Microeconomic Theory in a Nut Shell

iroeonomi Theor in a Nut Shell iroeonomi Theor in a Nut Shell Summer 6 ESP CONSUER THEORY: u (, ) some funtion suh that u, u, u, u U U, u > < law of diminishing marginal utilit law of diminishing marginal utilit γ u(, ) u(, ) u is homogeneous of degree γ solving u for ields the indifferene urve one a number for u is hosen, sa u u. f( u). The absolute value of the slope of the indifferene urve is the arginal Rate of Substitution (RS): d RS d whih is equivalent to u u du d + d U d + U d d U d U U RS U Consumer utilit is onstrained b the budget: P P + P or P P The absolute value of the budget line (with on the vertial ais) is sometimes referred to as the Eonomi Rate of Substitution (ERS): P ERS P The -interept is equal to /P while the -interept is equal to /P. 1

iroeonomi Theor in a Nut Shell EXPLE 1: Suppose onsumer utilit is given b u (, ).5.5 +. u( 4, 4 ) ( 4 ) + ( 4 ) ( ) + ( ) +.5.5.5.5.5. 5 ( ( ) ( ) ). 5 4 (, ) u Thus the utilit funtion is homogenous of degree.5. This means that if the onsumer quadruples her onsumption of goods and her utilit doubles. However, if she inreases her onsumption of goods and b a fator of 9, her utilit inreases b onl a fator of 3. The law of diminishing marginal utilit holds beause.5 u/ > ( > ) 1.5 u/.5 < ( > ) The slope of the indifferene urve is U RS U.5.5.5.5.5 Setting this equal to the absolute value of the slope of the budget line (P /P ) ields the optimal deision rule for this onsumer: Solving for ields:.5 P.5 P P (1) 4P aimize onsumer utilit given her inome is. The Lagrangian is given b L.5.5 (,, λ) + + λ( P P) The first order onditions (FOCs) are.5 L / λp.5 L /.5 λp L / λ P P Solving the first two FOCs of λ and then setting the resulting equations equal to eah other ields:

iroeonomi Theor in a Nut Shell P.5. P.5.5 This is sas the onsumer hooses and so that U /P U /P. Solving for ields P, 4P whih is idential to equation (1). Substituting this result into the final FOC ields P P + P 4P Solving for ields good s arshallian demand: P (, P, ) 4PP + P 4P Substituting this into equation (1) ields good s arshallian demand: P (, P, ) P 4P + PP lso reall than λ equals.5 / P and arshallian into either of these ields.5.5 / P 4P + P λ.5 PP. Substituting the appropriate Notie that the law of demand holds for the arshallian demands for goods and (i.e., and inrease in the prie of good results in less onsumption of good ). lso notie that inome and pries of related goods shift the arshallian demands. Finall, sine the 4 is the square of the oeffiient in the onsumer s utilit funtion and this represents onsumer preferenes, if the onsumer s preferenes shift, then so do the arshallian demands. The eponents of and in the onsumer s utilit funtion are a result of onsumer preferenes as well. Substituting the arshallian demands into the Lagrangian ields.5.5.5 4P P 4P P λ + + P P 4 4 4 4 PP P P PP PP P P PP + + + + u 3

iroeonomi Theor in a Nut Shell ( P) ( ).5.5 ( P ) P P λ.5.5.5.5 4P + P 4P + P 4 4 + + 4P + P P ( 4P + P ) ( P ) u.5.5 4 P P 4P + P + + λ.5.5.5.5 ( 4P ) ( ) ( 4 ) ( ) 4 P + P PP P + P PP + P u Whih redues to u 1.5 ( 4P + P) ( 4P + P) ( PP ).5.5 Further simplifiation and the replaement of u with v(p, P, ) ields the onsumer s indiret utilit funtion:.5 ( 4P + P) vp (, P, ) PP EXPLE : inimize onsumer ependitures given utilit equal to u. The Lagrangian is given b L The first order onditions (FOCs) are u.5.5 (,, ) P + P + u ( + ) λ λ L.5 / P u λ.5 / P.5 u λ L L + u.5.5 / λ u ( ) Notie that solving the first two FOCs for ields eatl what we found previousl: P, 4P onl this time we substitute this result into a different final FOC:.5 P u + 4P.5 4

