Profit Maximization Cont d Renting or buying capital Profit maximization and returns to scale Renting Capital If physical capital is one of the firm s inputs the firm can either rent capital or buy it E g firm can lease computers Problem solved by firm max pF K L wL L wK K Buying Capital What problem would the firm solve in the case it decides to buy rather than to rent capital Buying a machine has an impact on the firm s revenue for several years Q how do we compare revenue tomorrow to revenue today How do we account for risk No Uncertainty Firm can borrow and lend at interest rate r 0 10 Firm is considering how many computers to buy today Each computer Costs 10 000 Will be used for two years and then discarded zero resale value No Uncertainty Objective of the firm is to maximize the present value of profits present value of revenues minus the present value of costs buy computers t 0 R C t 1 R C t 2 Computing the Present Value What is the value today of having 1 one year from now if the interest rate is r 0 1 1 0 9 1 0 1 What is the value today of having 1 two years from now 1 0 82 2 1 0 1 Present Value of the Firm at t 0 R1 C1 R2 C2 PV pk K 2 1 0 1 1 0 1 R1 C1 p1 F K L1 wL1 L1 R2 C2 p2 F K L2 wL 2 L2 Maximizing the Present Value The firm should decide how many computers to buy and how much labor to hire in order to maximize its present value V max K L1 L2 PV Q how much would you be willing to pay to buy this firm at time t 0 V What are the Firms Profits Cost of buying computers must be amortized across their lifetime To construct cost as a flow consider 1 Annual economic depreciation 2 Opportunity cost due to foregone interest User Cost of Capital Year 1 1 Annual economic depreciation 5 000 2 Opportunity cost of funds 10 000 0 10 Year 2 1 Annual economic depreciation 5 000 2 Opportunity cost of funds 5 000 0 10 Profits Profits annual revenue labor cost user cost of capital Buy or Rent If the rental rate is larger than the user cost then it is convenient to buy capital If the rental rate is lower than the user cost then it is convenient to rent capital If the capital market is competitive the rental rate should equal the user cost firm indifferent between buying and renting Uncertainty Suppose there is uncertainty about price of the product firm is selling Problem gets more complicated because 1 Firm must take expectation of output price 2 Discount factor must be adjusted to take risk into account Profit Maximization and Returns to Scale Q How much profit does a competitive firm with a constant returns to scale technology make in the long run Profit Maximization and Returns to Scale A Zero Suppose it makes positive profits 1 1 py w x w2 x2 0 Double all inputs 2 p 2 y w1 2 x w2 2 x2 1 Profit Maximization and Returns to Scale Double all inputs 2 p 2 y w1 2 x w2 2 x2 1 This means that the firm was not choosing inputs optimally before Contradiction Thus zero profits is the only possibility Interpretation Suppose you are the owner of a firm that produces software with a constant returns to scale technology y f xL xM where xL represents workers and managers including yourself xM Interpretation Then this firm s profits in the long run are zero Pay wage to workers xL Pay salary to managers xM including yourself because of opportunity cost Nothing else is left