compute the “first order effect” of how q changes the objective function evaluated at the fixed maximizer x * ( q ). This observation is sometimes called the Envelope (Pseudo-)Theorem. If x and q are scalars, D q x * ( q ) becomes ∂x * ∂q = 2 φ ∂q∂x 2 φ ∂x 2 5.2.2 Comparative statics with constraints using the Implicit Function Theorem Now suppose there are k equality constraints, F i ( x ; q ) = 0, with each F i smooth, which we can stack to form the k -dimensional constraint F ( x ; q ) = 0 k . We are therefore considering the following 31
maximization problem: x * ( q ) = arg max φ ( x ; q ) subject to F ( x ; q ) = 0 k . We assume all constraint qualifications are met. Form the Lagrangian: L ( λ, x ; q ) = φ ( x ; q ) - λ > F ( x ; q ) . The derivative of the Lagrangian is: f ( λ, x ; q ) | {z } 1 × ( k + n ) = D ( λ,x ) L ( λ, x ; q ) = 1 × k z }| { - F ( x ; q ) ; D x φ ( x ; q ) | {z } 1 × n - λ > |{z} 1 × k D x F ( x ; q ) | {z } k × n . At the maximum ( λ * , x * ; q ), we know f ( λ * , x * ; q ) = 0 > k + n . The Jacobian of f is: D ( λ,x ) f | {z } ( k + n ) × ( k + n ) = k × k z}|{ 0 k × n z }| { - D x F ( x ; q ) - D x F ( x ; q ) > | {z } n × k D 2 xx φ ( x ; q ) - D x [ λ > D x F ( x ; q )] | {z } n × n .
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