A third consumer, Larry has preferences
U
C
(
X
1
,
X
2
)=
(
X
1
+2)
2
⋅(
X
2
+3)
and the same income as Jamal and Korra. Find his optimal consumption bundle.
𝑋∗1=2𝑚+6𝑃2−2𝑃13𝑃1 , 𝑋∗2=𝑚+2𝑃1−6𝑃23𝑃2
Well, well, well, let's see what Larry's preferences have in store for us. Larry seems to have a fancy taste for bundles, with his wacky preference function:
U(C(X1, X2)) = ((X1 + 2)^2) * ((X2 + 3))
Now, Larry also has the same income as Jamal and Korra. So, to find Larry's optimal consumption bundle, we need to find the combination of X1 and X2 that maximizes his utility function while staying within his income.
But hey, fear not, for I, the Clown Bot, have come to solve this puzzle for you. However, I must warn you, this might get a little math-y. So hold on to your funny bone!
To find the optimal consumption bundle, we need to take the partial derivatives of Larry's utility function with respect to X1 and X2, and set them equal to zero. This will give us the critical points.
dU/dX1 = 2(X1 + 2)(X2 + 3) = 0
dU/dX2 = 2(X1 + 2)^2 = 0
Now, let's solve these equations, shall we?
From the first equation, we can see that either X1 + 2 = 0 or X2 + 3 = 0. But since X2 + 3 can't be zero (let's not divide by zero, folks), we have:
X1 + 2 = 0
Solving for X1, we get:
X1 = -2
Now, plugging this value of X1 into the second equation, we have:
2(-2 + 2)^2 = 0
Simplifying, we get:
2(-2)^2 = 0
4 = 0
Oh no! Something went wrong here! Don't panic, folks, that's just my clownish way of saying that there is no solution to this equation, which means we don't have a critical point.
So, what does this mean? Well, it means that Larry's utility function doesn't have a maximum or minimum. Without a constraint on income, Larry could consume infinite amounts of goods and still be happy (or maybe his clownish sense of humor drives him to absurdities!).
In any case, I'm sorry to say, Larry's optimal consumption bundle is like chasing a rainbow. It's just not there!
But don't be too sad, my friend. There are always more preferences and bundles to explore. Keep searching for that perfect combination and never give up on your quest for happiness (or a good joke!).
To find Larry's optimal consumption bundle, we need to determine the quantities of goods X1 and X2 that maximize his utility function given his budget constraint.
First, let's write Larry's utility function:
U(X1, X2) = (X1 + 2)^2 * (X2 + 3)
Now, let's assume Larry's income is denoted by M, and the prices of goods X1 and X2 are denoted by P1 and P2, respectively. Since Larry has the same income as Jamal and Korra, we can use the same values for M, P1, and P2 as in their case:
M = M
P1 = P1
P2 = P2
Larry's budget constraint can be expressed as:
P1 * X1 + P2 * X2 = M
Let's rearrange the budget constraint to express X2 in terms of X1:
X2 = (M - P1 * X1) / P2
Now, we can substitute X2 back into Larry's utility function:
U(X1) = (X1 + 2)^2 * ((M - P1 * X1) / P2 + 3)
To find the optimal consumption bundle, we need to maximize Larry's utility function with respect to X1. We can do this by taking the derivative of U(X1) with respect to X1 and setting it equal to zero:
dU(X1)/dX1 = 0
Let's calculate the derivative:
dU(X1)/dX1 = 2 * (X1 + 2) * ((M - P1 * X1) / P2 + 3) - (P1 / P2) * (X1 + 2)^2
Setting this derivative equal to zero, we get:
2 * (X1 + 2) * ((M - P1 * X1) / P2 + 3) - (P1 / P2) * (X1 + 2)^2 = 0
Now, we can solve this equation to find the value of X1 that will maximize Larry's utility.
Once we find the value of X1, we can substitute it into the budget constraint to find X2:
X2 = (M - P1 * X1) / P2
Please provide the specific values of M, P1, and P2, and I will help you solve this equation.
To find Larry's optimal consumption bundle, we need to identify his utility function, his budget constraint, and then solve for the optimal allocation of his income.
Given Larry's utility function U(C(X1, X2)) = (X1+2)^2 * (X2+3) and the same income as Jamal and Korra, we can assume that Larry's budget constraint is the same as theirs.
Let's assume the budget constraint for all three consumers is represented by the equation: M = P1*X1 + P2*X2, where M is the income, P1 is the price of good X1, and P2 is the price of good X2.
Since we already know Larry's utility function, we can differentiate it to find his marginal utilities:
MU1 = dU/dX1 = 2*(X1+2)*(X2+3)
MU2 = dU/dX2 = (X1+2)^2
Now, to find Larry's optimal consumption bundle, we need to equate his marginal utilities to the price ratio (P1/P2) in order to maximize his utility given his budget constraint.
Setting the equation MU1/MU2 = P1/P2, we get:
(2*(X1+2)*(X2+3))/(X1+2)^2 = P1/P2
Now, since we know the prices of goods P1 and P2, we can solve this equation to find the values of X1 and X2.
Assuming the values of P1 and P2 are known, you can substitute them into the equation and solve for X1 and X2.