A firm’s production function is y = 2x1 + x2 ?
What would the firm’s conditional input demand functions be (given a fixed y)?
Is this a problem of interest? In other words: are there any detailed examples of a firm facing such a minimization problem? Downsizing end last 19, luxury goods
How would one explain intuitively what should happen in terms of conditional input demands for a rise in w1 (w2 being constant).
Is this intuition confirmed if you do the comparative statics with the firm’s conditional input demand you just derived.
How would an output expansion plan and both the conditional demand for input 1 and 2, be drawn.
Is the firm having constant return to scale? What happens to its costs and average costs if it wants to triple its output?
Yours,
James
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Assuming long-run:
A - resource A
B - resource B
w - price of resource A
v - price of resource B
MP(A) - marginal product of A
MP(B) - marginal product of B
Y = 2A+B
B = Y-2A
A = (Y-B)/2
TC = wA + vB
Optimum: MP(A)/w = MP(B)/v
MP(A) = MPA = ∂Y/∂A = ∂(2A+B)/∂A = 2
MP(B) = MPB = ∂Y/∂B = ∂(2A+B)/∂A = 1
2/w = 1/v
2v=w
v=w/2
Thus firm will use only one resource depending on it's relative prices.
In case if v>w/2 then only resource "A" will be used
In case if w>2v then only resource "B" will be used
And one case if 2v=w or v=w/2, in this case producer is indifferent which one to use (thus it can combine it in any proportions).
TC = wA + vB = w(Y-B)/2 + v(Y-2A) = (wY/2 - wB/2) + (vY - 2vA)
Thus depending on resource price condition only A or B resource will be used:
B→ TC[w>2v] = vY
B→ MC[w>2v] = v
A→ TC[v>w/2] = wY/2
A→ MC[v>w/2] = w/2
(Assuming perfect competition on product/output market)
MR=P
Thus resource demand satisfying MR=MC condition will be infinite (though depending on resource and/or output markets competition levels)
Demand between resources will switch discrete way at certain point, depending on resource price condition, over all other ranges demand will be unaffected.
Firm does have constant returns to scale because Y=2A+B fits constant return-required condition.
Y=2A+B
zY = 2Az + Bz = z(2A+B)
Dividing both parts by "z" we get back original function:
Y=2A+B
This is having similarities to my problem set 2, for intermediate economics 1 class, at Heriot Watt University.
I hope my students aren't resorting to these measures to solve my problem set, it is a rather sexy one.
Yours Philippe