How to find the substitution and income effects?

The usual definition of Substitution Effect (pg. 30; also found in Varian) tells that the Slutsky SE is $$x(p_x', I') - x(p_x,I) = x(2,15) - x(1,10) = \frac{15}{2 \cdot 2} - \frac{20}{2 \cdot 1} = -6.25$$ where $$I$$ is the initial income and $$I'$$ is the new (imaginary) income. Similarly, $$p_x$$ is the initial price of good $$X$$ and $$p_x'$$ is the new price.

There's another definition of the SE which we derive from the Slutsky equation and it says that the SE (substitution effect) is $$\frac{\partial x^c}{\partial p_x} = \frac{\partial x}{\partial p_x} \Big|_{u = \overline{u}}$$.

Following are my questions related to the second definition of SE:

1. The compensated demand is defined as $$x^c(p_x, \overline{p_y}, \overline{u}) := x(p_x, \overline{p_y}, E(p_x, \overline{p_y}, \overline{u}))$$. How does $$\frac{\partial x^c}{\partial p_x} = \frac{\partial x}{\partial p_x} \Big|_{u = \overline{u}}$$ follow from this? Also, do we assume $$u = \overline{u}$$ first and then calculate the partial derivative or calculate the partial derivative first and then fix the $$u$$ values to $$\overline{u}$$?

2. As I understand, Hick's SE can be written as $$x(p_x', p_y, E(p_x', p_y, \overline{u})) - x(p_x, p_y, I)$$ which is equal to $$x(p_x', p_y, I') - x(p_x, p_y, I)$$, the Slutsky SE when the income is set as the expenditure. But how is any of these equal to the other definition of SE: $$\frac{\partial x^c}{\partial p_x}$$?

One definition talks about the change in demand while the other talks about the slope of demand curve. How can they be equal in any way?

3. This brings me to my final question. Consider the following example problem:

Suppose a cup of coffee and a plate of beans are sold at € 1 and € 3 respectively during the winter. In summer, the government decides to remove the subsidy on coffee and its new price per cup goes up to € 2. If a customer has an income of € 10 and the utility function $$u(x,y)=xy$$, what is the substitution effect?

Using the usual definition, we can calculate it as $$x(p_x', p_y, I') - x(p_x, p_y, I) = -6.25$$. It's the Cobb-Douglas utility function, so I skipped the calculations.

How can I calculate the SE using the second definition: $$\frac{\partial x}{\partial p_x} \Big|_{u = \overline{u}}$$? I don't see where/how to consider $$u = \overline{u}$$.

• You asked a similar question earlier also. I think procedure is clearly explained here: economics.stackexchange.com/a/52138/11824
– Amit
Commented Aug 18, 2022 at 7:35
• @Amit Yes, I used the same example so it's easier for the answerer to refer to, if needed (as it's linked on the right side). I am interested in calculating using partial derivates which is not mentioned there. I also wanted to know how $\frac{\partial x^c}{\partial p_c} = \frac{\partial x}{\partial p_c} \Big|_{u = u_0}$ and how it's the Slutsky SE, hoping that it is.
– Isa
Commented Aug 18, 2022 at 10:03
• If I have understood your question correctly, are asking the difference between the derivative of a function $\frac{df}{dx}$ and the total change $\Delta f$ due to $\Delta x$ change in $x$?
– Amit
Commented Aug 18, 2022 at 10:20
• @Amit I partially understood certain things, so I changed my questions (somewhat). I am mainly curious about 1 and 2. Maybe after that, I will understand 3 myself.
– Isa
Commented Aug 19, 2022 at 6:01