Anonymous asked in Science & MathematicsMathematics · 9 years ago

apple banana consumer problem math questions?

a consumer measures the utility u of possessing a quantity of x of apples and y of bananas by the formula u=u(x,y)=x^α y^(1-α)

It is known that when the consumer budget for apples and bananas is £1 he will buy 1 apple and 2 bananas when they are equally priced. Use this information to find α? (answer α=1/3) (I am able to get this answer with Lagrange)

The price of apples falls to half that of bananas, with the price of bananas unchanged. How many apples and bananas will the consumer then buy for £10? (answer 20 of each)( I do not seem to be able to get this answer, if you would be able to help me, I would be really really thankful)

2 Answers

  • 9 years ago
    Favorite Answer

    (1) If A is the price of apples and B the price of bananas and the consumer buys x apples and y bananas, the total cost is c = xA + yB. We know that A=B and c=1 so

    xA+yB=1 and therefore x + y = 1/B and y = (1/B) - x so the utility is

    u = x^α [ (1/B) - x]^(1-α) and therefore the maximum occurs when du/dx = 0 i.e.

    α x^(α-1) [ (1/B) - x^(1-α)]^(1-α) - x^α (1-α) [ (1/B) - x]^(2-α) = 0 which simplifies to

    α [(1/B) - x] = x (1-α)

    (α/B) - αx = x - αx and so

    x = α/B which we know to be 1 so α=B, and

    y = (1/B) - x = (1/B) - (α/B) = (1-α) / B = 2 so 1-α = 2B = 2α and therefore 3α=1 and α=B=1/3

    (2) Now B remains 1/3 and A is 1/6. The cost of buying x apples and y bananas is therefore

    c = (x/6) + (y/3) = 10 so y = (60-x)/2. The utility is

    u = x^(1/3) [ (60-x)/2]^(2/3) so the maximum utility occurs when du/dx=0 i.e.

    (1/3) x^(-2/3) [ (60-x)/2]^(2/3) - x^(1/3) (2/3) (1/2) [ (60-x)/2]^(-1/3), so

    (60-x)/2 = x

    60 = 3x and therefore x=20. Then

    y = (60-20)/2 = 20

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  • 3 years ago

    ok thinking chocolate cake people have been one million/3 of the finished, which potential there have been 30 finished. it particularly is thinking you're able to multiply the ten (of the people who like chocolate cake) situations 3. Algebraically it particularly is: x = finished people surveyed 10 = one million/3x multiply the two sides by utilising ability of three 30 = x now you be attentive to there have been 30 people surveyed, in user-friendly terms subtract 6, 4, 6, and 10 from it. you have 4 leftover. So 4 is the respond.

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