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Expert Answer

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  vlma007answered this
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  Utility function is given by U=F^0.5C^0.5
  Find the MRS which is -MUF / MUC
  MUF = dU/dF = 0.5F^-0.5C^0.5
  MUC = dU/dC = 0.5F^0.5C^-0.5
  Now MRS = -(0.5F^-0.5C^0.5) / (0.5F^0.5C^-0.5)
  = -(C^0.5C^0.5)/(F^0.5F^0.5)
  = - C/F
  From the budget constraint, we have slope = -Price ratios = -coefficient of F / coefficient of C = -2/1 or -2.
  At the optimal choice, utility function is tangent to budget line so MRS = slope of budget line
  - C/F = -2
  C = 2F
  Use C = 2F in the budget equation
  120 = 2F + 2F
  120 = 4F
  F = 120/4 = 30 units
  Then C = 2F = 2*30 = 60 units
  Hence her optimal bundle of consumption should be 30F and 60C
  (This is the correct answer. For a generalized results, I have attached the derivation of formulas)
  A consumer consumes two commodities X and Y and the utility function of the consumer is given by U(X,Y) = X^αY^β, X≥0 and Y≥0. The consumer can purchase the required amount of good X at a price p>0 for each unit of X and the amount of good Y at a price q>0 for each unit of Y. The consumer has exogenously determined income I to spend on goods X and Y. First note that the consumer spends her entire income on purchasing both goods X and Y. Thus, her budget constraint is:
  pX + qY = I
  Consumer wishes to maximize her utility function given by U(X,Y) = X^αY^β. Use Lagrangian method to maximize the utility function with respect to the budget constraint:
  Max Z = X^αY^β + λ(I – pX – qY)
  To solve this equation, set the first order partial derivatives of this equation with respect to X, Y and λ equal to zero. This implies:
  Z’X = 0
  αX^(α-1)Y^β – λp = 0
  Z’Y = 0
  βX^αY^(β-1) – λq = 0
  Z’λ = 0
  pX + qY = I
  Solve the first two equations and note that they are reduced to:
  αY/βX = p/q
  Y = βpX/αq
  Use this relation in the third Lagrangian FOC which can be modified into:
  pX + q*βpX/αq = I
  pX(α + β)/α = I
  X* = (α/α + β) ×I/p
  Plug in this value of X* in Y = βpX/αq
  Y = (βp/αq)*(α/α + β) ×I/p)
  This gives the optimum value of Y* = (β/α + β)×I/q
  Hence we have the constant budget share demand function X* = (α/α + β) ×I/p and Y* = (β/α + β)×I/q
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