Review of some basic ideas about the THEORY OF THE FIRM. • How to derive individual and market demand functions • Production isoquants • The marginal rate of technical substitution (MRTS) of labor for capital is MRTSLK = − • MRTSLK = dK = negative of the slope of the isoquant dL M PL M PK
• To move from production function to an isoquant, ﬁx the level of output, Q. Solve for K as a function of L and your ﬁxed Q. Then ﬁnd the derivative of this function. The negative of the derivative is the MRTS. • Deriving long-run cost functions from production function – Recall the cost function is C(Q) = pL L+pK K. The cheapest, best bundle is represented by M PL pL = . M PK pK – Solve for the capital and labor in terms of output, i.e. K = K(pK , pL , Q) and L = L(pL , pK , Q).
– Plug these values into the cost function, C = pL L(pL , pK , Q) + pK K(pK , pL , Q) • Deriving short-run cost functions from production function when labor varies but capital is ﬁxed: – Solve for L = L(pL , pK , K, Q) using the given production function. – Plug into cost function: C(Q) = pL L(pL , pK , K, Q) + pK K • Cobb-Douglas production functions: Q = F (L, K) = dLa K b • Constant, increasing and decreasing returns to scale. • Graph of marginal and average functions – If x > 0, marginal function 0, marginal function > average function, when average is rising. – The marginal function = average function, when the average function is neither falling nor rising [at a maximum or minimum] 1 d, a, b > 0
... production process that manufactures 50 units per day, then the average product of variable labour input is 5 units per day. Average and Marginal ... average, and marginal product Total Product Curve The total product (or total physical product) of a variable factor of production ... Physical Product Curves The average product typically varies as more of ...
Short-Run Supply Curve
Assume that ﬁrms are perfectly competitive price takers. This is a reasonable assumption if there are a large number of competing ﬁrms, buyers and sellers know the price everyone is charging and one can eﬀortlessly ﬁnd a ﬁrm charging a lower price. proﬁt = total revenue − total cost π(Q) = P · Q − C(Q) where total cost is equal to variable cost and ﬁxed cost, i.e. C(Q) = V (Q) + F C.1
Three Important Conditions
Condition #1: Marginal Revenue = Marginal Cost. Proof. This is your ‘ﬁrst-order condition’ for ﬁnding a maximum from calculus. Since you are maximizing proﬁts, you need to ﬁnd the points where the derivative of the proﬁt function are zero. π (Q) = P − C (Q) = 0 Label this proposed quantity as Q∗ . Condition #2: marginal cost is rising. Proof. This is your ‘second-order condition’ for checking whether the point you found in Condition #1, Q∗ , is actually a maximum point. You need to check that the second derivative of the proﬁt function is negative.2 π (Q∗ ) = −C (Q∗ ) 0 Condition #3: Price is greater than Average Variable Cost. Proof. Check that your proposed quantity, Q∗ , satisﬁes this condition. Proﬁt at Q∗ must not be smaller than proﬁt at Q = 0, i.e. π(Q∗ ) ≥ π(0).
The proﬁt function is: π(Q) = P · Q − (V (Q) + F C)) π(Q∗ ) ≥ π(0) P · (Q∗ ) − V (Q∗ ) − F C ≥ −F C P · (Q∗ ) ≥ V (Q∗ ) V (Q∗ ) P ≥ = AV C Q∗ If the ﬁrm were to shut down, in the short run [no variable costs, no revenues], its losses would be equal to its ﬁxed cost since ﬁxed costs are incurred whether or not the ﬁrm produces any output or not. As long as price is greater than or equal to short-run average variable cost, the ﬁrm is doing at least as well as it could by shutting down.
The problem should specify C(0).
If it is not stated, then you need to consider two cases (i) C(0) = 0 and (ii) the speciﬁed cost function C(Q) at Q = 0. 2 At a maximum point, the derivative decreases, i.e. goes from positive to zero to negative.
... average cost of producing the good (P 2), then ... must accept the market price for their product. They are price takers. The Short Run and the Long Run The short run under perfect competition is ... point of its average cost curve. This can be seen in the diagram below. In the long run if the price falls below the ...
P = C (Q)
In most cases, we consider an industry such that the short-run supply curve for the industry is the horizontal sum of the individual supply curves. Assuming a ﬁxed group of potential buyers and a ﬁxed group of price-taking, proﬁt-maximizing ﬁrms, equilibrium price and quantity is determined when market demand equals market supply.
