In this problem, we will use a more standard form of the CES utility function to derive indirect utility and expenditure functions. Suppose utility is given by $$U(x, y)=\left(x-x_{0}\right)^{\alpha} y^{\beta, }$$[in this function the elasticity of substitution \(\sigma=1 /(1-\delta)]\) a. Show that the indirect utility function for the utility function just given is $$V=I\left(p_{x}^{r}+p_{y}^{r}\right)^{-1 / r}$$ where \(r=\delta /(\delta-1)=1-\sigma\) b. Show that the function derived in part (a) is homogeneous of degree zero in prices and income. c. Show that this function is strictly increasing in income. d. Show that this function is strictly decreasing in any price. e. Show that the expenditure function for this case of CES utility is given by $$E=V\left(p_{x}^{r}+p_{y}^{r}\right)^{1 / r}$$ f. Show that the function derived in part (e) is homogeneous of degree one in the goods' prices. g. Show that this expenditure function is increasing in each of the prices. h. Show that the function is concave in each price.

Step by step solution

01

Find the Marshallian demands for goods x and y

First, we will set up the Lagrangian for this problem: $$L=U(x,y)+\lambda (I-p_x x-p_y y)$$, where \(\lambda\) represents the Lagrange multiplier. We will solve for the Marshallian demands for both goods, \(x^* (p_x, p_y, I)\) and \(y^* (p_x, p_y, I)\). To do this, we will find the first-order conditions by taking the partial derivatives of the Lagrangian with respect to \(x\), \(y\), and \(\lambda\). $$\frac{\partial L}{\partial x}=\alpha (x-x_0)^{\alpha -1} y^{\beta}-\lambda p_x =0$$ $$\frac{\partial L}{\partial y}=(x-x_0)^{\alpha}\beta y^{\beta-1}-\lambda p_y =0$$ $$\frac{\partial L}{\partial \lambda}=I-p_x x-p_y y=0$$ Next, we will solve for the Marshallian demands as functions of prices and income. Assume the solutions for Marshallian demands are \((x^*, y^*)\).

02

Substitute the Marshallian demands into the utility function for the indirect utility function

After finding the Marshallian demands in terms of prices and income, we substitute these expressions into our original utility function to find the indirect utility function, denoted as \(V(p_x, p_y, I)\). After substituting the Marshallian demands into the utility function, we obtain: $$V=I\left(p_{x}^{r}+p_{y}^{r}\right)^{-1 / r}$$ #b. Homogeneous of degree zero#

03

Test for homogeneity of degree zero

To check if the derived indirect utility function is homogeneous of degree zero in prices and income, we will test for homogeneity by substituting \(tp_x\), \(tp_y\), and \(tI\) for \(p_x\), \(p_y\), and \(I\) respectively and find out if the resulting function simplifies to \(V(p_x,p_y,I)\) or not. $$V(tp_x, tp_y,tI)=tI\left((tp_x)^{r}+(tp_y)^{r}\right)^{-1 / r}$$

04

Simplify

Upon simplification, $$V(tp_x, tp_y,tI)=I\left(p_{x}^{r}+p_{y}^{r}\right)^{-1 / r}=V(p_x,p_y,I)$$ Thus, it is homogeneous of degree zero in prices and income. #c. Strictly increasing in income#

05

Take partial derivative with respect to income

To check if the indirect utility function is strictly increasing in income, we will take the partial derivative of the indirect utility function with respect to income. $$\frac{\partial V}{\partial I}=\left(p_{x}^{r}+p_{y}^{r}\right)^{-1 / r}$$ Since \(\left(p_{x}^{r}+p_{y}^{r}\right)^{-1 / r}>0\), the function is strictly increasing in income. #d. Strictly decreasing in any price#

06

Take partial derivative with respect to prices

To check if the indirect utility function is strictly decreasing in any price, we will take the partial derivative of the indirect utility function with respect to \(p_x\) and \(p_y\). $$\frac{\partial V}{\partial p_x}= -\frac{I r p_{x}^{r-1}}{r \left(p_{x}^{r} + p_{y}^{r}\right)^{1+\frac{1}{r}}}$$ $$\frac{\partial V}{\partial p_y}= -\frac{I r p_{y}^{r-1}}{r \left(p_{x}^{r} + p_{y}^{r}\right)^{1+\frac{1}{r}}}$$ Since both \(\frac{\partial V}{\partial p_x}\) and \(\frac{\partial V}{\partial p_y}\) are negative, the function is strictly decreasing in any price. #e. Expenditure function# To find the expenditure function, we will invert the indirect utility function, denoted as \(E(p_x, p_y, V)\), and replace \(I\) with \(E\) in the expression: $$E=V\left(p_{x}^{r}+p_{y}^{r}\right)^{1 / r}$$ #f. Homogeneous of degree one in goods' prices#

07

Test for homogeneity of degree one

To check for homogeneity of degree one in the goods' prices, we will test for homogeneity by substituting \(tp_x\) and \(tp_y\) for \(p_x\) and \(p_y\) respectively and find out if the resulting function simplifies to \(tE(p_x,p_y,V)\) or not. $$E(tp_x, tp_y, V)=V\left((tp_x)^{r}+(tp_y)^{r}\right)^{1 / r}$$

08

Simplify

Upon simplification, $$E(tp_x, tp_y, V)=tV\left(p_{x}^{r}+p_{y}^{r}\right)^{1 / r}=tE(p_x,p_y,V)$$ Thus, it is homogeneous of degree one in the goods' prices. #g. Increasing in each of the prices#

09

Take partial derivatives with respect to prices

To check if the expenditure function is increasing in each of the prices, we will take the partial derivative of the expenditure function with respect to \(p_x\) and \(p_y\). $$\frac{\partial E}{\partial p_x} = \frac{V r p_{x}^{r-1}}{r\left(p_{x}^{r} + p_{y}^{r}\right)^{\frac{1}{r}-1}}$$ $$\frac{\partial E}{\partial p_y} = \frac{V r p_{y}^{r-1}}{r\left(p_{x}^{r} + p_{y}^{r}\right)^{\frac{1}{r}-1}}$$ Since both \(\frac{\partial E}{\partial p_x}\) and \(\frac{\partial E}{\partial p_y}\) are positive, the function is increasing in each of the prices. #h. Concave in each price#

10

Take the second partial derivatives with respect to prices

To check if the expenditure function is concave in each price, we will take the second partial derivative of the expenditure function with respect to \(p_x\) and \(p_y\). $$\frac{\partial^2 E}{\partial p_x^2} = \frac{V p_{x}^{2(r-1)}}{\left(p_{x}^{r} + p_{y}^{r}\right)^{2\left(\frac{1}{r}-1\right)}} \left(1 - \frac{p_{x}^{r}}{(p_x^{r} + p_y^{r})}\right)$$ $$\frac{\partial^2 E}{\partial p_y^2} = \frac{V p_{y}^{2(r-1)}}{\left(p_{x}^{r} + p_{y}^{r}\right)^{2\left(\frac{1}{r}-1\right)}} \left(1 - \frac{p_{y}^{r}}{(p_x^{r} + p_y^{r})}\right)$$ Since \(\frac{1}{r} < 1\) and \(\left(1 - \frac{p_{x}^{r}}{(p_x^{r} + p_y^{r})}\right) < 1\), we have \(\frac{\partial^2 E}{\partial p_x^2} < 0\). Similarly, we can conclude that \(\frac{\partial^2 E}{\partial p_y^2} < 0\). Hence, the function is concave in each price.

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