Note: I asked in the Mathematics Meta regarding if it is permitted to ask Mathematics questions of economic nature. I also posted this question in the Economics Stackexchange but have not got any replies regarding the mathematical nature of this problem so I'm going to try it here. Let me know if I did not give enough information Mathematically
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.
Another Note: I'm no simply looking for someone to "do my homework" I'm primarily interested is in knowing how to apply the Slutsky equation when facing similar problems.
Thanks in advance.
Best Answer
In (c) you are asked to calculate the income and substitution effects for a discrete change in the price of good $1$ from $1$ to $3$. Thus you will not be calculating the effects using derivatives.
In (a) which demand functions were you asked to find? Just the Marshallian, or the Hicksian (compensated) demand function as well?
The Hicksian demand function (found by minimizing expenditure subject to attaining some level of utility $u$) is: $$x_1^h(p_1,p_2,u)=\left(\frac{2up_2}{p_1}\right)^{1/3}\qquad x_2^h(p_1,p_2,u)=\left(\frac{up_1^2}{4p_2^2}\right)^{1/3}.$$
At the income $m=12$ and initial prices (both $\$1$) the Marshallian demand functions you derived tell you that consumption of good $1$ is $8$ units and consumption of good $2$ is $4$ units. Thus initially utility is $U(8,4)=256$. After the rise in price of good $1$ to $\$3$ consumption of good $1$ is $8/3$ units. The total effect of the price rise on consumption of good $1$ is $$x_1^m(1,1,12)-x_1^m(3,1,12)=8-8/3=16/3.$$ The substitution effect can be found by considering the effect of the price change on consumption of good 1 keeping utility constant at each its initial level. This is where the Hicksian demand function is needed. The Hicksian demand for good $1$ at the new prices and old utility is $$x_1^h(3,1,256)=\left(\frac{2(256)1}{3}\right)^{1/3}=\frac{8}{3^{1/3}}.$$
It follows that the substitution effect is $$x_1^m(1,1,12)-x_1^h(3,1,256)=8-\frac{8}{3^{1/3}}\approx 2.45$$ while the income effect is $$x_1^h(3,1,256)-x_1^m(3,1,12)=\frac{8}{3^{1/3}}-8/3\approx 2.88.$$
Of course you could do that the other way and calculate the substitution effect by keeping utility constant at the new level (which corresponds to doing what I did above, but for a fall in price of good $1$ from $3$ to $1$). This would give you a slightly different answer (for marginal changes in the price there is not this problem of two different ways of calculating the substitution effect).
In terms of answering (d): have you seen a discrete version of the Slutsky equation?