Question

Jane's utility function is represented as U=F^0.5C^0.5, F is the quantity of food and C is the quantity of clothing. If her budget constraint is represented as 120= 2F +C, her optimal bundle of consumption should be .... (Please show clearly each mathematical step. Thank you so much!! I really appreciate it.)

A.​(50F, 50C).

B.​(45F, 20C).

C.​(40F, 40C).

D.​(20F, 60C)

Answer 1

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