Question

Problem Set 1 Due Date: Wednesday, January 25,2017 1. Consider the following demand function of an individual for good 1: where p" p2, ps are the prices of good 12, and 3, respectively, and Y represents the income the individual. Suppose good 1 and 2 are substitutes while good 1 and 3 are complements. (a) Describe in words what B,,B,, B, and B, measure. (b) Can you say anything about the expected signs of p.B.B, and B, in the demand equation? (c) Write out the formulae for the own price elasticity, cross price elasticities, and income elasticity of demand for good 1. 2. Now consider the following specific demand function for good 1: -20-2p,+4 (a) What are the price and income elasticities of demand when p, 3, and Y-4? (b) What are the price and income elasticities of demand if the demand function for good is given by q,-2p 3. Consider the following equation of a linear demand curve: q a-bp, Show demand is unit- price elastic at the midpoint of the demand curve. Show demand is elastic above the midpoint of the demand curve. (a) (b) Assume there are only two firms producing good 1. The is given by P,-5 while that of the second firm is p, 9,-5 4. inverse supply function of the first firm (a) Derive the market supply curve. (b) Suppose there is only one consumer of the good whose demand curve is given by q = 20-2p, . What is firm 1's and firm 2's price elasticity of supply at the equilibrium price? Suppose that the inverse demand curve of a good is given by p -10-1o g and the supply curve i represented byq 5p-10 (a) Graph the supply and demand functions 5.

Answer 1

We know that Good 1 and Good 2 are substitutes, while Good 1 and Good 3 are complements.

$\beta_{0}$ measures the intrinsic demand for Good 1 - when price of all goods and income is 0, the demand for the product is measured by  $\beta_{0}$. On the other hand, $\beta_{1}$ measures the slope (correlation) between price and demand of Good 1. $\beta_{2}$ measures the correlation between Good 1 and Good 2 and $\beta_{3}$ measures the correlation between Good 1 and Good 3. $\beta_{4}$ represents the relation between income and demand for Good 1.

Logically, if income increases, the demand for Good 1 should increase. Thus, $\beta_{4}$ should be positive. If price of Good 2 increases, consumers will switch to Good 1 ( a substitute ) increasing demand, and hence, the sign of $\beta_{2}$ should be positive. If the price of Good 3 increases, the demand for Good 1 will decrease - thus, the sign of $\beta_{3}$ should be negative. If price of Good 1 increases, demand for Good 1 will decrease - hence, $\beta_{1}$ should be negative.

Price elasticity is defined as $\frac{\Delta q}{\Delta p}$
We differentiate wrt p1 -
Price elasticity = $\beta_{1}$ (assuming that there is no correlation between p2,p3 and Y)
Cross price elasticity
wrt p2 = $\beta_{2}$
wrt p3 = $\beta_{3}$
income elasticity of demand = $\frac{\Delta q}{\Delta Y}$ = $\beta_{4}$

Similarly, the second question can be solved.

Hope this helped!