Part 2: Monopoly and Oligopoly Investment

Transcription

1 Part 2: Monopoly and Oligopoly Investment Irreversible investment and real options for a monopoly Risk of growth options versus assets in place Oligopoly: industry concentration, value versus growth, and risk Open loop investment equilibria Perfection and closed loop strategies Review The objective function is E 0 e rt π(x t, k 0 ) dt+e 0 e rt [ Kt k 0 π (X t, k) dk dt di t ] The first term is the value of assets in place The maximized value of the second term is the value of growth options The value of growth options is k 0 sup τ k E [ e rτ k {S(X τk, k) 1} ] dk, where [ S(x, k) = E e r(u t) π (X u, k) du t ] X t = x

2 Review continued Define the value of option k: V (x, k) = sup τ E [ e rτ {S(X τ, k) 1} X 0 = x ] When it is optimal to invest, we must have value matching: V (X t, k) = S(X t, k) 1 Given x, the largest capital stock such that it would be optimal to invest is κ(x) def = sup {k V (x, k) = S(x, k) 1} The optimal capital stock process is K t = k 0 sup κ(x s ) 0 s t Example Assume constant returns to scale and constant elasticity demand, so π(x, k) = xk 1 1/γ Assume X is a GBM with coefficients µ and σ, with µ < r The underlying asset for each perpetual call is a GBM: S(x, k) = 1 ( ) γ 1 k 1/γ x r µ γ The option values are given by Merton (BJE, 1973) as V (x, k) = 1 β 1 ( ) β 1 β S(k, x) β, where β is the positive root of the previous quadratic equation Assume β > γ Otherwise, the integral of growth option values is infinite β

3 Monopoly Example cont Value matching occurs when S(k, x) = β/(β 1) This is equivalent to k = κ(x) def = [ 1 r µ The optimal capital stock process is ( ) ( ) β 1 γ 1 γ x] β γ K t = k 0 [ 1 r µ ( ) ( β 1 γ 1 β γ )] γ sup 0 s t X γ s Reflection and q The monopolist invests when K t = κ(x t ), which is equivalent to ( ) ( ) X t K 1/γ γ β t = (r µ) γ 1 β 1 The output price P t = X t K 1/γ p m def = (r µ) t ( γ γ 1 is a GBM reflected at ) ( ) β β 1 Similar to perfect competition, marginal q is β P t 1 ( ) β Pt β 1 β 1 p m The risk of dq/q decreases as P t increases towards p m, vanishing at P t = p m, but J/K > q, so the risk of dj/j is different from the risk of dq/q p m

4 Growth is Riskier than Value The value of assets in place is proportional to K t P t : K t P t /(r µ) The value of growth options is proportional to K t P β t : ( γ ) (β 1)(β γ) ( ) β Pt K t pm Because β > 1, growth options are riskier As P t rises towards p m, growth options become a larger part of the total firm value, so the firm becomes riskier Here, assets in place are valued as if the firm never invests again Erosion of the value of assets in place due to growth is a debit to the value of growth options What About Oligopoly? Natural question: How does risk vary with value/growth under imperfect competition? Related question: Are firms in competitive industries riskier than firms in concentrated industries? Hou-Robinson (JF, 2006) show firms in competitive industries have higher average returns

5 Aguerrevere (JF, 2009) Industry demand P t = X t Y 1/γ t, with X a GBM Feasible output for firm i at date t is any y i [0, K it ] Variable cost cy i Firms play Cournot game Assume symmetric capital K it = K t for all i For X t above some critical level depending on K t, firms produce at capacity For lower X t, the capacity constraints are not binding and operating cash flows are a constant multiple of X γ t Firms play dynamic game Stage game payoffs are π(x t, K it ) dt dk it No depreciation Strategies are capital stock processes adapted to X Solution concept is Nash Industry Concentration and Risk Given K t, there is a switch point x below the investment boundary such that: For X t < x, perfect competition is riskier than oligopoly, which is riskier than monopoly For X t > x, monopoly is riskier than oligopoly, which is riskier than perfect competition

