How about the more interesting CES production function AK, LP= YaK+ lapa Again, as long as this is CRS, we can only do cost minimization not profit maximization This gives us r= Vaka? Yaka+ lApa? W= VbLa? Yaka+bLap K=Li ar from which L b W)12a ?1 R +bio K=(a+b需(最)市)
How about the more interesting CES production function fÝK,LÞ = ÝaKa + bLa Þ 1 a Again, as long as this is CRS, we can only do cost minimization not profit maximization. This gives us R = VaKa?1 ÝaKa + bLa Þ 1 a ?1 W = VbL a?1 ÝaKa + bLa Þ 1 a ?1 or K = L aW bR 1 1?a , from which L = a 1 1?a b ?a 1?a W R a 1?a + b ? 1 a Q K = a + b 1 1?a a ?a 1?a R W a 1?a ? 1 a Q
Holding quantity constant-the elasticity of labor with respect to wages equals L=(ai7ab17a()17a+bag 21+a and >aa 17a b 17aR1aW17aalab17a(w)17a+60 W "靠 1?a a7ablaR1aW 7a +1 This is increasing with a, decreasing with b, increasing with W and decreasing with r If the terms in the denominator don ' t move too much with a, then as a rises, the elasticity rises- this is Marshalls substitutability point
Holding quantity constant– the elasticity of labor with respect to wages equals: L = a 1 1?a b ?a 1?a W R a 1?a + b ? 1 a Q /L /W = 1 1?a a 1 1?a b ?a 1?a R ?a 1?a W 2a?1 1?a a 1 1?a b ?a 1?a W R a 1?a + b ? 1+a a Q and W L /L /W = 1 1 ? a 1 a ?1 1?a b 1 1?a R a 1?a W ?a 1?a + 1 This is increasing with a, decreasing with b, increasing with W and decreasing with R. If the terms in the denominator don’t move too much with a, then as a rises, the elasticity rises– this is Marshall’s substitutability point
To get law 3: we need to allow r to be a function of K, and then we get =米?袁斧p Where frwY a 1?ab 1?a(w)1?a 1?a.R R a古b需(")+b R This a little artificial, since we are holding overall output constant But nonetheless. from this it should be clear that as 公 gets bigger, the value of /gets smaller That is marshall's third law
To get law # 3: we need to allow R to be a function of K, and then we get /L /W = f W R 1 W ? 1 R /R /K /K /W Q where f W R = a 1 1?a b ?a 1?a 1?a W R a 1?a a 1 1?a b ?a 1?a W R a 1?a + b 1+a a This a little artificial, since we are holding overall output constant. But nonetheless, from this it should be clear that as /R /K gets bigger, the value of /L /W gets smaller. That is Marshall’s third law
To get law number 4, just go to a single input case maxL PALp? WL, which yields PfYLP=WL Now differentate this totally with respect to w, allowing P to change as well PflP L+w. /W Using the fact that do Lp ZD, we get ALP W PAL YLP+YLP?2 dp Higher demand elasticities make the denominator smaller. because O p do is one divided by the demand elasticity What does this mean?
To get law number 4, just go to a single input case: maxL PfÝLÞ ? WL, which yields Pfv ÝLÞ = WL Now differentate this totally with respect to W, allowing P to change as well dP dQ dQ dW + Pfvv ÝLÞ /L /W = L + W /L /W Using the fact that dQ dW = f v ÝLÞ /L /W , we get /L /W = fÝLÞ ? PfÝLÞ W f vv ÝLÞ + fÝLÞ ? Q P dP dQ Higher demand elasticities make the denominator smaller, because Q P dP dQ is one divided by the demand elasticity. What does this mean?
Dynamic Supply Issues The first thing that is useful to keep in mind is that everything is essentially dynamic, and that everything that we are doing is a static approximation to that Perhaps the most extreme version of this is when we try to say something about long run/short run distinctions using the basic model The essence of the Le Chatelier/Samuelson principle is that the long run response to a price change is larger than the short run response to the same change Long run/short run distinction is handled by just assuming that some inputs are fixed in the short run Typically in the k, L formulation -this means assuming that capital is fixed and labor is flexible
Dynamic Supply Issues: The first thing that is useful to keep in mind is that everything is essentially dynamic, and that everything that we are doing is a static approximation to that. Perhaps the most extreme version of this is when we try to say something about long run/short run distinctions using the basic model. The essence of the LeChatelier/Samuelson principle is that the long run response to a price change is larger than the short run response to the same change. Long run/short run distinction is handled by just assuming that some inputs are fixed in the short run. Typically in the K, L formulation– this means assuming that capital is fixed and labor is flexible