iroeonomi Theor in a Nut Shell Solving for ields good s Hiksian (or ompensated) demand: ( P, P, u ) 4P u ( 4P + P) Substituting this into equation (1) ields good s Hiksian (or ompensated) demand: ( P, P, u ) Pu ( 4P + P). Notie that the law of demand holds for the Hiksian demands for goods and. lso notie that utilit, pries of related goods, and the relative preferene of good to shift these demands (the oeffiient of in the utilit funtion and the eponents of and ). lso reall than λ u equals either of these ields.5 P and.5p.5 u PP λ u 4P + P. Substituting the appropriate Hiksian into Notie that if we substitute vp (, P, ) in for u into λ u the inverse of λ: ( 4P + ) PP P u λ 4P + P PP ( PP ) ( 4P + P).5 1 λ.5 u λ Substituting the Hiksian demands into the Lagrangian ields.5.5 4P u u 4P u Pu Pu P + P + λ u + ( 4P + P) ( 4P + P) ( 4P + P) ( 4P + P) Ependitures 4P P u 4P + P + PPu + λ u u ( 4P ) ( 4 ) 4P + P P + P + P Ependitures..5 5

iroeonomi Theor in a Nut Shell Further simplifiation ields the optimized onsumer ependiture funtion: PPu EP (, P, u) 4P + P The power of the ependiture funtion allows an eonomist to ompute the inome neessar to make the onsumer indifferent (u does not hange) after a hange in sa the prie of good (the prie of doesn t hange but s does from P to P ): P Pu PPu Δ E( P, P, u) E( P, P, u) 4P + P 4P + P lso, differentiating the ependiture funtion with respet to the prie of good ields the Hiksian demand equation (Shephard s Lemma): EP (, P, u) 4PPu ( 4 ) P P + P ( P, P, u ) n eample of a ompensating inome differential was the proposal b the Senate Republians to ompensate ommuters with a one-time $1 pament after the spike in gas pries in the spring of 6. Dualit between the two problems (utilit ma and ost min) is demonstrated b solving EP (, P, u) for u and then solve vp (, P, ) ufor : EP (, P, u) PPu 4P + P Properties of onsumer theor equations: (4 P + P) u PP ( 4P + ) P vp (, P, ) PP.5 (1) EP (, P, u) P whih means P E( P, P, u) 6

iroeonomi Theor in a Nut Shell () EP (, P, u ) is onave in P. (3) Propert () is equivalent to the Law of Demand holding for good : EP (, P, u) ( P, P, u) P (6) E( P, P, u) E( P, P, u) and E( P, P, u ) ( P, P, u ) P P (7) ( P, P, ) ( P, P, ) (8) ( P, P, v( P, P, ) ) ( P, P, ) (9) ( P,, (,, ) ) P E P P u ( P, P, u) (1) ( P, P, u) ( P, P, u) Using the arshallian and Hiksian demands and the indiret utilit and ependiture funtions, verif all of the properties above. Suppose the prie of good dereases from P to P : Final budget line with, P, P Hpothetial budget line with 1, P, P Original budget line with, P, P HOS FOS OOS u 1 u > 1 ΔΜ and P > P 7

iroeonomi Theor in a Nut Shell OOS: u v(p, P, ) E(P, P, u ) HOS: u v(p, P, 1 ) 1 E(P, P, u ) FOS: u 1 v(p, P, ) E(P, P, u 1 ) OOS: (P, P, ) (P, P, u ) HOS: FOS: (P, P, 1 ) (P, P, u ) (P, P, ) (P, P, u 1 ) Derive the Slutsk equation b differentiating with respet to P using the hain rule: ( ) ( ) P, P, u P, P, E( P, P, u) rgument of the arshallian Demand for good ( ) ( ) ( ) P, P, u P, P, E( P, P, u) P, P, E( P, P, u) EP (, P, u) + P P P ( ) ( ) ( ) P, P, u P, P, P, P, EP (, P, u) + P P P E P P P TE SE IE SE (P, P, 1 ) (P, P, ) SE (P, P, u ) (P, P, u ) TE (P, P, ) (P, P, 1 ) TE (P, P, u 1 ) (P, P, u ) TE (P, P, ) (P, P, ) TE (P, P, u 1 ) (P, P, u ) LONG RUN FIR THEORY: qlk (, ) some funtion suh that q PL, q, q P, q, q K L L K K L K law of diminishing marginal produtivit of Labor law of diminishing marginal produtivit of apital 8