Long-Run Supply Curve and Equilibrium
• Firms are perfectly competitive, price-taking ﬁrms • Entry and exit. In a market with entry and exit, a ﬁrm enters when it can earn a positive long-run economic proﬁt and exits when it faces a negative long-run economic proﬁt. • LRAC is U-shaped reﬂecting economies of scale; decreasing long-run average cost for smaller levels of production to some minimum cost and increasing thereafter. • Zero economic proﬁt means that the ﬁrm is earning a normal or competitive return. • Opportunity costs
The process of entry and exit in response to economic proﬁts continues until all opportunities to make further economic proﬁts have been completely exhausted. The industry is in LONG-RUN EQUILIBRIUM when all its ﬁrms make zero economic proﬁt, so that no ﬁrm wants to leave and no ﬁrm wants to enter. 1. The quantity that an individual ﬁrm will produce will correspond to the bottom of the LRAC curve, i.e. LRAC = LRM C 2. The price that the ﬁrms (and all other competing ﬁrms) will charge, will correspond to the height of the bottom of the LRAC curve, i.e. P = LRAC = LRM C We use the long-run competitive equilibrium idea to trace out the ultimate eﬀect of a change. In a constant-cost industry, input prices remain the same regardless of the number of ﬁrms. As a result, the ‘long-run industry supply curve’ is a horizontal line. [Can also consider increasing cost and decreasing cost industries]
Let R(Q) denote total revenue and let P = P (Q) = D(Q) denote the demand curve faced by the monopolist. proﬁt = total revenue − total cost π(Q) = P (Q) · Q − C(Q)
Three Important Conditions
Condition #1: Marginal Revenue = Marginal Cost. Proof. This is your ‘ﬁrst-order condition’ for ﬁnding a maximum from calculus. Since you are maximizing proﬁts, you need to ﬁnd the points where the derivative of the proﬁt function are zero. π (Q) = R (Q) − C (Q) = 0 ⇒ R (Q) = C (Q)
... environment? In a capitated environment Revenue and Costs ($) Total Costs Loss Profit Total Revenue Fixed costs 0 Volume (Number of Visits ... Patient services revenue is used as the cost driver? Allocation rate= cost pool/ cost driver = 100,000/ patient services revenue = 100,000 ... the direct method. Two cost drivers are under consideration: patient services revenue and hours of housekeeping services ...
Use the ‘product rule’ to ﬁnd the marginal revenue, R (Q).
Note that if you have a linear demand curve, i.e. P = A − BQ, then M R = A − 2BQ (“doubling shortcut”).
Label this proposed quantity as Q∗ . Condition #2: At Q∗ , Marginal Revenue must cut Marginal Cost ‘from above’. Proof. This is your ‘second-order condition’ for checking whether the point you found in Condition #1, Q∗ , is actually a maximum point. You need to check that the second derivative of the proﬁt function is negative.3 π (Q∗ ) = R (Q) − C (Q∗ )
Condition #3: Price is greater than Average Variable Cost. See Proof Short-Run. IMPORTANT: A monopolist will choose the Q∗ that satisﬁes these conditions, and the P ∗ that corresponds with that quantity on the demand curve, i.e. P ∗ = P (Q∗ ) = D(Q∗ ).
Relationship between marginal revenue and elasicity M R = R (Q) = P (Q) + Q · P (Q) using product rule Q = P (Q)[1 + P (Q)] by factoring out P (Q) P (Q) 1 P = P (Q)[1 − ] where E = −Q (P ) E Q
At a maximum point, the derivative decreases, i.e. goes from positive to zero to negative.
• A monopolist would never want to choose a level of output where MR is negative. • Deadweight loss • Fairness doctrine: economic proﬁt is zero, i.e. the quantity where AC curve crosses the demand curve. • Marginal cost doctrine: ideal output, i.e. quantity where the MC curve cuts the demand curve. • Monopolist with several PLANTS: The cheapest way to produce Q in multiple plants is to split Q among plants in such a way that MC is the same at each plant. • Monopolist in diﬀerent MARKETS
Applications for Short-Run, Long-Run, Monopoly
• Sales taxes vs. excise taxes • Shifts in demand and supply curves. This depends on the context of the question. If perfectly competitive industry, consider both the short-run and long-run equilibrium, if not otherwise stated. • Consumer surplus, producer surplus • Area under marginal revenue is total revenue; area under marginal cost is total cost • Application of long-run equilibrium: a lump-sum tax on each ﬁrm versus an excise tax on each ﬁrm (Handout 15, Topic 2, page 4a).
... the nightly brown outs are not acceptable. This tax revenue should be allocated and dispersed to help remedy ... Alluded to A 14. In this case tax revenues should be spent to correct the environmental problems ... way they work they are both there to curve pollution and to correct it after it occurs ... the exception of a lump-sum tax, all conventional taxes have distortion cost in the form of affecting the ...
– Consider the following Cost Function, C(Q) = q 2 + 25 – With the lump sum tax of T , C(Q) = q 2 + 25 + T . You’ll see that q1 = 5, q2 = √ P1 = 10 and P2 = 2 25 + T – The per-unit value of the lump-sum tax is
25 + T ,
√ T 25+T
– With the sales tax, we want to choose t such that P2 − P1 = t. Since P1 = 10 and √ √ P2 = 2 25 + T , that means that t = 2 25 + T − 10. The new cost function is C(Q) = q 2 + 25 + tq. – In order to ﬁnd which plan gives the government the most revenue, compare t and √ t = 2 25 + T − 10 Plug in values to see that t ≥ result?
T q2 . T q2
T T =√ q2 25 + T
[Easiest to do this in Excel.] What is driving this