Connecting Utility Maximization and Expenditure Minimization PDF

6 Connecting Utility Maximization and Expenditure Minimization The utility maximization problem and the expenditure minimization problem, and the Marshallian and Hicksian demands that result from them, have a close relationship. These notes help clarify the relationship be...

6 Connecting Utility Maximization and Expenditure Minimization The utility maximization problem and the expenditure minimization problem, and the Marshallian and Hicksian demands that result from them, have a close relationship. These notes help clarify the relationship between these concepts. Concepts Covered The Indirect Utility Function The Expenditure Function Connecting the Functions Recall that for the utility function U (x, y) = x1/2 y 1/2 , we have derived Marshallian demands and Hicksian demands of  1/2 ∗ 1 I ∗ py xm = xh = Ū 2 px px  1/2 ∗ 1I px ym = yh∗ = Ū 2 py py It may not be obvious how to connect these functions (apart from the fact that we found both by setting the MRS equal to the price ratio). The rest of these notes will show their connection. First, we need to define two more concepts 6.1 Indirect Utility Function The indirect utility function tells us what the value of utility will be given any set of prices and income. It is found by plugging the Marshallian demand functions into the utility function. Formally, V (px , py , I) = U (x∗m , ym ∗ ) For the utility function above, the indirect utility function would be  1/2  1/2 ∗ 1/2 ∗ 1/2 1 I 1I 1 V (px , py , I) = (xm ) (ym ) = = Ip−1/2 py−1/2 2 px 2 py 2 x Once we have the indirect utility function, it makes it easier to see how changes in prices and income will affect the consumer. For example, if prices went up (bad for the consumer), but income also went up (good for the consumer), it might not be obvious whether the consumer is better or worse off overall. The indirect utility function could quickly answer that question without needing to solve for quantities demanded directly. 1 6.2 Expenditure Function The analog of the indirect utility function for the expenditure minimization problem is called the expenditure function. The expenditure function tells us how much we need to spend to achieve utility Ū given prices E(px , py , Ū ) = px (x∗h ) + py (yh∗ ) For the utility function above, we get   1/2    1/2  py px E(px , py , Ū ) = px Ū + py Ū = 2Ū p1/2 x py 1/2 px py Like the indirect utility function, the expenditure function makes it easier to compare outcomes when parameters change. In this case, the expenditure function is showing how much it will cost the consumer to maintain a given utility level Ū 6.3 Connecting the Functions The Connection Between Indirect Utility and Expenditure Function It turns out that the indirect utility function and the expenditure function are in some sense two sides of the same coin. The indirect utility function takes income as an input and figures out how much utility you can get with that income. The expenditure function takes utility as an input and figures out how much it costs to achieve that level of utility (in other words, how much income you need to get that level of utility). If we plug the expenditure function into the indirect utility function, we get the level of income such that the utility maximization problem would have given us Ū and if we plug the indirect utility function into the expenditure function, we get the utility such that the expenditure minimization problem would give us I V (px , py , E(px , py , Ū )) = Ū E(px , py , V (px , py , I)) = I What does this really mean? Consider the following procedure: 1. Pick an income I 2. Find the Marshallian demands that maximize utility (x∗m , ym ∗ ) ∗ 3. Solve for the utility that you get from this bundle (call it U ) 4. Now solve the expenditure minimization problem to find (x∗h , yh∗ ) when Ū = U ∗ 5. Plug these values into expenditure to find the expenditure function 6. You will find that the expenditure needed to achieve this bundle is the original I Alternatively 1. Pick a utility level Ū 2. Find the Hicksian demands that minimizes expenditure (x∗h , yh∗ ) 3. Solve for the expenditure that is required to buy this bundle (call it E ∗ ) 4. Now solve the utility maximization problem to find (x∗m , ym ∗ ) when I = E ∗ 5. Plug these values into utility to find the indirect utility function 6. You will find that the utility received from the bundle is the original Ū 2  This logic gives us a few nice implications. First, it means that we can always change an indirect utility function into an expenditure function and vice versa by inverting the functions. Let’s look at our indirect utility 1 V = Ipx−1/2 p−1/2 y 2 Now replace V with Ū and I with E and solve for E 1 Ū = Epx−1/2 p−1/2 y =⇒ E = 2Ū px1/2 py1/2 2 Which is the same equation we derived before. Had we started with the expenditure function, E = 2Ū px1/2 p1/2 y We would just replace E with I and Ū with V and solve for V to recover our indirect utility 1 I = 2V p1/2 x py 1/2 =⇒ V = Ipx−1/2 p−1/2 y 2 The Connection Between Marshallian and Hicksian Demand We also have an easy shortcut to convert Marshallian into Hicksian demand and back. From above we have our Marshallian demands of 1 I 1I x∗m = ym∗ = 2 px 2 py Now let’s take our expenditure function (which we could have derived from inverting the indirect utility) E(px , py , Ū ) = 2Ū p1/2 x py 1/2 And plug it into the Marshallian demands for I 1/2 1/2  1/2 1 I 1E 1 2Ū px py py x∗m = = = = Ū = x∗h 2 px 2 px 2 px px 1/2 1/2  1/2 ∗ 1I 1E 1 2Ū px py px ym = = = = Ū = yh∗ 2 py 2 py 2 py py Or, going the other way, let’s start with the Hicksian demands  1/2  1/2 ∗ py ∗ px xh = Ū yh = Ū px py And take our indirect utility (derived by inverting the expenditure function) 1 V = Ipx−1/2 p−1/2 y 2 Plug this into the Hicksian demands for Ū  1/2  1/2  1/2 ∗ py py 1 −1/2 −1/2 py 1 I xh = Ū =V = Ipx py = = x∗m px px 2 px 2 px  1/2  1/2  1/2 ∗ px px 1 −1/2 −1/2 px 1I ∗ yh = Ū =V = Ipx py = = ym py py 2 py 2 py 3 Discussion These results make it relatively quick to move back and forth between the utility maximization problem and its solutions (Marshallian demand and indirect utility) and the expenditure mini- mization problem and its solutions (Hicksian demand and expenditure function) without having to solve each problem directly. This duality can be useful mechanically. Sometimes Marshal- lian demand or Hicksian demand are computationally easier to solve for directly. The process discussed above gives a way to use one to derive the other, which can sometimes be easier. More importantly, the process above helps us see the close connection between utility maxi- mization and expenditure minimization. We have already seen that the two problems result in the same initial result that MRS equals the price ratio (assuming no corner solutions). The process described above shows that the solutions to the two problems will actually be exactly the same if we choose a specific level of Ū and I The statement above is a bit subtle and can be difficult to understand at first. In particular, it does not mean that Marshallian and Hicksian demands will always be equal to each other. In general they will not, because they depend on different parameters. They will only be equal at one specific point, where the Ū from the expenditure minimization problem is exactly equal to the maximum value of utility achieved for a given income level I (or, equivalently, when the value chosen for I in a utility maximization problem is exactly equal to the minimum expenditure to reach a given utility level Ū ). Why do we need two different conceptions of demand at all? We will show how to use these different functions in the coming weeks. 4