Find the values of (x,x) that maximize the function f(x,x) = 80x – 2x2- XV.– 3y² + 100y subject to the constraint x + y = 12 a. using the substitution method. b. using the Lagrange multiplier method.
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- Q1) Sales are a function of advertising in newspapers and magazines (X, Y). S = XY2 The price of advertising in newspapers and magazines is Rs.5 and Rs.10 respectively. The total budget for advertising is Rs.105. For maximizing sales, find out the best combination of advertisements in newspapers and magazines by using the Lagrangian multiplier. (Hint: Make equation of the budget line with the help of the above information).For the multiplier to be positive what condition must be satisfied?Sales are a function of advertising in newspapers and magazines (X, Y). 05 S = XY2 Price of advertising in newspapers and magazines are Rs.5 and Rs.10 respectively. The total budget for advertising is Rs.105. For maximizing the sales, find out the best combination of advertisement in newspapers and magazines by using lagrangian multiplier. (Hint: Make equation of budget line with the help of above information).
- Prove that the multiplier is equal to 1/(1-mpc)1. Use the Method of Lagrange to solve this problem. To do so, construct the La- grangean function for this problem. Use λ1 as the Lagrange multiplier attached to the period 1 budget constraint and λ2 as the Lagrange multiplier attached to the period 2 budget constraint.1.2) Suppose that a consumer’s utility function is U=10 lnx+20 lny. A) Find the marginal utility of x, MU_x, and the marginal utility of y, MU_y.B) Suppose that the consumer has R3600 to spend on x and y, while p_x=200 and p_y=400. C) Use the Lagrange multiplier (LM) method to find the levels of x and y that will maximise the consumer’s utility, subject to her budget constraint. D) Find and interpret the value of the Lagrange multiplier. Show that MRCS=p_x/p_y at the utility maximising levels of x and y.
- A consumer is maximising her utility function: U(x, y) = (x¹/³+y¹/³)³, subject to the budget constraint x + 3y = 100. (a) Set up the Lagrangian function of this utility maximisation problem and derive the first-order conditions. (b) What are the utility maximizing amounts of x and y? Also, calculate the Lagrange multiplier. (c) What are the utility maximising amounts of x and y if the budget constraint changes to x + 3y = 50? Also, calculate the Lagrange multiplier.Write down the Planner’s problem, as well as the Lagrangian. (Hint: There should be two constraints because aggregate consumption of each good cannot exceed the economy’s endowment.) Label the multipliers ϕ1 and ϕ2. What are the first order conditions of the Planner’s problem? (Hint: There should be 6.). Find the solution to the Planner’s problem and label the quantities (xpA; ypA) and (xpB; ypB). If we compare the first order conditions of the Planner’s problem to the first order conditions of a corresponding competitive equilibrium (CE), we can obtain a relationship between ϕx, ϕy, λ, px, and py so that the CE and the Planner’s problem give us the same solution. This is called decentralizing the Planner’s problem.Decentralize the Planner’s problem. (find the equilibrium prices and the value of λ so that the two problems have the same solution.) Can you relate this to the two welfare theorems?The utility derived by a consumer from the consumption of two commodities is given by the function U (A, B) = 0.5In (A) + 0.5 In (B) where A are the number of units of the first commodity consumed and B are the number of units of the second commodity consumed each month. A unit of the first commodity costs $8 and a unit of the second commodity costs $ 4 using the Lagrange multiplier method determine the optimal quantity of each of the commodities consumed each month given that consumer has $32 to spend on both commodities each month
- What is constrains optimization, explain and discuss the significance of langrage multiplier.If the consumption function is given by C=20+3/4 Y then the value of multiplier will be???Zeynep lives on an island where she produces two goods, apples (x) and bananas (y), according to the production possibility frontier 200 = x² + y², and she consumes all the goods herself. Her utility function is U(x,y) = xy³. Find her utility maximizing x and y as well as the value of ? (Lagrange multiplier).