iroeonomi Theor in a Nut Shell DRS if γ < 1 γ q( L, K) q( L, K) CRS if γ 1 IRS if γ > 1 solving q for K ields the isoquant urve one a number for q is hosen, sa q q. K f( L q). The absolute value of the slope of the isoquant is the arginal Rate of Tehnial Substitution (TRS): dk RTS dl whih is equivalent to q q dq dk + dl K L U dk + P dl K dk dl L P P L K P RTS P If the firm produes q units of output it requires K and L units of apital and labor whih osts C w C wl+ rk or K L The absolute value of the isoost is sometimes referred to as the Eonomi Rate of Tehnial Substitution (ERTS): w ERTS r The K-interept is equal to C /r while the L-interept is equal to C /w. EXPLE 3: Suppose the firm s prodution funtion is given b L K ( ) qlk (, ) e L + K.1t.5.5 r r t time t, this simplifies to qlk (, ) L + K.5.5 if is equal to 1. Thus the prodution funtion is homogenous of degree.5, whih means the prodution funtion ehibits dereasing returns to sale: q( 4, 4 ) ( 4 L) + ( 4 K).5.5 9

iroeonomi Theor in a Nut Shell ( L) + ( K).5. 5.5.5 ( ( L) ( K) ) + 4.5 qlk (, ) The law of diminishing marginal produtivit of labor holds beause.5 q/ L L > ( L > ) 1.5 q/ L.5L < ( L > ) The slope of the isoquant urve is PL L K RTS P.5K L K.5.5.5.5 Setting this equal to the absolute value of the slope of the isoost line (w/r) ields the optimal deision rule for this firm:.5 K w.5 L r Solving for K ields: w K L. () 4r Notie that equation () is analogous to equation (1). Over time (t inreases) the apital and labor that is neessar to produe a given level of output (q ) redue beause an inrease in t shifts the isoquant orresponding to q in toward the origin. Over the long run firms hoose apital and labor to minimize the ost of produing given output equal to q. The Lagrangian is given b u.5.5 L ( LK,, λ) wl+ rk+ λ q ( L + K ) The above problem assumes the firm ompetes in perfetl ompetitive fator markets (K and L) beause w and r are given to the firm. That is, w is not a funtion of L and r is not a funtion of K. The first order onditions (FOCs) are u.5 L / L w λ L u.5 L / K r.5λ K u.5.5 L / λ q ( L + K ) Notie that solving the first two FOCs for ields eatl what we found previousl in equation (): 1

iroeonomi Theor in a Nut Shell K w L, 4r onl this time we substitute this result into a different final FOC:.5 w q L + L 4r Solving for L ields labor s ontingent fator demand: L ( w, r, q ) 4rq.5 ( 4r + w) Substituting this into equation () ields apital s ontingent fator demand: K ( w, r, q ) wq ( 4r + w) Notie that the law of demand holds for the ontingent fator demands for apital and labor (e.g., higher w results in less L). lso notie that quantit, pries of other inputs, and the relative produtivit of labor to apital (oeffiient in the output funtion and the eponents of L and K) shift these demands. Finall, it an be shown that over time the ontingent fator demand for apital dereases. lso reall than λ u.5.5 equals wl and rk. Substituting the appropriate ontingent fator demand into both of these ields u rwq λ 4r + w Substituting the ontingent fator demands into the Lagrangian ields.5.5 4rq wq u 4rq wq w + r + λ q + ( 4r + w) ( 4r + w) ( 4r+ w) ( 4r+ w) ost 4r w u 4r + w + wrq + λ q q ( 4r w) ( 4r w) + + 4r + w Further simplifiation ields the optimized firm ost funtion: ost. 11