6 970 The Journal of Finance Figure 2 The beta of the firm as a function of Y for different numbers of firms in the industry This figure shows the beta of the firm as a function of Y when K depends on the number of firms Source: in the market Aguerrevere, for 1 firm, F, 2009, 2 firms, Real 5 Options, firms, 10 Product firms, and Market perfect Competition, competition and Asset The assumed parameter values are I = 1, c = 006, γ = 16, r = 006, δ = 005, and σ = 02 Returns, Journal of Finance 64, of in place Also notice that, except for perfect competition, the beta of the Open assetsvs in place Closed is theloop same for any number of firms in the industry when Y = 0 It follows from equations (19) and (20) that β F (0, Y) = γ Thus, for any number of competitors the beta of the assets in place is equal to the demand elasticity Terminology when Y = comes 0 from engineering The beta of the growth options is Example: Y G i (K, Y ) β G (K, Y ) = = λ, (21) G i (K, Y ) Y 1 where theair lastconditioner equality follows at a cheap from equation motel: You (16) turn Equation it High (21) Cool shows that the beta ofand the that s growthit options is a constant independent of industry capacity, the demand 2 At factor, a reasonable and themotel number or hotel: of firms There in the is aindustry thermostat 8 The and reason is that the option you set to the invest temperature in additional you capacity want The is a AC perpetual then goes American on call option While the beta of the growth options is the same constant irrespective and off based on the current room temperature of the number of firms in the industry, the level of demand and the degree of competition affect the value of these options and thus the firm beta Figure The2 second illustrates system the effect is closed of competition loop Closed on themeans beta of an there individual is firm usingfeedback the samefrom parameters the output of Figure to the 1 input As inthe Figure first system 1, Y c denotes is open the level of demand loop above which firms will produce at full capacity, and Y n (K n ) denotes the investment threshold The relationship between the degree of competition

7 Open Loop Investment Equilibria Assume the operating cash flow of firm i is π(x, k i, k i ), where X is a diffusion and k i denotes the aggregate capital of firms j i Assume there is no depreciation and each firm has the same initial capital stock k 0 Open-Loop Equilibrium: Each firm chooses a nondecreasing capital stock process K it adapted to X to maximize E 0 e rt {π(x t, K it, K i,t ) dt dk it }, taking the stochastic process K i def = j i K j as given Solving Symmetric Games Each firm i chooses y i to maximize taking y i as given The FOC is max y i π(y i, y i ) y i, π yi (y i, y i ) = 1 Assuming concavity, we obtain a symmetric equilibrium y i = a by solving Define ˆπ(a) def = π yi (a, (n 1)a) = 1 (1) a 0 π yi (b, (n 1)b) db The equilibrium condition (1) is the first order condition for the problem : max ˆπ(a) a

8 Open Loop Investment Equilibria We can find an open-loop equilibrium by maximizing E 0 e rt {ˆπ(X t, K t ) dt dk t }, where ˆπ(x, k) = k k 0 π ki (x, a, (n 1)a) da See Steg, J-H, 2012, Irreversible Investment in Oligopoly, Finance & Stochastics 16, An Example of Perfection Two firms choose quantities q i to maximize pq i, where p = 1 (q 1 + q 2 )/2 Cournot game: simultaneous moves, equilibrium is q 1 = q 2 = 2/3 Stackelberg game: player 1 moves first, equilibrium is q 1 = 1, q 2 = 1 q 1 /2 Is the Cournot equilibrium Nash in the Stackelberg game? Is the Cournot equilibrium subgame perfect in the Stackelberg game? Player 1 is more aggressive in the subgame perfect equilibrium than in the open loop Nash equilibrium (benefit of commitment)

9 Another Example Suppose your significant other is willing to get married Should you? You have an American call on marriage You should not exercise early unless there is a dividend Commitment may be the source of the dividend Closed-Loop Continuous-Time Strategies Consider a game with one player who has two choices (U and D) at each time Consider strategy: Play U at date 0 At t > 0, play U if played U at all s < t and play D if played D at any s < t Following this strategy, which direction will the player go? Pick any τ 0 Playing U for all t τ and D for all t > τ is consistent with the strategy For any ε > 0, at t = τ + ε the strategy says to play D if D was played at τ + ε/2 The problem is that there is no smallest t > τ

10 Stochastic Differential Game Players choose investment rates Strategies are functions f i defining investment rates as dk it dt = f i (t, K 1t,, K nt, X t ) Markov perfect equilibrium: strategies form Nash equilibrium starting at any state (t, k 1,, k n, x) Could assume quadratic adjustment costs, for example