1. The utility function is given by:

U=x+y
and the budget line is
x+2y=100.
Then the price of good x goes up to 4.

Find the Hicksian substitution effect, income effect, and total change in demand for good x from the change

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Ok i read the theory but i cant solve the problem at all. I just need to a little and maybe i can apply the theory into it. Any help would be appreciated.

I can see why you are confused. Your marginal rate of substitution in utility is constant, and therefore linear. Which means you will always have a corner solution. That is, the slope of any and all indifference curves is -1.

The slope of the initial budget constraint is -1/2, and the person consumes x only. With the price increase, the slope of the budget constraint becomes -2 and the person consumes all y.

With this, you should be able to calculate the hicksian substitution effect and the income effect. (Hint, its all substitution).

To find the Hicksian substitution effect and income effect for good x, we need to analyze how the price change affects the consumer's consumption and purchasing power.

First, let's find the initial optimal consumption bundle before the price change.

Given the utility function U = x + y and the budget constraint x + 2y = 100, we can solve for the optimal consumption bundle using the Lagrangian method:

L = x + y + λ(100 - x - 2y)

Taking the partial derivatives with respect to x, y, and λ:

∂L/∂x = 1 - λ = 0
∂L/∂y = 1 - 2λ = 0
∂L/∂λ = 100 - x - 2y = 0

From the first equation, λ = 1. Substituting this into the second equation, we get 1 - 2(1) = 1 - 2 = -1, implying that y = -1/2. By substituting the values of λ and y into the third equation, we can solve for x:

100 - x - 2(-1/2) = 0
100 - x + 1 = 0
x = 99

The initial optimal consumption bundle is x = 99 and y = -1/2.

Now, let's consider the price change for good x. After the price change, the new price of good x is 4. We need to find the new optimal consumption bundle under the new budget constraint.

The new budget constraint becomes 4x + 2y = 100. Solving this constraint, we find that y = 50 - 2x.

Substituting the value of y into the utility function, we have U = x + (50 - 2x) = 50 - x.

To find the new optimal consumption bundle, we maximize this new utility function subject to the new budget constraint:

L = 50 - x + λ(4x + 2(50 - 2x))

Taking the partial derivatives with respect to x, y, and λ:

∂L/∂x = -1 + 4λ = 0
∂L/∂y = -2λ = 0
∂L/∂λ = 4x + 2(50 - 2x) = 0

From the first equation, 4λ = 1, so λ = 1/4. Substituting this into the third equation and solving, we find x = 25. Substituting this value into the y equation, we get y = 50 - 2(25) = 0.

The new optimal consumption bundle is x = 25 and y = 0.

Now, we can calculate the Hicksian substitution effect and income effect for good x.

Hicksian Substitution Effect:
The Hicksian substitution effect measures the change in consumption of good x due to the price change, while keeping utility constant. To find it, we compare the initial and new optimal consumption bundles for good x. In this case, the Hicksian substitution effect is:
Δx_H = 25 - 99 = -74

Income Effect:
The income effect measures the change in consumption of good x due to the change in purchasing power caused by the price change. To isolate the income effect, we need to adjust the consumer's income to keep them at the initial utility level. As the price of good x increased, the consumer's purchasing power decreases, leading to a decrease in consumption. In this case, the income effect is:
Δx_I = 99 - 25 = 74

Total Change in Demand for Good x:
The total change in demand for good x is the sum of the Hicksian substitution effect and the income effect:
Δx_Total = Δx_H + Δx_I = -74 + 74 = 0

Therefore, the total change in demand for good x is zero. This means that the price increase for good x does not lead to any change in the consumer's demand for it.

I hope this explanation helps you understand how to solve the problem and apply the theory. If you have any further questions, feel free to ask!