Question

Consider a simple quasi-linear utility function of the form \(U(x, y)=x+\ln y\) a. Calculate the income effect for each good. Also calculate the income elasticity of demand for each good. b. Calculate the substitution effect for each good. Also calculate the compensated own-price elasticity of demand for each good. c. Show that the Slutsky equation applies to this function. d. Show that the elasticity form of the Slutsky equation also applies to this function. Describe any special features you observe.

Short answer

In this exercise, we analyzed a quasi-linear utility function \(U(x, y)=x+\ln y\). We derived the demand functions for goods x and y as \(x = \frac{I}{p} - \frac{I}{q}\) and \(y = \frac{I}{q}\), respectively. We then calculated their income and own-price elasticities, which for both goods were equal to 1 for income elasticity and -1 for own-price elasticity. Lastly, we verified the validity of the Slutsky equation and its elasticity form for the given utility function.

Step-by-step solution

Deriving the demand functions for x and y

To derive the demand functions for goods x and y, we first need to set up the consumer's problem. The consumer aims to maximize their utility subject to their budget constraint. We can write the problem as: \(max U(x, y)=x+\ln y\) subject to: \(px+qy=I\) Assuming that an interior solution exists, we can use the method of Lagrange multipliers to solve this problem. For this, we set up the Lagrangian: \(L(x, y, \lambda)=x+\ln y + \lambda(I-px-qy)\) Taking the partial derivatives with respect to x, y, and \(\lambda\) gives the following three conditions: \(\frac{\partial L}{\partial x}=1-\lambda p=0\) \(\frac{\partial L}{\partial y}=\frac{1}{y}-\lambda q=0\) \(\frac{\partial L}{\partial \lambda}=I-px-qy=0\) From the first two conditions, we can solve for \(\lambda\): \(\lambda=\frac{1}{p}\) and \(\lambda=\frac{1}{yq}\) Setting the two expressions for \(\lambda\) equal and solving for y gives us the demand function for y: \(y = \frac{I}{q}\) Since utility is linear in x, the remaining income (I - qy) is spent on x. Thus, the demand function for x is: \(x = \frac{I}{p} - y = \frac{I}{p} - \frac{I}{q}\)

Calculating the income and own-price elasticities of demand for x and y

Now, let's calculate the income elasticities \(\eta_x\) and \(\eta_y\) and the own-price elasticities \(\epsilon_x\) and \(\epsilon_y\) for both goods x and y: The income elasticity \(\eta\) is given by: \(\eta_i = \frac{dD_i}{dI} \frac{I}{D_i}\), where \(i \in \{x, y\}\) For x, we have: \(\eta_x = \frac{d}{dI} \left(\frac{I}{p} - \frac{I}{q}\right) \frac{I}{\frac{I}{p} - \frac{I}{q}} = 1\) Similarly, for y: \(\eta_y = \frac{d}{dI} \left(\frac{I}{q}\right) \frac{I}{\frac{I}{q}} = 1\) The own-price elasticity \(\epsilon\) is given by: \(\epsilon_i = \frac{dD_i}{dp_i} \frac{p_i}{D_i}\), where \(i \in \{x, y\}\) For x, we have: \(\epsilon_x = \frac{d}{dp} \left(\frac{I}{p} - \frac{I}{q}\right) \frac{p}{\frac{I}{p} - \frac{I}{q}} = -1\) Similarly, for y: \(\epsilon_y = \frac{d}{dq} \left(\frac{I}{q}\right) \frac{q}{\frac{I}{q}} = -1\)

Verifying the Slutsky equation

Now let's verify the Slutsky equation, which states that: \((\frac{dD_x}{dp} \frac{p}{D_x}) = (\frac{dh_x}{dp} \frac{p}{D_x}) + (\frac{dh_x}{dI} \frac{\partial{E}}{\partial{p_y}})\) Where \(D_x\) and \(D_y\) are demand functions, \(h_x\) and \(h_y\) are the Hicksian demand functions, \(E\) is the consumer's expenditure. Using the demand functions derived above, we have: \(-1 = -1 + \frac{d}{dI} \left(\frac{I}{p} - \frac{I}{q}\right) \frac{I}{q}\) This simplifies to: \(0 = 0\), verifying the Slutsky equation for this utility function.

Verifying the elasticity form of the Slutsky equation

Finally, let's verify the elasticity form of the Slutsky equation, which states that: \(\epsilon_{x_px} = \epsilon_{x_p}^c + \eta_x \times \frac{p_x\epsilon_{xp_y}}{p_y}\) For the function at hand, we have: \(-1 = -1 + \eta_x \times \frac{p\epsilon_{xp_y}}{q}\) As both \(\eta_x\) and \(\epsilon_{x_px}\) are equal to 1, this simplifies to: \(0 = 0\), verifying the elasticity form of the Slutsky equation for this utility function. In conclusion, we have derived the demand functions for both goods x and y, calculated their income and own-price elasticities, and verified the Slutsky equation and its elasticity form for the given quasi-linear utility function.