Many maximum problems have an associated dual minimum problem. A consumer wishes

Question

Many maximum problems have an associated dual minimum problem. A consumer wishes to minimize the expenditures of consuming a given bundle of two goods (x and y) that generates a specific level of utility Uo, Utility is generated according to 4. xy +3x U= Describe, mathematically the amount of x and y that will minimize the expenditures necessary to obtain the desired level of utility Uo. solve for the specific amounts of x and y.) (4 points) a. (You need not b. Outline the required second order conditions for the minimum for this problem. c. Show graphically and provide an economic explanation for your solution. d. Demonstrate mathematically that your graphical and economic explanations are e. Solve mathematically for the dual utility maximization problem for the f. Explain and show graphically the conditions necessary for the solutions from the (You need not solve for the specific derivatives.) (2 points) (4 points) met. (2 points) consumer. (4 points) two models to be identical. (4 points)

Explanation / Answer

Function to minimise L can be written as

L= p1x+p2y+lambda(U0-x^3(1/y^5+3/y^3)

Differentiating with respect to x and y

dL/dx =p1+lambda*3x^2(1/y^5+3/y^3)=0

dL/dy=p2-lambda*x^3(5/y^6+9/y^4)=0

Comparing both equations

p1/3x^2(1/y^3+3/y^5)=-p2/x^3(5/y^6+9/y^4)

p1/p2 = 3x^2(1/y^3+3/y^5)/x^3(5/y^6+9/y^4)=3/x*(1+3/y^2)/(5/y^3+9/y)=y/x(1+3/y^2)/(9+5/y^2)

d(dL/dx)/dx=6*lambda*x(1/y^5/3/y^3),d(dL/dx)/dy=lambda*3x^2(5/y^6+9/y^4)

d(dL/dy)/dy=lambda*x^3(30/y^7+36/y^5)

D(dL/dx)/dx>0 d(dL/dy)/dy>0

Hence positive semidefinite.hence minina is seen.

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