Sketch the demand for good 1 if we have u(x1,x2) = min{x1,x2}
To determine the demand for good 1, we need to find the quantity of good 1 that maximizes utility given the utility function u(x1, x2) = min{x1, x2}.
First, let's understand the meaning of this utility function. The utility function u(x1, x2) = min{x1, x2} indicates that the consumer's utility is determined by the lesser of the quantities of goods 1 and 2 consumed. In other words, the consumer values both goods equally, and their overall satisfaction is determined by the minimum quantity of the two goods.
Now, let's draw the demand curve for good 1 under this utility function. We will assume that the prices of goods are constant.
To start, we can set up the consumer's problem as a utility maximization subject to a budget constraint. Let's say the price of good 1 is p1 and the price of good 2 is p2. Also, let's assume the consumer has a fixed budget B.
The consumer's problem can be written as:
Maximize u(x1, x2) = min{x1, x2}
subject to p1*x1 + p2*x2 = B
To find the demand for good 1, we need to solve this optimization problem for different values of p1 and p2, while keeping the budget B constant.
Unfortunately, the optimization problem for this utility function does not have a unique solution since the utility is the same for any combination of x1 and x2 where x1 = x2. Thus, the demand for good 1 will not have a typical downward-sloping demand curve. Instead, it will be a step function.
The demand for good 1 using this utility function can be represented as follows:
If p1 < p2, then the consumer would buy only good 1, and the demand for good 1 would be equal to the budget divided by the price of good 1.
If p2 < p1, then the consumer would purchase only good 2, and the demand for good 1 would be zero.
If p1 = p2, then the consumer would be indifferent between buying good 1 or good 2, and any combination of x1 and x2 where x1 = x2 would provide the same utility level.
In summary, the demand for good 1 when utility is given by the function u(x1, x2) = min{x1, x2} is not a typical demand curve. Instead, it follows a step function where the consumer either buys only good 1 (when its price is lower) or does not buy it at all (when its price is higher than good 2).
1. Let 𝑢(𝑥1, 𝑥2) = 𝑥1 − (2𝑥1)^2 + (𝑥2)^2
a. Find the function that defines the marginal rate of substitution of good 1
b. Sketch the indifference curves for utility level 1 and 2
a. The marginal rate of substitution (MRS) of good 1, denoted as MRS(1,2), is the rate at which a consumer is willing to substitute good 2 for good 1 while maintaining the same level of utility. To find the MRS, we need to calculate the partial derivatives of the utility function with respect to x1 and x2.
Taking the partial derivative with respect to x1:
∂u/∂x1 = 1 - 4x1
Taking the partial derivative with respect to x2:
∂u/∂x2 = 2x2
So the MRS of good 1 is given by:
MRS(1,2) = (∂u/∂x2) / (∂u/∂x1) = (2x2) / (1 - 4x1)
b. To sketch the indifference curves for utility levels 1 and 2, we need to equate the utility function to the respective utility levels and solve for x2 in terms of x1.
For utility level 1:
Setting u(x1, x2) equal to 1:
x1 - (2x1)^2 + x2^2 = 1
Simplifying the equation, we get:
(2x1)^2 + x2^2 - x1 + 1 = 0
This equation represents the indifference curve for utility level 1.
Similarly, for utility level 2:
Setting u(x1, x2) equal to 2:
x1 - (2x1)^2 + x2^2 = 2
Simplifying the equation, we get:
(2x1)^2 + x2^2 - x1 + 2 = 0
This equation represents the indifference curve for utility level 2.
To sketch the indifference curves, plot points that satisfy the respective equations for different values of x1 and x2. Connect these points to form the curves. Keep in mind that the curvature of the curves will depend on the specific values of x1 and x2.
Let 𝑢(𝑥1,𝑥2) = 𝑥1(𝑥2)^2
a. Find the demand functions for goods 1 and 2
To find the demand functions for goods 1 and 2, we need to maximize the utility function u(x1, x2) = x1(x2)^2 subject to a budget constraint.
Let p1 be the price of good 1 and p2 be the price of good 2. Assuming the consumer has a fixed budget B, the budget constraint can be written as p1x1 + p2x2 = B.
We can set up the consumer's problem as follows:
Maximize u(x1, x2) = x1(x2)^2
subject to p1x1 + p2x2 = B
Using the Lagrangian method, we can form the Lagrangian function L as follows:
L(x1, x2, λ) = x1(x2)^2 + λ(B - p1x1 - p2x2)
Now, we can take partial derivatives of L with respect to x1, x2, and λ, and set them equal to zero to find the optimal values.
