p 1 _ x 1 (p 1 _, p 2, I ) x 1 X 1 X 2

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1 Today we will cover some basic concepts that we touched on last week in a more quantitative manner. will start with the basic concepts then give specific mathematical examples of the concepts. f time permits the lecture will conclude with the concept of how consumer react to changes in income and price. Marshallian Demand revisited. Last lecture we saw graphically that we could derive the Marshallian demand curve from the utility function by was of the indifference curve and budget constraint. 2 p _ x (p _, p 2, ) f we can do it graphically we can certainly do it mathematically, assuming we have an explicit utility function and budget constraint. All we have to do is find a function for x in terms of prices and income. How can we do this? Take the first order conditions in the constrained utility maximization problem them solve for the xs in terms of prices and income. x

2 ndirect Utility Function Last lecture we made the assumption that indifference curves had to be monotonic. This means we can make a monotonic transformation of the utility function and keep the preference ordering between different baskets of goods. So instead of writing the utility function in terms of the goods consumed we can re-write it in terms of the price of those goods and the income of the consumer. We can do this since the amount of goods consumed will depend not only on the consumers direct utility function but also on the prices of the goods and the consumers income. Example: Below we will derive the Marshallian demand function and indirect utility function of a consumer in a two good model. This consumer face price p and p Y for goods and Y respectively. She has income and her utility function (which is known as a Cobb-Douglas function) is as follows. U(, 2 )= a Y b First set up the Lagragian function to solve this problem. L= U(,Y)+l(- Y) L= a Y b + l(- Y) Now solve for the FOCs First order conditions: = aa- Y b - l = 0 b a Y b - - l = 0 l = Next use the first two FOCs solve for l and set then equal. fi l = aa- Y b = b a Y b - P x fi = a a- Y b P x P y P y b a Y b - = aby b -(b -) b a-(a-) = ay b Now we can solve for Y in terms of a Y b = fi b a Now plug in the equation for Y into the last FOC

3 l = b a = 0 Now solve for b = - a = + b x a fi = + b a This is the Marshallian demand function for (notice it does not include, is will not always be the case.) We can now solve for Y using the equation we calculated above for Y in terms of. Just plug our answer for into this expression. y = b a y = b a b a + b a a + b a a a + b b a + b = a + b b ndirect Utility function. + a b Now we need to re-write the Utility function in terms of, and. To do this all we have to do is plug in our solutions for and Y into the direct Utility function.

4 U = a Y b V = + b a V = V = a a +b P a P b Y + b a a +b a a b b + a b a a b (a + b) a +b + a b b b Now consider the another utility function. U(,Y)= d +Y d Note: This is known as a Constant Elasticity of Substitution (CES) utility function. The variables for price and income are the same. First set up the Lagrangian. L= U(,Y)+l(- - Y) L= d +Y d +l(- - Y) Now solve for the FOCs First order conditions: = dd - - l = 0 = dd - - l = 0 l = Next use the first two FOCs solve for l and set then equal. fi l = dd - = dy d - fi = dd - d d - d - Y d -

5 Now we can solve for Y in terms of P Y d - d - = d - d - d - = d - Now plug in the equation for Y into the last FOC l = l = - - Now solve for = + d - = 0 d - = + d - fi = + d - This is the Marshallian demand function for. We can now solve for Y using the equation using the equation we calculated above for Y in terms of. Just plug our answer for into this expression. d - d - + d -

6 P Y P + since d - d - - d - P d - = we can re-write the expression as d - P d - P + Y P = d - P + + ndirect Utility function. d - Now we need to re-write the Utility function in terms of, and. To do this all we have to do is plug in our solutions for and Y into the direct Utility function. U = d + Y d V = - d - d + + d - d

