Term 1 Week 10: Profit Maximisation Flashcards
What is the 2 step problem for profit maximisation (2)
-Minimising costs for a given output level
-Choosing output to maximise π
How can we show profit maximisation occurs where price = marginal cost (4)
-Profit: π = pq – C(w, r, q)
-To maximise profit, we use the FOC, differentiating the profit function with respect to quantity (choosing output to maximise profit)
-∂π/∂q = p - ∂C(w, r, q)/∂q = 0
-Price = marginal cost for profit maximisation
What is the profit function with optimal output (1)
-π(p, w, r) = pq* - C(w, r, q*)
How do we find the profit maximising quantity/profit level for f(L, K) = L^(1/3)K^(1/3) (3,2,2) with the two step problem
-TC = wQ^(1/a+β)(ar/βw)^(β/a+β) + rQ^(1/a+β)(βw/ar)^(a/a+β)
-Now put a = β = 1/3 into this
-TC = wQ^(3/2)(ar/βw)^(1/2) + rQ^(3/2)(βw/ar)^(1/2) = 2Q^(3/2)(rw)^(1/2)
We want to maximise profit:
-π = pq – C(w, r, q) = pq - 2Q^(3/2)(rw)^(1/2)
-FOC: p = MC = 3Q^(1/2)(rw)^(1/2)
-Q(p, r, w) = p^2/9rw
-π(p, r, w) = p^3/27rw
What is the one step problem for profit maximisation (4)
-L and K are used to produce Q = f(L, K) (priced at p), with Pl = w, Pk = r
-The firms profit maximisation problem involves choosing Q, L, K to: max π = pq - wL - rK = pf(L, K) - wL - rK
-The FOC’s then yield the optimal input demands L(p, w, r) and K(p, w, r) (these now don’t depend on Q, but P), known as the unconditional input demand functions
-We can then derive the supply function Q(p, w, r) = f(L, K*) and the profit function
How do we find the profit maximising quantity/profit level for f(L, K) = L^(1/3)K^(1/3) (3,2,2) with the one step problem (2,6,2)
-F(L, K) = L^(1/3)K^(1/3)
-π = pq - wL - rK = pL^(1/3)K^(1/3) - wL - rK
-Use the FOC’s to find the optimal stock of labour and capital
-∂π/∂L = (1/3)pL^(-2/3)K^(1/3) - w = 0
-∂π/∂k = (1/3)pL^(1/3)K^(-2/3) - r = 0
-Using (1), k = (27(w/p)^3L^2
-Subbing this into (2) gives us L* = p^3/27w^2r
-Then, K* = p^3/27wr^2
-Q* = L^(1/3)K^(1/3) = p^2/9wr (identical supply function from 2 step
-π* = pQ* - wL* - rK* = p^3/27rw (identical profit function from 2 step)
Where does profit maximisation occur (2)
-Profits are maximised at Q* where MC = MR
-Profits are given by: π = pQ* - c(Q*)
How can we diagramatically show where profits occur (3,3)
-One one diagram, have quantity on the x axis, costs on the y axis
-On this diagram have a linear upward sloping TR=PQ function, and aa convex upward sloping TC(q) curve
-Q* occurs where the distance between those 2 curves is at its greatest, and is TR-TC
-One another diagram have quantity on the x axis, price on the y axis
-Have a horizontal AR=MR curve, a convex AC(Q) u shaped curve, and an upwards convexx sloping MC(q) curve
-Profit maximisation occurs where MR=MC, and is (P-AC)(AC(Q*))
What is the relationship between MPL and a profit maximising choice of inputs (4)
-When more of one input is used, ΔL, output rises by ΔQ = MPL(ΔL)
-The value of this extra output is p(MPL)(ΔL), the cost is w(ΔL)
-Hence, at a profit maximising choice of inputs, the value of the MPL should equal the price of labour
-pMPL = w
What are isoprofit curves (3)
-Isoprofit curves are all combinations of the input goods and output good which give the same level of profit
-A bit like a firms indifference curve
-The higher the isocost curve, the higher the level of profit
How do we draw isoprofit curves (2,2,2)
-Assume K is held fixed and x denotes a firm’s output level
-The graph has labour (l) on the x axis, and output (x) on the y axis
-If we remember profits: π = px - wL - rK, solving yields x = (π/p) + (r/p)K + (w/p)L
-The isoprofit curve is hence an upwards sloping linear curve with slope (w/p) and vertical intercept (π/p + rK/p)
-Now draw the producer choice set, a convexly increasing function starting from the origin, all the area under it filled in
-Profit maximisation occurs where there is tangency between the PCS and isoprofit, where w/p = MPL
How can we derive the supply curve based on an MC and AC diagram (2,1)
-With MC = MR = P, firms produce when P ≥ AC, as profits ≥ 0
-The MC curve at MC>AC shows how much will be produced at any given price
-But if TR<TC, or P<aC, Q = 0
What is the difference between long run and short run supply vs homogeneuity (2,1)
-In the short run, supply is where MC ≥ AVC, but MC ≥ AC in the long run
-LR supply is flatter than SR supply, due to fixed inputs and capital
-Supply function is homogeneous of degree 0
