Part IIA, Paper 1 Consumer and Producer Theory

1. Part IIA, Paper 1Consumer and Producer Theory Lecture 8 Profit Maximisation and Supply Functions Flavio Toxvaerd

2. Recap and Today’s Outline • Last lecture looked at problem of cost minimisation, and obtained Cost function - C(p,y). • We maintained throughout the analysis that factor prices were unaffected by the firm’s employment decisions – that is, the firm was a price taker in the input markets • Today we look specifically at the problem of profit maximisation – maintaining the assumption that the firm is a price taker in the factor markets • Note: the mathematical requirements for ‘unique’ solutions become more complicated if the assumption is dropped – but the basic properties of the cost function remain unchanged

3. Profit Maximisation Two ways to represent profit maximisation problem: 1. with FOC: MC curve cuts MR curve from below and SOC: 2.

4. Profit Maximisation Simplifying Assumption: Firm is a price taker in the output market – thus p(y)=py Thus maximisation problem 1 becomes: and solution depends on both input and output prices This is the supply function for the firm As solution obtained from FOC: have that supply function is the inverse of the marginal cost curve

5. Profit Maximisation Formulated the second way – we have MR=MC First order conditions: These are the same conditions as those from cost minimisation

6. Profit Maximisation Once again we have that: y Graphically – have isoprofit line tangental to production function PPS x

7. Supply and Factor Demand Functions Solution to maximisation problem gives: Supply Function Factor Demand Functions

8. Properties of Supply and Factor Demand Functions Homogeneous degree zero. Proof: Increasing all prices by the same proportions leaves the maximisation problem unchanged. Output is increasing and factor demand decreasing in own price: Proof: Follows from second order conditions of maximisation prob.

9. Profit Function Evaluating the maximisation problem at the optimal solution gives the Profit Function From Envelope Theorem:

10. Properties of Profit Function Homogeneous of degree 1 in prices Proof: No change in relative prices, so no change in optimal solutions - so profit changes by same proportion. Non-decreasing in output price and non-increasing in input prices Proof: Envelope Theorem. Convex in prices Proof: If prices change, profits will increase linearly if inputs and outputs are unchanged. Thus any change must be to increase profits - and so profits increases more than linearly.

11. Recovering Production Function from Profit Function y y0 Generates concave production function y=F(x) y1 x1 x0

12. Short Run Profit Function We may want to consider the profit function when some inputs are not variable: Clearly: with equality if and only if

13. LeChatelier Principle The price elasticity of supply is greater in the long run than in the short run. Alternatively, thus the slope of the Supply Function is lower in the long run. Proof: Let be the optimal solutions when prices are

14. LeChatelier Principle We know that h(py) is minimised at p*y

15. Alternative Specification Throughout we have kept the vectors of inputs and output separate. However we could also consider inputs to be nothing more than negative outputs – and we could consider a vector of net outputs from a production process z = y - x Profit maximisation becomes maxzp.zsubject to z  Z , where Z is the set of feasible (net) production plans.

16. Graphically Isoprofit line y PPS x

17. General Equilibrium We have developed methods to generate demand functions for individuals and supply functions for firms – GIVEN a specific vector of prices. We have made no effort to see whether or not supply and demand are equal at those prices – if the prices are EQUILIBRIUM prices.

18. Readings • Varian, Intermediate Economics, chapter 19 • Varian, Microeconomic Analysis, chapters 2,3