Question

1. (a) Find the value of that maximizes the following function. Note thatェ2 0. f(r) 102 +45 Hint: For ar2 + br + c = 0, the values of r which are the solutions of the equation are given by: 2a (b) At the value of r you found in (a), does it satisfy the second-order condition for the maximum? Explain your answer 2. Suppose that a consumer has income of $12 to spend on i and Also suppose that the price of ri įs $2 per unit and the price ofェ2 İs $1 per unit. You will be asked to draw the three budget line in one diagram. Please label each (as (a), (b) and (c)) carefully (a) Draw the consumer's budget line on a diagram where zi is on the horizontal axis and 12 is on the vertical axis. the same graph as (a). Draw the new budget line on the same graph as (a) (b) Suppose that the government imposes a tax of 50% onエ2, Draw the new budget line on (c)Suppoe, instead, that the government imposes a lump-sum tax of $2 to this consumer

Answer 1

As per HOMEWORKLIB RULES first question is answered

Answer:

a) To find the maximum value we have equate

f'(x) = 0

f(x) = 10x - ( 4x + 5 -2x​​​​2​​​ + (1/3)x​​​​​​3​)

So differentiate f(X) wrt X

f'(x) = 10 - (4-4x + 2x2) = -x2 + 4x +6

So now

f'(x) = 0

-x​​​​​​2 +4x +6 = 0

x​​​​​2​​​​​ -4x -6 = 0

x =( -(-4) sqrt (42 - 4(1*-6) )/ (2*1)

x = (46.32)/2

x = 2.064 or -1.16

Maximum value of X = 2.064

2) Second order condition of maximum

f"(X) = 0

f'(X) = -x​​​​​2​​​​+4x+6

f"(X) = -2x + 4

f"(X) = 0

-2x +4 =0

x = 2

Yes the value for second order is satisfying the first order maximum value (approximately)