9.7 Appendix B: Deriving the Hicks Decomposition Bundle
In this Appendix we’ll derive the solutions for the Hicks decomposition bundles for two utility functions: one an example of complements, one an example of substitutes.
Example 1: Complements
The following diagram shows the effect of an increase in the price of good 1 from $p_1 = 2$ to $p_1^\prime = 8$, holding $p_2 = 2$ and $m = 48$ constant, for a consumer with the CES utility function \(u(x_1,x_2) = (x_1^{-1} + x_2^{-1})^{-1}\) which has the associated marginal rate of substitution \(MRS(x_1,x_2) = \left(x_2 \over x_1\right)^2\)
[ See interactive graph online at https://www.econgraphs.org/graphs/consumer/income_substitution/hicks_decomposition_complements ]
To find the coordinates for bundle B, we follow these steps:
- Utility maximization with initial prices: Budget line $BL_1$ represents the initial budget line (before the price change):\(BL_1: 2x_1 + 2x_2 = 48\)The price ratio along this budget line is $p_1/p_2 = 2/2 =1$, so the tangency condition is\(MRS = {p_1 \over p_2} \Rightarrow \left(x_2 \over x_1\right)^2 = {2 \over 2} \Rightarrow x_2 = x_1\)Plugging this into the budget constraint gives us the optimal bundle $A = (12,12)$.
- Evaluate utility at the initial bundle: Plugging the coordinates $A = (12,12)$ into the utility function gives us:\(U_1 = u(12,12) = \left({1 \over 12} + {1 \over 12}\right)^{-1} = \left({1 \over 6}\right)^{-1} = 6\)
- Find tangency condition at the new prices: After the price of good 1 rises to $p_1^\prime = 8$, the price ratio becomes $p_1^\prime/p_2 = 8/2 =4$, so the tangency condition is \(MRS = {p_1 \over p_2} \Rightarrow \left(x_2 \over x_1\right)^2 = {8 \over 2} \Rightarrow x_2 = 2x_1\)
- Cost minimization with initial utility and final prices: The cost-minimizing way of achieving $U_1 = 6$ when the price ratio is $p_1^\prime/p_2 = 4$ will satisfy two conditions:\(\begin{aligned} \text{Tangency condition: \ } & x_2 = 2x_1\\ \text{Utility constraint: \ } & \left({1 \over x_1} + {1 \over x_2}\right)^{-1} = 6 \end{aligned}\) Plugging the tangency condition $x_2 = 2x_1$ into the utility constraint and solving gives us \(\begin{aligned} \left({1 \over x_1} + {1 \over 2x_1}\right)^{-1} &= 6\\ \left({3 \over 2x_1}\right)^{-1} &= 6\\ x_1^c &= 9\\ \end{aligned}\) and therefore \(x_2^c = 2x_1^c = 18\)
In the diagram above, if you check the box “show income offer curves” it shows the two tangency conditions (that is, the IOC’s) before and after the price change.
Example 2: Substitutes
The following diagram shows the effect of an increase in the price of good 1 from $p_1 = 2$ to $p_1^\prime = 4$, holding $p_2 = 2$ and $m = 36$ constant, for a consumer with the CES utility function \(u(x_1,x_2) = \sqrt{x_1} + \sqrt{x_2}\) which has the associated marginal rate of substitution\(MRS(x_1,x_2) = \left(x_2 \over x_1\right)^{1 \over 2}\)
[ See interactive graph online at https://www.econgraphs.org/graphs/consumer/income_substitution/hicks_decomposition_substitutes ]
To find the coordinates for bundle B, we follow these steps:
- Utility maximization with initial prices: As in the example above, the tangency condition is $x_2 = x_1$. Plugging that into the budget constraint\(2x_1 + 2x_1 = 36\) yields the optimal bundle $A = (9,9)$.
- Evaluate utility at the initial bundle: Plugging the coordinates $A = (9,9)$ into the utility function gives us:\(U_1 = u(9,9) = \sqrt{9} + \sqrt{9} = 6\)
- Find tangency condition at the new prices: After the price of good 1 rises to $p_1^\prime = 4$, the price ratio becomes $p_1^\prime/p_2 = 4/2 =2$, so the tangency condition is \(MRS = {p_1 \over p_2} \Rightarrow \left(x_2 \over x_1\right)^{1 \over 2} = {4 \over 2} \Rightarrow x_2 = 4x_1\)
- Cost minimization with initial utility and final prices: The cost-minimizing way of achieving $U_1 = 6$ when the price ratio is $p_1^\prime/p_2 = 4$ will satisfy two conditions:\(\begin{aligned} \text{Tangency condition: \ } & x_2 = 4x_1\\ \text{Utility constraint: \ } & \sqrt{x_1} + \sqrt{x_2} = 6 \end{aligned}\) Plugging the tangency condition $x_2 = 4x_1$ into the utility constraint and solving gives us \(\begin{aligned} \sqrt{x_1} + \sqrt{4x_1} &= 6\\ 3\sqrt{x_1} &= 6\\ x_1^c &= 4\\ \end{aligned}\) and therefore \(x_2^c = 4x_1^c = 16\)
As before, if you check the box “show income offer curves” it shows the IOC’s before and after the price change.