Question

U = 8x₁^0.5+ 2x₂, where x₁ is the quantity of good 1 consumed, and x₂ is the quantity of good 2 consumed. (Yes the x is raised) 8x₁^.5

Suppose that the consumer has a budget of M = $400 to spend and that good 1 has a price of p₁= 2, and good 2 has a price of p₂= 8.

Answer the following questions, and write your answers in the Answer Sheet.

• Write the person’s budget constraint as an equation, with two variables (x₁ and x₂).

• Write the utility maximization problem. This involves rewriting the utility function with the budget constraint substituted in.

• Find the first-order condition.

• Find the combination of goods (x₁ and x₂) that maximizes the consumer’s utility at these prices.

  • What is the optimal x₁?

  • What is the optimal x₂?

    Budget Constraint Equation	400 = 2x1 + 8x2
    Utility Maximization Problem	Max U = 8x10.5 − 0.5x1 + 100
    First-order condition	4x1−0.5 − 0.5 = 0
    Optimal x1	x1=
    Optimal x2	x2=

Answer 1

Hue utility fruction jor. U= 8x,0.5 + 2X2 aget of m= ice of 4400, also, good X2 i PL Now, the con sues has a budget of m= 4400 Price of good xi is P and Price of good Hue P1 = 2 and P2 = 8 Hence the budget constraint will be? M = Pi XI + P2 X2 =) 400 - 2x1 + 8 m2 Now the lag raugian for utility manimization problem will be L 8 x, 0.5 + 2X2 + 1 400-281-8417 Now the first order condition for utility manim sation Problem will be? al x,-0.5. 21 = 0 -0.5 2) =) 2 x,-0.5 =) 2*? ---Fil 8 (0.5) x,-0.5. ai 4x10.5 ca = and 2 - 81 -0 =1 2 x,-0.5 --6) and 400-24, - 8 X2=0 -- Gií) Now equating both values of a from equation is and lis) we get? a> - - => x1=64 now substituting x1=64 in a t=0 equation [equation ins/ we gut :- O a în 400-2X1-8X1 - 0 400 - 2(64) - 8X2 272 - 882 0 272 = 9 X 2 X2 = 34 Scanned with CamScanner

Budget constraint Equation 400 = 2x1 + 8X2 Utility Manimization Problem L= 84,014 2X2 + 1 [460- 2X1 - 8 X first-order Condition *-0 => 43,-0.5.21 = 0 | 22 -0 => 2 -89 = 0 2x2 -034 00 - 2 x,-8X2=0 - optimal Xi 4 34 optimal X2 cs Scanned with CamScanner