iroeonomi Theor in a Nut Shell wrq Cwrq (,, ). 4r + w Notie that I replaed q with simpl q. Differentiating this with respet to q ields the firm s long run marginal ost funtion: wrq C( w, r, q) 4r + w Dividing the ost equation b q ields the firm s long run average ost funtion: wrq C( w, r, q) 4r+ w Differentiating the ost funtion with respet to the fator pries ields the ontingent fator demand equations (Shephard s Lemma): Cwrq (,, ) 4rq L ( w, r, q) w r+ w ( 4 ) Cwrq (,, ) wq K ( w, r, q) r r+ w ( 4 ) PROLE 4: The following profit maimizing problem assumes the firm ompetes in perfetl ompetitive output (q) and fator markets (K and L) beause w, r and p are given to the firm. That is, w is not a funtion of L, r is not a funtion of K and p is not a funtion of q..5.5 ma π ( LK, ) p ( L + K ) ( w L+ r K) The FOC are π.5 L( LK, ) p L w RPL.5 K RPK revenue π ( LK, ).5 p K r Notie that solving the first two FOCs for ields eatl what we found previousl: ost K w L, 4r onl this time we substitute this result into the seond profit ma FOC, and then solve this for L. Thus the fator demand for labor is given b 1

iroeonomi Theor in a Nut Shell p Lwr (,, p). w Substituting this into equation () ields good apital s fator demand: p Kwrp (,, ). 4r Notie that the law of demand holds for the fator demands for apital and labor (e.g., higher w results in less L). lso notie that the relative produtivit of labor to apital (oeffiient in the output funtion and the eponents of L and K) shifts these demands. Finall, it an be shown that over time the ontingent fator demand for apital dereases due to inreases in tehnolog. Substituting the fator demands results into the firms profit equation ields.5.5 p p p p p + w r w 4r w 4r p p p p p + + w r w 4r p 4r p w + w 4r 4r w Further simplifiation ields the optimized firm profit funtion: ( rwp,, ) 4r+ w p 4rw Π. The profit funtion is ver powerful. Differentiating the profit funtion with respet to the p, r, and w ield the suppl funtion and the negative of the fator demands for apital and labor, respetivel, (Envelope Theorem): Π (, rwp, ) (4 r+ wp ) qrwp (,, ) p 4rw Π( rwp,, ) p Krwp (,, ) r 4r Π( rwp,, ) p Lrwp (,, ) w w In a perfetl ompetitive market the firm annot influene the prie of its output. If it tries to sell above the going prie, it sells nothing. ut, it an sell all it wants at the going prie, but it maimizes profit when it hooses an output level suh that R equals C: 13

iroeonomi Theor in a Nut Shell p C( w, r, q) Dualit between the two problems (profit ma and ost min) is demonstrated b solving qwrp (,, ) q for q and then solving C( wrq,, ) pfor p: Solving p C( w, r, q) for q ields the suppl equation: p C( w, r, q) wrq p 4r + w wrp qwr (,, p) 4r+ w Thus the C equation is the firm s inverse suppl equation. n inrease in C is a derease in suppl, a derease in C is an inrease in suppl. Properties of firm theor equations: Cwrq (,, ) Cwrq (,, ) (1),, w r w, r, q C( w, r, q ) () Cwrq (,, ) is onave in w. Cwrq (,, ) C q whih mean (3) Propert () is equivalent to the Law of Demand holding for L: Cwrq (,, ) L ( w, r, q ) w Cwrq (,, ) L( wrq,, ) and w w (4) C( w, r, q) C( w, r, q) C( w, r, q ) C( w, r, q ) C( w, r, q ) C( w, r, q ) (5) L( wrqwrp,, (,, )) L( wrp,, ) (6) L( wrcwrq,, (,, )) L( wrq,, ) Suppose rent (r) dereases from r to r 1 : in a perfetl ompetitive market C 1 < C < C and r > r 1 14