7 ncome and Substitution Effects Y Py A C B P x Figure The Slutsky Equation The compensated demand is a function of the prices and a utility level. h x (p x, p y, U) is the point on the indifference curve for utility level U where the marginal rate of substitution equals the ratio of prices (MRS = p y / p x ). So a better notation for the substitution effect is: h x ( p,p,u( d (p,p,),d (p,p,) ) x y x x y y x y Expenditure function. To do the derivation of the Slutsky equation need to explain first the notion of the expenditure function. The arguments of the expenditure function are the prices of the two goods and a utility level E(p x,p y,u). The expenditure function tells you the minimum expenditure you need in order to achieve a given level of utility U with the given prices p x and p y. n figure we can start from considering the budget set. The slope of the budget set is determined by the ratio of the prices and the income tells you how far the budget set goes. To find the demand we find the point where the indifference curve is tangent to that budget line (point A). P x 2 A Figure

8 We can also start with a given level of utility and prices without knowing the income level. The budget line that is tangent to the indifference curve has the property that it represents the cheapest amount of expenditure needed to achieve the given level of utility. So we can think about having a fixed budget and choosing the consumption point that maximizes utility. We know that it is the tangency point that has the property that the MRS is equal to the ratio of the prices. We can look at this from the opposite direction. With the utility level fixed, try to find the cheapest income that would allow me to achieve that utility and find the same conditions: want to find the income level so that the budget set is tangent to the indifference curve. At that tangency point, the MRS is equal to the ratio of the prices. So the expenditure function is the smallest income,, such that you can achieve utility level U. can compute the compensated demand in terms of the expenditure function: h x is the consumption of good x if p x and p y are the prices and am constrained to a utility level U. That has to equal the usual demand for x if the prices are p x and p y and my income is the minimum income that would let me achieve utility level U. Derivation of the Slutsky equation f we take the derivative (of the equation above) with respect to the price of x, we obtain: The Slutsky equation says that the whole derivative of the garden variety demand, d x with respect to p x is the sum of two terms so let s solve for it: dx hx dx E = - The first term on the right hand side is the substitution effect. The first part of the second term ( d x / ), is the one that is positive if we have a normal good and negative if we have an inferior good. The last part ( E/ p x ), is the additional expenditure needed to keep utility constant. am claiming that: E = x will not go into the mathematical proof but let s go over the intuition. Suppose am consuming an amount x of a good. f the price of a unit of the good increases, in order to consume the same amount x, need to increase my expenditure by an amount equal to the number of units was consuming (x) times the increase in price. That is: [change in expenditure] = [number of units was consuming] times [increase in price] or DE = xdp.

9 What am claiming is that, this is also equal to the amount that have to be compensated for as a result of the price change (this comes from mathematical result called the Envelope theorem). Suppose originally consumed five units of good x at a price of $0 per unit of x. My total expenditure on good x was $50. Now suppose that the price of good x increases from $0 to $. Five units of good x will now cost $55. Therefore, to continue to consume five units of good x (i.e., maintain the same level of utility), my expenditure has to increase by $5. The change in expenditure ($5) is obtained by multiplying the change in price ($), times the number of units was originally consuming (5). There is a first and a second order effect. The first order effect is just this additional expenditure. The second order effect matters only if you are making large changes. t does not matter if you are making small changes. n summary, ( d x / ) ( E/ p x ) is just: x -x and therefore, dx hx = - x x We know that the substitution effect is always negative. f the price of a good increases, the compensated demand for that good decreases. The demand may increase or decrease but the compensated demand decreases. We also know that x is positive and that if x is a normal good, the derivative with respect to income will be positive so the derivative of the total demand with respect to its own price will be negative. However, if x is an inferior good the derivative of the total demand with respect to its own price may be negative or positive. t would seem that if x is an inferior good, there is not much to say about it. However, in recent years there is an area of research that looks at conditions under which we can say something about the relative sizes of the income effect and the substitution effect. The book gives an example (example 5.4) about how to compute these terms (read it through). t calculates the substitution effect by going through the calculation of the compensated demand function. There is an easier way to calculate the substitution effect. n practice it is often easier to compute the total derivative (the left-hand side of the equation) and the income effect and obtain the substitution effect by adding the total derivative and the income effect.