2. Predictions of Utility Theory About the Nature of Demand Suplementary References: Layard, P & A.Walters: Microeconomic Theory p135-137 Deaton A., & J. Muellbauer, Economics and Consumer Behavior p14-16 p43-46

3. Predictions of Utility Theory About the Nature of Demand Given the axioms (or properties) 1-7 we have assumed about the utility function, these imply certain things about the demand function. We can now want to derive a set of core properties we would expect any demand function we estimated to exhibit

4. Testable Predictions and the Theory We can then test the demand functions we derived and ask if they exhibit these properties. If they don’t then we either have a problem with our data, or we have used the wrong functions or estimation method OR (more seriously), with our theory!

5. PROPERTIES OF DEMAND FUNCTION 1.The Adding-Up Condition The effect of a change in income on the demand for all goods P x x + P y y = m P x dx + P y d y = dm (  dm)

6. PROPERTIES OF DEMAND FUNCTION 1.The Adding-Up Condition The effect of a change in income on the demand for all goods P x x + P y y = m P x dx + P y d y = dm

7. PROPERTIES OF DEMAND FUNCTION 1.The Adding-Up Condition The effect of a change in income on the demand for all goods P x x + P y y = m P x dx + P y d y = dm

8. PROPERTIES OF DEMAND FUNCTION 1.The Adding-Up Condition The effect of a change in income on the demand for all goods P x x + P y y = m P x dx + P y d y = dm

9. Property 1: S x  x + S y  y = 1 (adding-up condition) PROPERTIES OF DEMAND FUNCTION 1.The Adding-Up Condition The effect of a change in income on the demand for all goods P x x + P y y = m P x dx + P y d y = dm

10. Testable property 2. Homogeneity Now  ’ing PRICES AND INCOME If demand is unaffected by an equi- proportional change in all prices and income then there is an absence of money illusion That is, if I choose the bundle (x, y) with prices P x, P y and income m, then I will choose the same bundle with 2 P x, 2P y and 2m.

11. Formal Statement Formally, a function is homogeneous of degree t, if when all prices and income change by  x = f (  P x,  P y,  m) =  t f ( P x, P y,m) Claim: Demand functions are homogeneous of degree zero in prices and income, that is x = f (  P x,  P y,  m) =  t f ( P x, P y,m) =  0 f ( P x, P y,m) = f (P x, P y,m)

12. Other examples of homogeneous Functions Production Function: Q = ƒ (L, K) What happens if we scale up all inputs by a factor of  ƒ(  L,  K) = ? Homogeneity of degree t implies ƒ(  L,  K) =  t ƒ(L, K) What is t ? If we have CRS, that is, if the production function is homogeneous of degree 1, then t=1 and =  1 ƒ(L, K) =  ƒ(L, K) e.g. ƒ(2L, 2K) = 2 ƒ(L, K)=2Q

13. Homogeneity of degree zero in prices and income seems a reasonable property, after all it simply implies the absence of money illusion. What does homogeneity imply about our demand functions in general? Taking the total derivative of x= x(P x, P y, m)we get:

14. Homogeneity of degree zero in prices and income seems a reasonable property, after all it simply implies the absence of money illusion. What does homogeneity imply about our demand functions in general? Taking the total derivative of x= x(P x, P y, M)we get:

15. Homogeneity of degree zero in prices and income seems a reasonable property, after all it simply implies the absence of money illusion. What does homogeneity imply about our demand functions in general? Taking the total derivative of x= x(P x, P y, m)we get:

16. Will imply no change in the demand for x If the demand function is homogeneous of degree zero in prices and income then changing P x, P y, and m in the same proportion:

17. Cournot Condition:

18. Property 2: The Cournot Condition Homogeneity of the demand function for x requires Similarly for the demand function for y, homogeneity requires