Mathematical derivation of consumer equilibrium Lagrange approach Dr

Mathematical derivation of consumer equilibrium (Lagrange approach) Dr Osakede Bowen University

• Equation 9 simply tells us that in equilibrium the MRS = the slope of the budget line. • We then make q 1 or q 2 the subject of the equation 9 and • Insert in the constraint to obtain the optimal bundle • Lets see an illustration using real values

• QUESTION • Okon’s utility function for R and C is given as U =R 0. 25 C 0. 5. His pocket allowance is N 1, 000 per day. If a unit of R costs N 100, and a unit of C N 350 • a. What is the utility-maximizing consumption bundle for Okon? • b. Show graphically the equilibrium position of Okon as obtained from b above.

• R= 2. 47 • Recall from equation 10 • C= 0. 87 R • Therefore C= 0. 87 * 2. 47 10 C= 2. 15 The utility-maximizing consumption bundle for Okon is to consume 2. 47 units of R and 2. 15 units of C.

Show graphically the equilibrium position of Okon as obtained from b above

Problem set Consumer Choice Model 1. Elizabeth has the following utility function for goods X and Y: U =X 2 Y. Her income is $300 per unit of time, the price of X equals $10 per unit, and the price of good Y equals $2 per unit. a. Find the MRS. b. Calculate and sketch the budget constraint. c. What is the utility-maximizing consumption bundle for Elizabeth? d. How would your answer to part (c) change if the price of X increased to $20 per unit? e. Derive Elizabeth’s demand curve for good X. • . 2. Find the • • • MUx , MUy , and MRS equations for each of the following utility functions. U = x 0. 6 y 0. 4. U = x 2 + y 2 U = 2 x + 4 y. U = x 2 y 2. U = xayb. x, y>0.