An individual consumer consumes two commodities X₁ & X₂. The utility function is
The price of commodity one is P₁ = Rs.3.00, the price of commodity two is P₂ = Rs.4.00, the individual's income per period is Rs.108. Determine the utility maximizing level of X₁ & X₂ and derive the demand curves for the two commodities.
The consumer maximizes utility subject to the budget constraint. The problem is: Utility function: [ U = X_1^{0.4} X_2^{0.6} ] Budget constraint: [ P_1 X_1 + P_2 X_2 = I \quad \Rightarrow \quad 3X_1 + 4X_2 = 108 ] Step 1: Set up the Lagrangian: [ \mathcal{L} = X_1^{0.4} X_2^{0.6} + \lambda (108 - 3X_1 - 4X_2) ] Step 2: First-order conditions (FOC) by partial derivatives: [ \frac{\partial \mathcal{L}}{\partial X_1} = 0.4 X_1^{-0.6} X_2^{0.6} - 3\lambda = 0 \quad \Rightarrow \quad \lambda = \frac{0.4 X_1^{-0.6} X_2^{0.6}}{3} ] [ \frac{\partial \mathcal{L}}{\partial _______ ____ _________ __________ _____ ______ ______ ___ __________.
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Consider the following two matrices
A =
B =
(i) Find the rank of 'A' and 'B'
(ii) Show that (AB)⁻¹ = B⁻¹ A⁻¹
(iii) Show that (A⁻¹)⁻¹ = A
(iv) Show that (B⁻¹)⁻¹ = B
Singular matrix
Let Z = f(x,y) = 3x³-5y²- 225x + 70y + 23.
(i) Find the stationary points of z.
(ii) Determine if at these points the function is at a relative maximum, relative minimum, infixion point, or saddle point.
Explain the concept of maximum value function.
Solve the following differential equation
given Y(0) = 4
5. If Z = f(x,y) = xy
Find the maximum value for f(x,y) if x and y are constrained to sum to 1 (That is, x+y = 1). Solve the problem in two ways: by substitution and by using the Lagrangian multiplier method.
Denote by a, b and c the column vectors
Calculate
2a - 5b, 2a - 5b + c, at.b,
Let the production function by . Find the elasticity of production with respect to labour (L).