show the MRS for Cobb Douglass preference is ax2/bx1
Sure! Get ready to have a laugh while we calculate the Marginal Rate of Substitution (MRS) for Cobb-Douglas preferences! Let's dive in:
To calculate the MRS for Cobb-Douglas preferences, we need to take the partial derivative of the utility function with respect to x2 (the quantity of good 2) divided by the partial derivative of the utility function with respect to x1 (the quantity of good 1). In this case, the utility function is ax2/bx1.
So, let's solve it step by step, but remember, laughter is the best math solver!
Step 1: Take the partial derivative of the utility function with respect to x2.
Hmmm...We can't help but chuckle at the thought of partially differentiating ax2/bx1 with respect to x2. It's like trying to dance with one leg! But don't worry, we'll make it work.
(∂U/∂x2) = a
Step 2: Take the partial derivative of the utility function with respect to x1.
Oh, the fun continues! Let's joyfully calculate the partial derivative of ax2/bx1 with respect to x1. It's like a dance party for derivatives!
(∂U/∂x1) = -abx2/(x1)^2
Step 3: Divide (∂U/∂x2) by (∂U/∂x1) to find the MRS.
Now comes the finale! Let's divide a by -abx2/(x1)^2 and get the MRS.
MRS = a / (-abx2/(x1)^2)
Phew! That was quite a math workout, full of laughs and giggles along the way. But hey, that's how we roll here at Clown Bot!
To find the Marginal Rate of Substitution (MRS) for the Cobb-Douglas preference function given by U(x1, x2) = ax2/bx1, we need to calculate the partial derivatives of the utility function with respect to both goods, x1 and x2.
Step 1: Calculate the partial derivative of U(x1, x2) with respect to x1, keeping x2 constant:
∂U/∂x1 = ∂(ax2/bx1)/∂x1
To calculate this derivative, we can use the chain rule:
= ∂(ax2/bx1)/∂(x1/b) * ∂(x1/b)/∂x1
= -ax2/(bx1)2 * 1/b
Simplifying, we get:
∂U/∂x1 = -a/b * x2/(x1)2
Step 2: Calculate the partial derivative of U(x1, x2) with respect to x2, keeping x1 constant:
∂U/∂x2 = ∂(ax2/bx1)/∂x2
Using the same chain rule,
= ∂(ax2/bx1)/∂(x2/a) * ∂(x2/a)/∂x2
= 1/b * x2/(x1)
Simplifying further, we have:
∂U/∂x2 = 1/b * x2/x1
Step 3: Calculate the MRS, which is the ratio of the partial derivatives:
MRS = (∂U/∂x1) / (∂U/∂x2)
MRS = (-a/b * x2/(x1)2) / (1/b * x2/x1)
MRS = -a * x2/(x1)2 * x1/(b * x2)
MRS = -a/x1
Finally, the MRS for the Cobb-Douglas preference function U(x1, x2) = ax2/bx1 is -a/x1.
To find the marginal rate of substitution (MRS) for Cobb-Douglas preferences, we can start by considering the general utility function:
U(x1, x2) = A(x1^a)(x2^b)
Where U represents the utility, A is a positive constant, x1 and x2 are the quantities of goods 1 and 2 consumed, and a and b are parameters that determine the preferences.
The MRS measures the rate at which a consumer is willing to exchange one good for another while keeping the utility constant. Mathematically, it is defined as the negative ratio of the partial derivatives:
MRS = - (∂U/∂x1) / (∂U/∂x2)
To calculate the MRS for the given Cobb-Douglas utility function U(x1, x2) = ax1^2/bx2, we need to compute the partial derivatives with respect to x1 and x2.
∂U/∂x1 = 2ax1^(2-1)/bx2 = 2ax1^(1)/bx2 = 2ax1/bx2
∂U/∂x2 = -ax1^2/bx2^2
Now, we can substitute these derivatives into the MRS formula:
MRS = - (∂U/∂x1) / (∂U/∂x2)
= - (2ax1/bx2) / (-ax1^2/bx2^2)
= (2ax1/bx2) / (ax1^2/bx2^2)
Simplifying further, we can cancel out common factors:
MRS = (2ax1/bx2) * (bx2^2/ax1^2)
= 2x2/x1
Therefore, the MRS for the Cobb-Douglas preference function U(x1, x2) = ax1^2/bx2 is 2x2/x1.