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39,568 answers
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