iroeonomi Theor in a Nut Shell Isoost with Isoost with C, w, r 1 C 1, w, r 1 K Isoost with C, w, r FOS HOS q 1 OOS q OOS: L (w, r, q ) and K (w, r, q ) HOS: L (w, r 1, q ) and K (w, r 1, q ) FOS: L (w, r 1, q 1 ) and K (w, r 1, q 1 ) SE L L (w, r 1, q ) L (w, r, q ) SE K K (w, r 1, q ) K (w, r, q ) OE L L (w, r 1, q 1 ) L (w, r 1, q ) OE K K (w, r 1, q 1 ) K (w, r 1, q ) TE L L (w, r 1, q 1 ) L (w, r, q ) TE K K (w, r 1, q 1 ) K (w, r, q ) SHORT RUN FIR THEORY: K is fied in the short-run, sa at K 36. Therefore, at time t with 1 and r 1 the short-run prodution funtion is given b ( ) qlk (, ) e L + K ql ( ) 6.1t.5.5.5 + L. PROLE 5: The prie taking, profit maimizing firm hooses L to solve.5 ( ) ma π ( L) p 6 + L w L 1 36 Rev VC FC 15

iroeonomi Theor in a Nut Shell Step 1: Compute the FOC (one equation and one unknown): π ( L) p P ( L) w L L Step : The equation w p P ( L) is the short run labor demand equation. L PROLE 6: Solve ql ( ) 6.5 + L for L whih ields a funtion in q: Lq q q+ ( ).5 3 9 The profit maimizing problem with hoie L: ma π ( q) p q w.5q 3q+ 9 1 36 ( ) Rev Step 1: Compute the FOC (one equation and one unknown): π ( q ) q C ( q ) p Step : The equation p C( q) is the short-run suppl equation. GGREGTION: Let q D equal the quantit of good demanded, let p be the prie of good, and let N be the number of onsumers in this market VC FC Q D N ( p, P, ) D Q ( p, P,, N) 4N P 4Pp + p Notie that we used the arshallian demand equation for good. We use this one rather than Hiksian beause the arshallian is observable with firm level date (OOS to FOS). That is the onsumer reveals her preferenes with the purhases she makes. However, the Hiksian ould be derived via a surve. State preferenes are aptured with surves. Thus we would use the Hiksian to aggregate from stated preferene studies. Suppose there are n firms. Thus, the market suppl equation is S Q n q( w, r, p) S nwrp Q ( w, r, p, n) 4r+ w Notie that the laws of demand and suppl hold for the aggregated demand and suppl equations, respetivel. n inrease in the prie of good (p) inrease the quantit 16

iroeonomi Theor in a Nut Shell supplied (Q S ) but redues the quantit demanded (Q D ). Shifters of aggregate suppl inlude the number of firms (n), and the pries of inputs (w and r). Less obvious shifters of aggregate suppl inlude the relative produtivit of labor to apital (oeffiient in the output funtion and the eponents of L and K) inreases in tehnolog (e θt ). Shifters of aggregate demand inlude the population (N), pries of other goods (P ), onsumer inome ( ). less obvious shifter of aggregate demand inludes onsumer preferenes (oeffiient in the utilit funtion and the eponents of and ). EQUILIRIU UNDER PERFECT COPETITION: Given the prie of good (P ), number of firms (n), number of onsumers (N), onsumer inome ( ), wage (w), and rent (r), the equilibrium prie of good is the solution of setting Q D equal to Q S : Q S Q D nwrp 4N P 4r + w 4P p+ p N 3 P p + 4 P p (4 r+ w). nwr The solution to the above ubi polnominal (p ) has three values, two of whih are probabl either omple or negative roots. We would simpl pik the positive valued p. For eample suppose there are 1,, onsumers, 1 firms, P $, w $16 per hour, r $6 per hour, and $. Then p equals $74.7 or -141.34 ± 4.16i. Thus p $74.7. Substituting this into either Q D or Q S : Q 6,5 (units sold). ONOPOLY EQUILIRIU: There is onl one firm (n 1). Note beause there is onl one firm in this market, q in the ost equation beomes Q: C C( w, r, Q). Given values for wages (w) and rents (r), the ost equation simplifies to C C( Q). D Solving Q N ( P, P, ) for P ields the inverse market demand equation, sa (,,, ) P P Q N P. Given values for the prie of good (P ), number of onsumers (N) and onsumer inome ( ) the inverse demand equation simplifies to ( ) P P Q. 17

