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I know that the Slutsky equation is defined as:

$\frac{\partial x_1^s}{\partial p_1} = \frac{\partial x_1^m}{\partial p_1} + x_1^o \frac{\partial x_1^m}{\partial m}$

My problem is right now is making use of this information given (I am aware of how to take partial derivatives) but cannot seem to understand how to apply it to problem sets.

Here's an example (I'm more concerned about the steps on how to get to the answer not just the answer);

A consumer has preferences given by $U(x_1,x_2)= x_1^2x_2$

(a) Derive the demand curves for $x_1, x_2$ when prices and income are given by $p_1, p_2$ and $m$

$x_1^*=2m/3p_1$ and $x_2^*=m/3p_2$ -I think I understood how to do that

(b) Illustrate the equilibrium on a diagram when $p_1$ = $p_2$ $ =$ 1 and $I$ = $12

  • the way I did this was by graphing and simply finding the equllibrium point graphically based on the Demands for goods $1$ and $2$ on the budget line

(c) Calculate the exact income and substitution effects for $x_1$ when $p_1$ rises to $3.

-The only way I'm currently able to do this is without Calculus, as described in this video which doesn't seem to sit well with me being that the Slutsky Equation is defined very clearly with use of calculus. I just don't know how to apply it.

(d) Explain your exact results using the appropriate Slutsky equation.

  • same problem here.

Note: I'm no simply looking for someone to "do my homework" I'm primary interest is in knowing how to apply the Slutsky equation when facing similar problems.

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Utility function $u(x_1, x_2) = x_1^2x_2$.

Q. Derive the demand for $x_1$ and $x_2$ as a function of $p_1$, $p_2$ and $m$.

Here are the demand functions for $x_1$ and $x_2$: $$x_1(p_1, p_2, m) = \frac{2m}{3p_1}$$ $$x_2(p_1, p_2, m) = \frac{m}{3p_2}$$

Q. Illustrate the equilibrium in a diagram when $p_1=1$, $p_2=1$ and $m=12$.

enter image description here

Q. Suppose $p_1$ rises to 3. Calculate the substitution effect and income effect.

If $p_1$ rises to 3, the new equilibrium choice is $\left(\frac{8}{3}, 4\right)$. To find the Substitution effect and Income effect using Slutsky approach, we will find the equilibrium at new set of prices when the consumer has just enough money to buy the old equilibrium bundle i.e. we will find the demand at prices $(3,1)$ when income is $m' = 3(8) + 1(4) = 28$. Substituting this data in the demand functions, we get the equilibrium choice as: $\left(\frac{56}{9}, \frac{28}{3}\right)$. Here is how the situation looks in graph: enter image description here

Substitution effect = $\displaystyle\frac{56}{9} - 8 = -\frac{16}{9}$

Income effect = $\displaystyle\frac{8}{3} - \frac{56}{9} = - \frac{32}{9}$

Q. Explain your exact results using the appropriate Slutsky equation.

Slutsky equation: Change in Demand = Change in Demand due to substitution effect + Change in Demand due to income effect

$$\Delta x_1 = \Delta^s x_1 + \Delta^i x_1 = -\frac{16}{9} - \frac{32}{9} = -\frac{16}{3}$$

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The Slutsky equation links Hicksian and Marshallian demand functions.

Hicksian demand minimizes the cost necessary to reach a certain utility. In your question, you've labeled this $x_1^s$ (although i'm not familiar with this notation). Because Hicksian demand holds utility constant, it measures the pure substitution effect.

Marshallian demand maximizes utility given a fixed income. In your question, you've labeled this $x_1^m$. Because Marshallian demand holds income constant, price affects Marshallian demand both because a price increase for one good makes the other good more attractive (substitution effect) and because it decreases the different kinds of good baskets you can buy (income effect).

Note that, in order to find $x_1^* = 2m/3p_1$ and $x_2^* = m/3p_2$, you maximized utility given a fixed income, by solving $MU_{x_1}/MU_{x_2} = p_1/p_2$ with $m=p_1x_1+p_2x_2$. And so, what you've found and labeled $x_1^*$ and $x_2^*$ are Marshallian demand.

That takes us back to the Slutsky equation

$$\frac{\partial x_1^s}{\partial p_1} = \frac{\partial x_1^m}{\partial p_1}+x_1\frac{\partial x_1^m}{\partial m}$$

What is this saying? Remember, Hicksian demand is pure substitution effect. So $\frac{\partial x_1^s}{\partial p_1}$ is the substitution effect. And Marshallian demand includes the substitution effect and the income effect.

So what this is saying is that the substitution effect ($\frac{\partial x_1^s}{\partial p_1}$) is what you get if you start with the total effect ($\frac{\partial x_1^m}{\partial p_1}$), including income and substitution effects, and then take out the income effect ($x_1\frac{\partial x_1^m}{\partial m}$).[note below]

So where does that leave you?

You can calculate the income effect as $x_1\frac{\partial x_1^m}{\partial m}$. In other words, as the price rises, it rises for every one of the $x_1$ units you're buying, and so the effect on income is the price change (1) times the number of units ($x_1$). And income affects your demand for $x_1$ through $\frac{\partial x_1^m}{\partial m}$, so the income effect is $x_1\frac{\partial x_1^m}{\partial m}$. Since you have Marshallian demand, just take the derivative of $x_1^m$ with respect to $m$ to get $\frac{\partial x_1^m}{\partial m}$, and then multiply by $x_1$ to get the income effect.

You can calculate the substitution effect, then, in one of two ways. You can use the Slutsky equation: calculate the total effect $\frac{\partial x_1^m}{\partial p_1}$ by taking the derivative of $x_1^m$ with respect to $p_1$, and then plug that into the Slutsky equation with the income effect to get the substitution effect, $\frac{\partial x_1^s}{\partial p_1}$.

Or, you can calculate Hicksian demand $x_1^s$ directly by solving $MU_{x_1}/MU_{x_2} = p_1/p_2$ with $U(x_1,x_2) = s$. Then, take the derivative of $x_1^s$ with respect to $p_1$ to get the substitution effect.

[Note: It might seem weird to "take out" the income effect by adding it. But remember that the total effect is negative, since a higher $p_1$ leads to less $x_1$. And so adding on a positive income effect (higher income leads to more consumption of normal goods) is indeed "taking out" the income effect.]

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