Taking the partial derivative with respect to x1:
∂L/∂x1 = (x2)^2 - λp1 = 0 ...(1)
Taking the partial derivative with respect to x2:
∂L/∂x2 = 2x1x2 - λp2 = 0 ...(2)
Taking the partial derivative with respect to λ:
∂L/∂λ = B - p1x1 - p2x2 = 0 ...(3)
Solving equations (1) and (2) simultaneously, we get:
(x2)^2 - λp1 = 0 ...(4)
2x1x2 - λp2 = 0 ...(5)
Dividing equation (4) by equation (5), we have:
(x2)^2 / (2x1x2) = (λp1) / (λp2)
x2 / (2x1) = p1 / p2
(x2 / x1) = (2p1 / p2) ...(6)
Substituting equation (6) into equation (3), we get:
B - p1x1 - (p2)(2p1 / p2)x1 = 0
B - (3p1)(x1) = 0
x1 = B / (3p1) ...(7)
Further substituting the value of x1 obtained from equation (7) into equation (6), we have:
(x2 / (B / (3p1))) = (2p1 / p2)
x2 = (2p1^2 / p2)(B / (3p1))
x2 = (2p1B) / (3p2)
Thus, the demand functions for goods 1 and 2 are:
x1 = B / (3p1)
x2 = (2p1B) / (3p2)
These demand functions represent the optimal quantities of goods 1 and 2 that a consumer would purchase to maximize utility subject to the budget constraint.
To sketch the demand for good 1 with the utility function u(x1,x2) = min{x1,x2}, we can use the concept of indifference curves.
An indifference curve represents all the combinations of goods (in this case, x1 and x2) that give the same level of utility to the consumer. In our case, the utility function u(x1,x2) = min{x1,x2} implies that the consumer derives utility from the smaller of the two goods' quantities.
To start, let's consider some combinations of x1 and x2 that yield different levels of utility.
1. When x1 = 0 and x2 > 0:
- Since x1 is zero, the utility will also be zero since the minimum value between zero and x2 is always zero.
- This implies that the consumer does not derive any utility from consuming only good 2.
2. When x2 = 0 and x1 > 0:
- Similar to the previous case, if x2 is zero, the utility will also be zero.
- Thus, the consumer does not derive any utility from consuming only good 1.
3. When x1 = x2 > 0:
- In this case, both goods have the same quantity.
- Since u(x1,x2) = min{x1,x2}, the utility will be the minimum value between x1 and x2.
- As both goods have the same quantity, the utility will be equal to x1 (or x2), which is also the quantity of the good 1 (or good 2).
- This implies that the consumer derives utility from consuming equal quantities of both goods.
Based on these observations, we can sketch the demand for good 1 as follows:
- For x1 = 0, the demand for good 1 will be zero.
- As the quantity of good 1 increases, the demand will increase linearly until it reaches the point where x1 = x2. At this point, the demand for good 1 will be equal to the quantity of good 1.
- Finally, after x1 = x2, the demand for good 1 will be equal to x2 (since the utility is determined by the smaller quantity).
- This results in a kinked demand curve where the slope changes abruptly at the point where x1 = x2.
Note: The demand curve for good 2 would be symmetric to the demand curve for good 1, as the utility function is symmetric.
To sketch the demand for good 1 when the utility function is u(x1, x2) = min{x1, x2}, we need to understand how this utility function represents consumer preferences and how it relates to the demand for good 1.
The utility function u(x1, x2) = min{x1, x2} represents a consumer's preference for consuming the minimum of x1 and x2. In other words, the consumer derives utility from consuming the smaller quantity of the two goods.
To find the demand for good 1, we need to analyze the consumer's optimization problem, which is to maximize utility subject to a budget constraint. Let's assume that the consumer has a fixed budget and the prices of x1 and x2 are given.
To start, let's consider two extreme scenarios:
1. If the price of good 1 (p1) is very low compared to the price of good 2 (p2), the consumer would want to consume as much of good 1 as possible and as little of good 2 as possible. In this case, the demand for good 1 would be high.
2. If the price of good 1 (p1) is very high compared to the price of good 2 (p2), the consumer would prefer to consume more of good 2 and less of good 1. In this case, the demand for good 1 would be low.
Now, let's analyze the intermediate cases:
To find the demand for good 1 at different price levels, we need to compare the consumer's marginal utility (MU) of good 1 with its price (p1). If the MU of good 1 is higher than its price, then the consumer will demand more of good 1. If the MU of good 1 is lower than its price, then the consumer will demand less of good 1.
In this case, since the utility function is u(x1, x2) = min{x1, x2}, we can see that the consumer's preference is biased towards consuming the smaller quantity of the two goods. This implies that when the consumer consumes more of good 1, the utility would be limited by the quantity of good 2. Similarly, when the consumer consumes more of good 2, the utility would be limited by the quantity of good 1.
As a result, the demand for good 1 would be determined by the demand for good 2. If the consumer's income and the price of good 2 remain fixed, the demand for good 1 will depend on the consumer's preference for consuming good 1 relative to good 2.
To summarize, the demand for good 1 can vary depending on the price of good 1 relative to the price of good 2 and the consumer's preference for consuming good 1 relative to good 2. To get a precise sketch of the demand curve for good 1, you would need more information such as specific prices and preferences, and then apply the principles of consumer choice and budget optimization to determine the demand at different price levels.