FREC 834
Demand Theory and Welfare Analytics
Let utility U be a Cobb-Douglas function of quantities of goods X1 and X2 consumed: U = 10X10.5X20.5 where total consumption is constrained by money income or budget M: P1X1 + P2X2 = M.
Set up a spreadsheet model of this income-constrained demand system. At the top of the spreadsheet, enter cell values for P1, P2, and budget M of 1, 1 and 10 respectively. Below in the first column, enter values for X1 ranging from 0 to 10 in increments of 0.25.
Based on the budget and prices specified in the top cells and the quantities of X1 specified the first column, enter the formula to calculate the corresponding maximum affordable quantities of X2 in the adjacent column.
In the third column, enter the formula to calculate the utility obtained from each of these just-affordable combinations of X1 and X2. The values in the second and third columns should recalculate whenever you change either price or the budget at the top of the spreadsheet.
When P1 = P2 = 1 and M = 10, what combination of X1 and X2 yields maximum utility? What is that utility level?
Now change P1 to 2 while holding P2 = 1 and M = 10. What are the new utility-maximizing levels of X1 and X2? What is the new utility level U1?
Suppose you wanted to increase the budget M to compensate for the utility loss caused by the increase in P1. What adjustment to M restores the base level of utility (U0 = 50) if consumption of X1 is held at X1 = 5? (This income adjustment is known as compensating surplus.)
Note that with this income adjustment the consumer can actually obtain U > U0 by substituting some X1 for X2. So what is the adjustment to M that yields a unique maximum of U = U0 = 50? What are the consumption quantities of X1 and X2 at that point? (This income adjustment is known as compensating variation.)
Note that these compensation schemes imply that the consumer with budget M =10 is entitled to utility level U0 = 50 resulting from prices P1 = P2 = 1, and is entitled to compensation if the price increases to P1 = 2.
But suppose the consumer is only entitled to the higher prices P1 = 2 and P2 = 1, and the corresponding lower utility level. When P1 = 2, P2 = 1 and M = 10, what combination of X1 and X2 yields maximum utility. What is that corresponding base level of utility U0?
Now examine the effects of the reverse price change. Reduce P1 to 1, while holding P2 = 1 and M = 10 (so the consumer moves to X1 = X2 = 5 at U1 = 50). Now reduce the budget M to offset this utility gain. What reduction in M restores the base level of utility (U0 = 35.355) if consumption of X1 is held at X1 = 5? (This income adjustment is known as equivalent surplus.)
Note that with this income adjustment the consumer can still obtain U > U0 by substituting some X2 for X1. So what is the adjustment to M that yields a unique maximum U = U0 = 35.355? What are the consumption quantities of X1 and X2 at that point? (This income adjustment is known as equivalent variation.)
Derivation of demand schedules
Set up the Lagrangean to maximize utility U = 10X10.5X20.5 subject to the income constraint M - P1X1 - P2X2 = 0.
Take the partial derivatives with respect to X1, X2 and the Lagrangean multiplier and set these equal to zero.
Solve this set of first-order conditions to obtain the conventional (constant-income) demand functions X1 = X1(P1,P2,M) and X2 = X2(P1,P2,M).
Calculate the elasticity of demand E = (∂X1/∂P1)(P1/X1) for X1(P1,P2,M) with respect to P1.
Set up the dual Lagrangean to minimize expenditure P1X1 + P2X2 subject to the utility constraint U0 - 10X10.5X20.5 = 0.
Take the partial derivatives with respect to X1, X2 and the Lagrangean multiplier and set these equal to zero.
Solve this set of first-order conditions to obtain the income-compensated (i.e., constant-utility) demand functions X1 = X1(P1,P2,U0) and X2 = X2(P1,P2,U0).
Calculate the elasticity of demand E = (∂X1/∂P1)(P1/X1) for X1(P1,P2,U0) with respect to P1
Starting from initial conditions P1 = P2 = 1 and M = 10, use the elasticity measures to estimate the consumer surplus and compensating variation changes when P1 changes to P1 = 2.
Starting from initial conditions P1 = 2, P2 = 1 and M = 10, use the elasticity measures to estimate the consumer surplus and equivalent variation changes when P1 changes to P1 = 1.