iroeonomi Theor in a Nut Shell The monopolist s problem is ( ) ma π ( Q) P Q Q C( Q) Step 1: Compute the FOC (one equation and one unknown): ( ) ( ) P Q Q C( Q) π Q Q Q+ P, Q Q Q R( Q) C ( Q) If inverse market demand urve is for eample equal to PQ ( ) 1 Q, then P Q P ( Q ) { } R( Q) Q + P ( ) { } R( Q) Q+ 1 Q R( Q) 1 4Q This is idential to differentiating the monopolist revenue equation R( Q) 1Q Q : dr dq 1 4Q Step : The FOC is equivalent to R C. Solving this one equation for the one variable Q ields the monopolist s output Q. Step 3: Substitute Q into the inverse market demand urve beause the monopolist an harge up to the market demand urve. This ields the monopolist s prie P., C P P R C R D Q Q 18

iroeonomi Theor in a Nut Shell onopol Eample: Suppose there is one firm in a market and the market inverse demand urve and the monopolist s ost urve are given b p( Q) 5 3Q CQ ( ) 15 + 5 Q+.5Q The monopol hooses Q to maimize its profit: ma π ( Q) p( Q) Q C( Q) The general FOC is dp( Q) dq π Q dq Q+ p( Q) dq C( Q) R dp dq Q + p C p p dp Q dq p p C p 1 dp Q C p ε dq p p 1 p C Lerner Inde ( LI) ε p The monopolist s revenue equation is ( ) ( 5 3 ) 5 3 R p Q Q Q Q Q The monopol hooses Q to maimize its profit: ma ( Q) 5Q 3Q 15 + 5 Q+.5Q π The FOC is [ Q] π Q 5 6Q 5 + R C or mkt S 5 6Q 5 + Q 5 5 7Q Q 7 8.57 C 5 + 8.57 $78.57 19

iroeonomi Theor in a Nut Shell R 5 6 8.57 $78.57 p 5 3 8.57 164.9 The Lerner Inde for the monopol is LI p C 164.9 78.57 1% 1% 5.18% p 164.9 onopol profit is π 5(8.57) 3(8.57) 15 5(8.57).5(8.57) $167.14 Prie disrimination: 1. Uniform monopol priing involves harging all onsumers the same prie (see above) CS PS DWL. In first degree prie disrimination the monopolist harges eah ustomer different pries to apture all of the CS and all of the DWL (no pink and blak areas remain) PS

iroeonomi Theor in a Nut Shell 3. In seond degree prie disrimination the monopolist realizes it annot harge eah ustomer different pries, so it tries to harge two or more pries given the same demand urve to apture some of the CS (pink area) or the DWL (blak triangle) CS PS DWL 4. In third degree prie disrimination the monopolist has been able to disaggregate the market demand urve into two or more separate demand urves, and it harges the uniform monopol prie in both of the disaggregated markets arket 1 arket Competitive Duopol Eample: Suppose that firms and form a duopol in some market where firm is not onerned with what firm is doing and vie versa. Let q and q denote firm and s output, respetivel. Suppose the fae following inverse demand urve: p 5 3Q 1

iroeonomi Theor in a Nut Shell and the C urves are C 65 + 5q + q and C 65 + 5q + q Firms and solve the following problems in a ontestable (ompetitive) duopol: ( ) ma π ( q ) p q 65 + 5q + q ( ) ma π ( q ) p q 65 + 5q + q Firm s FOC is π 5 q + p ( q ) R C p 5 + q Firm s FOC simplifies to p 5 q p 5 q Sine firm and firm make up the entire market p 5 p 5 Q q + q + p 5 Q p 5 (1) Substituting this into the demand urve ields ( ) ( ) p 5 3 Q 5 3 p 5 p 5 3p + 15 4p 4 p 1 Substitute this bak into equation (1) or into the demand equation ields

iroeonomi Theor in a Nut Shell Q 1 5 5 or 1 5 3Q 3Q 15 Q 5. nother wa to think about this problem is to onvert the marginal ost urves (when C p these are the individual inverse suppl urves) of the firm into the inverse market suppl urve: p 5 + q q q q S Q q q.5q S S p 5 + (.5 Q ) p 5 + Q The suppl urve and demand urve are then S S Q p 5 D Q 5 1 3 3 p t the ompetitive equilibrium we have Q S Q D p 5 5 1 3 3 p 3p 15 5 p 4 p 4 p 1 and Q 5 3

