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(a) Differentiate the equations. (b) Find the differentials:dQanddP. (c) What are thepartial derivatives:∂Q/∂t,∂Q/∂s,∂P/∂tand∂P/∂s?
53. (10 marks) Consider the utility maximization problemmaxU(x1, x2, x3)subject top1x1+p2x2+p3x3=mwhereU(x1, x2, x3) = ln(x1)+2 ln(x2)+ln(x3) andp1,p2,p3,mare all positive constants.(a) Write down the Lagrangian. (b) Derive the first order conditions. (c) Find the onlypossible solutions forx1,x2,x3andλ, the Lagrangian Multiplier.(d) LetU*denotethe optimal value function for the problem, which meansU*=U*(p1, p2, p3, m) is themaximum value ofU(x1, x2, x3) as a function ofp1,p2,p3,m. Show by direct computationthat∂U*/∂m=λ. (e) Putp1= 5,p2= 2,p3= 5 andm= 100. Estimate the change inthe value ofU*ifmis changed from 100 to 101.
64. (10 marks) Find the tangent plane at (x0, y0, z0) = (1,1,3) to the graph of:f(x, y) = 4 + 3x+x2y2-5y.