Expenditure Minimization PDF

5 Expenditure Minimization So far we have phrased the consumer’s problem as a utility maximization problem given a budget constraint. However, there is another way that we could express a similar idea. Let’s assume that the consumer wants to achieve a certain level of utility...

5 Expenditure Minimization So far we have phrased the consumer’s problem as a utility maximization problem given a budget constraint. However, there is another way that we could express a similar idea. Let’s assume that the consumer wants to achieve a certain level of utility and wants to find the least expensive way of achieving that level. Concepts Covered The Consumer’s Expenditure Minimization Problem Graphical Representation of Expenditure Minimization Hicksian Demand Special Utility Functions 5.1 The Consumer’s Expenditure Minimization Problem Setup Assume that a consumer wants to choose consumption of x and y to reach a utility level of at least some arbitrary constant Ū in the cheapest possible way. Their total expenditure is given by px x + py y. Therefore, we can write a minimization problem as min px x + py y x,y subject to U (x, y) = Ū This problem is once again just a constrained optimization problem (in this case a minimization), so we can use our Lagrangian   L = px x + py y + λ Ū − U (x, y) Taking first order conditions ∂L ∂U ∂U = px − λ = 0 =⇒ px = λ ∂x ∂x ∂x ∂L ∂U ∂U = py − λ = 0 =⇒ py = λ ∂y ∂y ∂y ∂L = Ū − U (x, y) =⇒ Ū = U (x, y) ∂λ 1 MRS and Price Ratio (Again) If we divide the first equation by the second we get ∂U px ∂x = ∂U py ∂y Which tells us that, once again, the solution to the problem occurs exactly when MRS = price ratio! Why must this relationship hold in both the expenditure minimization problem and the utility maximization problem? We will spend more time on the connection next time, but it is worth briefly touching on the intuition here. Basically, the idea is the same as before. If the MRS were greater than the price ratio, then the relative benefit of x (per dollar) is greater than the relative benefit of y. Before, we interpreted this result by concluding that the consumer would get more utility by spending their income on x instead of y, but the flip side of that statement is that the consumer could stay at the same utility and spend less money if they consumed more x relative to y. Looking at some graphs makes this idea even more clear. 5.2 Graphical Representation When we showed the utility maximization problem on the graph, we fixed a budget constraint and then tried to find the point on that budget constraint that was on the highest indifference curve. For expenditure minimization we are doing the exact opposite. Now our constraint is an indifference curve where utility is fixed at Ū. Let’s fix the utility level at 4 (we will stick with the utility function U = x1/2 y 1/2 ) 10 8 6 y 4 2 U =4 0 0 2 4 6 8 10 12 14 16 x The shaded area shows all combinations of x and y that give at least Ū = 4 utility. We want to find the cheapest possible bundle that will be within this set. Note that like the utility maximization problem, we will always have a binding constraint (i.e. it would never make sense to reach more than 4 utility because then we could just purchase a little less of one good and still get to 4 in a cheaper way). 2  Now let’s plot expenditure. Each level of expenditure gives us a different feasible set of goods. To plot an expenditure line, we can fix an arbitrary level of expenditure (let’s call it Ē for now, and solve for y Ē px barE = px x + py y =⇒ y = − x py py Note that this expenditure line is extremely similar to the budget constraint we plotted before (in particular note that the slope is still the negative price ratio). The only difference is that we will allow expenditure to vary while before we looked at a fixed income. Keeping the prices for the example used in the utility maximization graphs of px = 1, py = 4, the graph below adds expenditure lines (in blue) for a few different levels of expenditure to the graph of the indifference curve given above. 10 8 6 y 4 2 0 0 2 4 6 8 10 12 14 16 x We want to find the point that is on the lowest expenditure line while being at least as high as our indifference curve. As in the utility maximization problem, the optimal point occurs when the expenditure line is tangent to the indifference curve (where price ratio equals MRS). In this case, that point occurs at (8,2), which represents an expenditure of 16. If you look back at the graphs from utility maximization, you will notice that when we maxi- mized utility with an income of 16 we also got (8,2) as the solution to the utility maximization problem. In fact, the graphs we drew in the utility maximization problem and expenditure min- imization problem look strikingly similar. The only difference is that in the utility max problem, we had a fixed income (constant expenditure) and varied utility to find the maximum achievable utility. Here, we held utility fixed, and varied expenditure to find the cheapest possible bundle. 3 5.3 Hicksian Demand Definition As we did with the utility maximization problem, we want to find the optimal value of x and y as a function of parameters. However, instead of taking income as a parameter, we take the utility level Ū. The optimal consumption as a function of prices and utility will be called Hicksian Demand (from now on I will use an m subscript for Marshallian demand, and an h subscript for Hicksian) x∗h = x(px , py , Ū ) yh∗ = y(px , py , Ū ) From our Lagrangian, we already have the result that these values must satisfy px M RS = py Although this condition is the same as the Marshallian demand, in this case we plug into the utility constraint instead of the budget constraint. Example Using the utility function U (x, y) = x1/2 y 1/2 , that results in y px px = =⇒ y = x x py py To this point, this result is the same as the utility maximization problem. However, for that problem, we plugged into the budget constraint. Now, we have to plug into the utility constraint  1/2  1/2 1/2 1/2 1/2 px px Ū = x y =x x =x py py Solve for x to get the Hicksian demand  1/2 py x∗h = Ū px Which gives  1/2 px yh∗ = Ū py To reiterate, these functions tell us the consumption bundle of x and y that achieves utility Ū. We will dig deeper into the close relationship between Marshallian and Hicksian demands next time. 4 5.4 Special Utility Functions Most of the intuition for the “special” utility functions we talked about last time also carries over to expenditure minimization and Hicksian demand. We will briefly touch on each of these issues here. Corner Solutions The exact same process for finding Marshallian demands with corner solutions also applies when finding Hicksian demand. Since we are still relying on the same idea of finding the point tangent to an indifference curve, we still have the same issue when indifference curves are concave. Once again, with concave indifference curves we just have to check both corner points (consuming all x or all y to reach the given Ū ). In this case we will be trying to see which is cheaper rather than which gives higher utility. Assuming convex indifference curves, we again can run into scenarios where one of the Hicksian demands comes out negative. In this case, we again set that value equal to zero and consume only the other good (plug into the utility function to determine how much) Perfect Substitutes Perfect substitutes preferences also work similarly. Once again, it will only make sense for a consumer to consume one good or the other. In particular, a consumer will consume only x when M RS > px /py and consume only y when M RS < px /py. The difference is that rather than use the budget constraint to find the demands, we need to use the utility function. Given preferences U (x, y) = Ax + By if the consumer consumes only x we set yh∗ = 0 so Ū Ū = Ax∗h =⇒ x∗h = A Similarly, when x∗h = 0 and the consumer consumes only y Ū Ū = Byh∗ =⇒ yh∗ = B Perfect Complements Perfect complements preferences of the form U (x, y) = min(Ax, By) will once again follow from the condition that Ax = By. Notice that this condition means that both terms in the utility function will be equal to each other. We can then solve for Hicksian demand Ū Ū = min(Ax, By) = min(Ax, Ax) = Ax =⇒ x∗h = A Ū Ū = min(Ax, By) = min(By, By) = By =⇒ yh∗ = B 5