iroeonomi Theor in a Nut Shell The Lerner Inde for this duopol is P C LI 1% % P The ompetitors in this duopol produe q 5 and q 5 Firm and profit are ( ) ( ) ( ) ( ) π ( q ) 1 5 65 5 5 5 ( ) ( ) ( ) ( ) π ( q ) 1 5 65 5 5 5 Zero profits in a ompetitive duopol disourage other firms from entering this market. Cartel Eample: two firms and zero profits enourage the firms to ollude (form a artel like OPEC). eause q + q is equal to Q, the Cartel s total ost equation is C 65 + 5q + q + 65 + 5q + q Dropping the subsripts (firms are idential so q q q) ields The Cartel s revenue equation is ( ) ( ) ( ) C 65 + 5 q+ q + q + q ( ) ( ) C 65 + 5 q+ q + q 1 ( ) ( ) C 65 + 5 q + q 1 ( ) ( ) C 65 + 5 Q + Q The Cartel hooses Q to maimize its profit: C 65 + 5 Q+.5Q ( ) ( 5 3 ) 5 3 R p Q Q Q Q Q ma ( ) 5 3 65 + 5 +.5 π Q Q Q Q Q The FOC is Q [ Q] π Q 5 6 5 + R artel C artel 4

iroeonomi Theor in a Nut Shell Solving for Q ields Q 7 8.57 C 5 + 8.57 $78.57 R 5 6 8.57 $78.57 P 5 3 8.57 164.9 The Lerner Inde for this duopol is LI P C 164.9 78.57 1% 1% 5.18% P 164.9 Thus eah member of the artel agrees to produe Firm and profit are 8.57 q 14.85 q 14.85 ( ) ( ) ( ) ( ) π ( q ) 164.9 14.85 65 5 14.85 14.85 $83.57 π ( q ) $83.57 Is there an inentive to heat on the artel s output quota agreement? Suppose firm does not heat on its agreement of q 14.85. If firm knows this, then an it inrease its profit b inreasing output b 1 additional unit? First, how is the prie affeted b s heating? Thus, firm and profits are ( + ) P 5 3 8. 57 1 161.9 ( ) ( ) ( ) ( ) π 161.9 + 1 + 1 4.85 + 1 $84.44 ( q ) 14.85 65 5 14.85 1 ( 161.9) ( ) ( ) ( ) π ( q ) 14.85 65 5 14.85 14.85 $76.7 Firm s profit is lower than it had epeted while Firm s is higher. Thus Firm knows that Firm heated on its artel prodution quota. Thus an inentive to heat on the quota eits. Cournot arket Eample: Cournot arket is haraterized b the following: arriers to entr and eit 5

iroeonomi Theor in a Nut Shell Impliit ollusion: firms make the same priing deisions even though the have not onsulted with one another (not illegal and ommon) Oligopolists deisions are based on the deisions of their fellow oligopolies Eah firm supplies q and q, and so demand in the market is p 5 3( Q) p 5 3( q + q ) Given firm s deision q, firm s revenue would be R q p R q (5 3q 3 q ) R q q q q 5 3 3 Firm s hooses q given firm s deision q ma π ( q) 5q 3q 3qq 65 + 5q + q Firm s FOC is Solving this for q ields [ ] π q 5 6q 3q 5 + q R q 3 8 8 q C Dropping the bar above q ields s est Response urve: q br 5.375q smmetr, s est Response urve is q 5.375q Solving this for q allows us to graph both urves together with q on the vertial ais: q br 66.7.667q 6

