We consider once again the utility function of Example 1:
$$U(x,y) = 6\sqrt{xy}.$$
A consumer has 100 of good $x$ and 2.25 of good $y$. Then his utility equals 90. Assume that the consumer has to give up 2 units of good $x$. How much more should he get of good $y$ to keep his utility at the same level?
According to the approximation of the change of the function value
$$\Delta U \approx U'_x(x_0,y_0) \Delta x + U'_y(x_0,y_0) \Delta y.$$
We have to determine $\Delta y$; we find this by rewriting this approximation to
$$\Delta y \approx \dfrac{\Delta U - U'_x(x_0,y_0)\Delta x}{U'_y(x_0,y_0)}.$$
It is known that
$$(x_0,y_0) = (100,2.25), \qquad \Delta x = -2 \qquad \text{and} \qquad \Delta U = 0.$$
We have determined the partial derivatives at $(x_0,y_0)=(100,2.25)$ in Example 1:
$$U'_x(100,2.25) = 0.45\qquad \text{and} \qquad U'_y(100,2.25) = 20.$$
Hence, we have all the input to find the change in $y$ such that the utility of the consumer stays approximately the same if $x$ decreases by 2:
$$\Delta y \approx \dfrac{\Delta U - U'_x(100,2.25)\Delta x}{U'_y(100,2.25)} = \dfrac{0 - 0.45 \cdot(-2)}{20} = \dfrac{-0.90}{20} = 0.045.$$
Consequently, the consumer should get approximately $2.25 + 0.045 = 2.295$ of $y$ to keep his utility constant when $x$ decreases by 2.
$$U(x,y) = 6\sqrt{xy}.$$
A consumer has 100 of good $x$ and 2.25 of good $y$. Then his utility equals 90. Assume that the consumer has to give up 2 units of good $x$. How much more should he get of good $y$ to keep his utility at the same level?
According to the approximation of the change of the function value
$$\Delta U \approx U'_x(x_0,y_0) \Delta x + U'_y(x_0,y_0) \Delta y.$$
We have to determine $\Delta y$; we find this by rewriting this approximation to
$$\Delta y \approx \dfrac{\Delta U - U'_x(x_0,y_0)\Delta x}{U'_y(x_0,y_0)}.$$
It is known that
$$(x_0,y_0) = (100,2.25), \qquad \Delta x = -2 \qquad \text{and} \qquad \Delta U = 0.$$
We have determined the partial derivatives at $(x_0,y_0)=(100,2.25)$ in Example 1:
$$U'_x(100,2.25) = 0.45\qquad \text{and} \qquad U'_y(100,2.25) = 20.$$
Hence, we have all the input to find the change in $y$ such that the utility of the consumer stays approximately the same if $x$ decreases by 2:
$$\Delta y \approx \dfrac{\Delta U - U'_x(100,2.25)\Delta x}{U'_y(100,2.25)} = \dfrac{0 - 0.45 \cdot(-2)}{20} = \dfrac{-0.90}{20} = 0.045.$$
Consequently, the consumer should get approximately $2.25 + 0.045 = 2.295$ of $y$ to keep his utility constant when $x$ decreases by 2.