Question

Price indifference curves are iso-utility curves with the prices of two goods on the \(X\) - and \(Y\) -axes, respectively. Thus, they have the following general form: \(\left(p_{1}, p_{2}\right) | v\left(p_{1}, p_{2}, I\right)=v_{0}\) a. Derive the formula for the price indifference curves for the Cobb-Douglas case with \(\alpha=\beta=0.5 .\) Sketch one of them. b. What does the slope of the curve show? c. What is the direction of increasing utility in your graph?

Short answer

Based on the given step-by-step solution, the correct short answer for this topic would be: When α = β = 0.5 in a Cobb-Douglas utility function, the corresponding price indifference curve equation can be derived as \(\frac{I - p_2y}{p_1} \cdot y = v_0^2\). Graphing the indifference curves gives us a concave shape with a negative slope, indicating that the consumer has to trade-off between the consumption of goods 1 and 2 in order to maintain the same level of utility. The direction of increasing utility on the graph is up and to the right, showing that consuming more of both goods maximizes the consumer's utility.

Step-by-step solution

Cobb-Douglas Utility Function and Budget Equation

In the Cobb-Douglas case, the utility function is given by: \(U(x,y) = x^\alpha \cdot y^\beta\). Since, \(\alpha = \beta = 0.5\), the utility function becomes, \(U(x,y) = x^{0.5} \cdot y^{0.5}\) The budget equation is given by: \(p_1x + p_2y = I\)

Solving for x and y

In order to derive the price indifference curve, we first need to solve for x and y in terms of the budget equation. Rearranging the budget equation for x, we get: \(x = \frac{I - p_2y}{p_1}\) Now, plug the value of x into the utility function: \(U(x,y) = (\frac{I - p_2y}{p_1})^{0.5} \cdot y^{0.5}\) Now, we need to find the equation that represents the indifference curve, which is represented by \(v(x,y,I)=v_0\).

Deriving the Price Indifference Curve Equation

Rewrite the utility function as: \(v(x,y,I)= (\frac{I - p_2y}{p_1})^{0.5} \cdot y^{0.5}=v_0\) Now, solve it for \(v(x,y,I)=v_0\), and we get: \((\frac{I - p_2y}{p_1})^{0.5} \cdot y^{0.5}=v_0\) Square both sides of the equation to eliminate the square root, \(\frac{I - p_2y}{p_1} \cdot y = v_0^2\) Now, we have an equation that represents price indifference curves.

Sketch the Price Indifference Curve

To sketch one of the price indifference curves, we can take the above equation and substitute some fixed value for \(v_{0}\): \(\frac{I - p_2y}{p_1} \cdot y= v_0^2\) Upon sketching the graph with x and y axes representing the quantities of good 1 and good 2, we would get a curve with a negative slope, intercepting the axes, and having a concave shape.

Analyze the Slope of the Price Indifference Curve

The slope of the price indifference curve represents the ratio of prices of the two goods. The negative slope of the curve means that in order to achieve the same level of utility, the consumer must trade-off between the consumption of goods 1 and 2. A steeper slope indicates that good 1 is relatively more expensive than good 2. On the curve, the tangent represents the Marginal Rate of Substitution (MRS), showing how many units of good 2 one is willing to give up for an additional unit of good 1 in order to maintain the same utility level.

Determine the Direction of Increasing Utility

On the graph of the price indifference curve, the direction of increasing utility is indicated by the direction that moves towards higher price indifference curves. Since the curves have a negative slope and are concave, the direction of increasing utility would be moving up and to the right, indicating that the consumer would maximize their utility by consuming more of both goods.