iroeonomi Theor in a Nut Shell The Cournot Equilibrium: q br q br Thus firm profits are equal: 5.375q 66.7.667q.91667q 41.667 CE 41.6667 q 18.1818.916667 ( ) q 5.375 18.1818 18.1818 CE ( ) q 66.7.667 18.1818 18.1818 CE CE Q (18.1818)() 36.3636 CE P 5 3(36.3636) $14.91 π CE ( Q / ) (14.91)(18.18) 65 5(18.18) (18.18) $697. Even though the firms in a Cournot market do not epliitl ollude, both firms harge the same pries that are greater then the marginal ost of produing the 18.18 th unit of output: Thus the Lerner inde is not equal to zero: CE C ( q ) 5 + (18.18) 86.36 LI 14.91 86.36 1% 38.7%. 14.91 Thus, using this information, the government would inorretl onlude that the firms in this duopol are epliitl olluding to jak up pries (engage in prie gouging). Comparison of the Contestable, Cartel and Cournot equilibriums: The artel produes the least amount of output at the highest prie The ontestable (ompetitive) duopol produes the most output at the lowest prie. Firm profit in eah tpe of market is 7

iroeonomi Theor in a Nut Shell duopol P Q q Contestable 1 5 5 5 Cournot 14.91 36.36 18.18 18.18 697. 697. Cartel 164.9 8.57 14.85 14.85 83.57 83.57 q π π 3. ertrand Equilibrium is similar to the Cournot eept the firm hooses its prie given its ompetitor s hosen prie. 4. Stakelberg Equilibrium is an equilibrium that arises when one firm announes its quantit (or prie) first. One this is done the other firms make their hoies on quantit (or prie) LEDER-FOLLOWER. Construt the Paoff atri for the Duopol Game: Eah firm hooses to produe 5, 18.18 or 14.85 units of output. There are 9 ombinations of these output sets. Therefore there are 9 different market pries for eah of these ombinations and 9 different sets of firm profits. The pries and the profits for deision sets (5, 5), (18.18, 18.18), and (14.85, 14.85) are alread omputed. Thus we must ompute the pries and profits for deision sets (18.18, 14.85), (18.18, 5), (14.85, 5): p 5 3(5 + 18.18) 1.46 π (5) (1.46)(5) 65 5(5) (5) $511.5 π (18.18) (1.46)(18.18) 65 5(18.18) (18.18) $35.45 p 5 3(5 + 14.85) 13.15 π (5) (13.15)(5) 65 5(5) (5) $83.63 π (14.85) (13.15)(14.85) 65 5(14.85) (14.85) $344.38 8

iroeonomi Theor in a Nut Shell p 5 3(18.18 + 14.85) 15.61 π (18.18) (15.61)(18.18) 65 5(18.18) (18.18) $99.85 π (14.85) (15.61)(14.85) 65 5(14.85) (14.85) $636.65 Firm 's output deision 14.85 18.18 5 Firm s output deision 14.85 18.18 5 83.57 99.85 83.63 83.57 636.65 344.38 636.65 697. 511.5 99.85 697. 35.45 344.38 35.45 83.63 511.5 If Firm deides to produe 14.85 units, then Firm should produe 18.18 units underline Firm s profit of $99.85 given these deisions. If Firm deides to produe 18.18 units, then Firm should produe 18.18 units underline Firm s profit of $697. given these deisions. If Firm deides to produe 5 units, then Firm should produe 14.85 units underline Firm s profit of $344.38 given these deisions. If Firm deides to produe 14.85 units, then Firm should produe 18.18 units underline Firm s profit of $99.85 given these deisions. If Firm deides to produe 18.18 units, then Firm should produe 18.18 units underline Firm s profit of $697. given these deisions. If Firm deides to produe 5 units, then Firm should produe 14.85 units underline Firm s profit of $344.38 given these deisions. The above illustrates that neither firm has a dominate strateg. The artel agreement (both firms produing 14.85 units of output) ields the maimum level of industr profit. However, Firm has an inentive to heat on this agreement. If it believes Firm will not heat, Firm an generate $99.85 in profit b produing 18.18 units of output. Similarl, Firm an generate $99.85 in profit b produing 18.18 units of output if it believes Firm will not heat. Thus both Firms heat and both produe 18.18 units of output (the Cournot equilibrium), whih ields lower firm and industr profits had neither firm heated on its artel quota this wh John Nash won the Noble Prize in eonomis!!! If both profits are underlined in a bo, then the orresponding deision set is a Nash Equilibrium (NE). Thus the deision set (18.18, 18.18) is a NE. Using the definition of NE ields this same onlusion: q 18.18 is optimal given q 18.18 q 18.18 is optimal given q 18.18 9