Producer theory PDF

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Notes from handout 2...

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Producer theory Production functions ● Production functions refer to the conversion process (inputs to outputs) ● Shows the maximum a firm can produce using various combinations of inputs ○ Inputs are used as efficiently as possible and there is no waste ● Assume 2 inputs Q = F(K,L) and that Q is concave and monotonic Short-run / 1 input model ● In the short run, one factor of production is fixed - Q = F(L) ● The producer’s choice set of production plans are technologically feasible ○ Increasing MPL implies non-convexity ● The production function is the set of production plans with no waste of inputs ● There is diminishing marginal returns as labour increases Marginal and average product ● Marginal product of labour - additional output produced by one more worker ○ = change in total product / change in quantity of labour ○ A linear production function has a constant MPL ○ Assume diminishing marginal productivity ○ MPL = the slope of the production function ● The average product of labour = total product / quantity of labour ● Maximum output occurs when MP = 0 ● The peak of MP is where diminishing marginal returns begin ● If MPL > APL, the average is rising ● If APL > MPL, the average is falling ● Negative MPL implies a downward sloping production frontier Isoquants ● Isoquants show all combinations of labour and capital that produce a given level of output ● They are downward sloping, monotonic, thin, convex and do not cross ● They represent output (measurable) ● They arise from production functions (tech constraints faced by producers) ● Doubling inputs alters the production technology - producing 2x as much output as before (returns to scale) The marginal rate of technical substitution ● The 𝑀𝑅𝑇𝑆 (of labour for capital) is the slope of the isoquant 𝐿,𝐾

○ Diminishes as you move down the isoquant ● Differentiate the production function to find the MRTS 𝑀𝑃

𝐿

○ 𝑀𝑅𝑇𝑆𝐿,𝐾 =

𝑀𝑃

𝐾

● Shows how many units of capital the firm can substitute for one unit of labour whilst keeping output constant Returns to scale (long-run) ● Returns to scale = % △ in output / % △ in inputs 𝑘

● 𝑓(𝑡𝐾, 𝑡𝐿) = 𝑡 𝑓(𝐾, 𝐿) = 𝑡𝑄(homogeneous production function) 𝑎

𝑏

● 𝑓(𝐾, 𝐿) = 𝐴𝐿 𝐾 ○ K or a+b >1 = increasing returns ○ K or a+b =1 = constant returns ○ K or a+b 1 shows increasing marginal product ○ a+b>1 shows IRS Linear production functions ● Q = f(L,H) = aL + bH ● MRTS = -a/b (constant along linear isoquants) Fixed proportion production functions ● Q= f(L,H) = min (aH + bO) ● If aH w/r ● All money spent on labour is more productive than any money spent on capital ● If you obtain a negative quantity of one of the inputs, it is a corner solution Comparative statics ● If there is an increase in the price of labour, the isocost curve becomes steeper and pivots around the isoquant ○ Quantity of labour decreases and capital increases ● This can be used to derive input demand curves ○ An increase in the wage decreases the quantity of labour - causes a movement along the demand for labour curve ○ The more curved the isoquant, the smaller the decrease in labour for a given wage rise ○ It shifts if there is an external factor affecting demand for labour Cost functions ● Total cost = wL*(r,w,Q) + rK*(r,w,Q) ● Average cost = total cost / Q ● Marginal cost = the derivative of the total cost with respect to Q ○ Shows the change in total costs following a change in output ● Just because an output level is technically efficient, does not mean it is economically efficient (least cost way of producing) ○ Is usually 1 economically efficient way where MRTS = -w/r and the isocost is tangential to the isoquant ● When technology is homothetic, the economically efficient bundles lie on the same ray from the origin ● A concave production function has a convex cost function ● IRS production functions have falling AC ● Production functions that are concave for positive values need a fixed cost

● ● ● ● ●

○ AC is u shaped Non-concave production functions have decreasing then increasing TC Slope of the TC curve = the MC curve Allocative efficiency occurs where MC=AR EoS occur where MC < AC , DoS occur where MC > AC Minimum efficient scale occurs where MC = AC (costs are minimised)

Properties of cost functions ● Homogeneous of degree 1 in input prices ● Non decreasing in Q and input prices ● If output is convex, it has IRS and costs are concave, the firm has EoS ● If output is concave, it has DRS and costs are convex, the firm has DoS ● With constant returns, output increases proportionately and AC is constant Shifting cost curves ● A change in technology or input prices shift the isocost / cost curve ● Technology / cyberspace can reduce LRTC Short-run costs ● In the short run, the firm has constraints so capital is fixed -𝑇𝐶 = 𝑤𝐿 + 𝑟𝐾 ● The cost of labour is the firm’s total variable cost ● The cost of capital is the firm’s total fixed cost ● The technically efficient level of output occurs on the highest isoquant which 𝐾 intersects ● Costs are normally higher in the short-run Long-run cost curves ● The LRAC curve forms a boundary around the SRAC curves ○ An envelope curve ● Each SRAC curve has a different level of fixed capital Profit maximisation 1 step problem ● Q= f(L,K) and π = pq - wL - rK ● Differentiate with respect to L and K to yield the optimal inputs (shows the expansion path) ○ Unconditional demands (does not depend on Q) ● Q*(p,w,r) = f(L*,K*) ○ Q gives the supply function 2 step problem

● Choosing output to maximise profits - an unconstrained maximisation problem ● Max profits = pq - C(w,r,q) ○ 𝑝 =

𝑑𝑐(𝑤,𝑟,𝑞) 𝑑𝑞

= 𝑀𝐶(𝑞)

● If revenue from the last unit > cost, produce more ○

𝑑𝑐(𝑤,𝑟,𝑞) 𝑑𝑞

> 0 to ensure MC is upward sloping at the optimal point

● Find the optimal quantity and profit Profit and supply Profit ● Profit is maximised where MC=MR ○ Profit = pQ* - c(Q*) = TR - TC = (p - AC(Q*))Q* ● Profit increases in p and decreases in w and r ● It is homogeneous of degree 1 in p, r and w Isoprofit curves ● An isoprofit curve is like a firm’s indifference curve ○ A firm is indifferent between inputs as long as they produce the desired profit - only profit changes between curves ● When one more input is used, output rises by 𝑀𝑃𝐿∆𝐿 ○ The value of this is 𝑝𝑀𝑃𝐿∆𝐿and the cost of this is 𝑤∆𝐿 ● When profits maximised, a change in labour used will decrease profits ● At π*(Q), the value of the MPL = w ● Profits = 𝑝𝑥 − 𝑤𝐿 − 𝑟𝐾(fixed price of capital) ○ 𝑥 =

π 𝑝

+

𝑟 𝑝

𝐾+

𝑤 𝑝

𝐿, which describes the isoprofit lines

● The slope of the isoprofit line is ● The vertical intercept is

π 𝑝

+

𝑟 𝑝

𝑤 𝑝

(constant so parallel curves)

𝐾

● Profit maximisation occurs at the intersection of the production function and the highest isoprofit curve - is tangential to the production function Supply ● Firms aim to maximise profits - P=MC=MR ● MC above AC is the long run supply curve ○ Q=0 for any level where AC>MC ● In the short run, S = MC above AVC ○ Only needs to cover variable costs ○ Lower shutdown point ● The supply curve is homogeneous of degree 0 ● LR supply is flatter than SR supply as capital is fixed

● AVC gets closer to ATC as output increases as AFC decreases...