Derive the cost function for a Homothetic production function - Economics Stack Exchange most recent 30 from economics.stackexchange.com 2022-01-23T00:47:33Z https://economics.stackexchange.com/feeds/question/26158 https://creativecommons.org/licenses/by-sa/4.0/rdf https://economics.stackexchange.com/q/26158 1 Derive the cost function for a Homothetic production function Robin Liao https://economics.stackexchange.com/users/20326 2018-12-27T22:48:52Z 2019-01-02T10:53:00Z <p>I'm having trouble understanding the steps in showing that a Homothetic production function's cost function must be expressible in the form <span class="math-container">$C(w, q) = a(w)b(q)$</span>.</p> <p>Since the production function is homothetic, I know that the optimal cost minimizing input ratio given the same input costs must be exactly the same for different quantities of output i.e. </p> <p><span class="math-container">$$\frac{H^j(w,q)}{H^i(w,q)} = \frac{H^j(w,q')}{H^i(w,q')}$$</span></p> <p>Rearranging this we arrive at the ratio:</p> <p><span class="math-container">$$\frac{H^i(w,q)}{H^i(w,q')} = \frac{H^j(w,q)}{H^j(w,q')}$$</span></p> <p>This makes sense because in order to the maintain the input ratio constant, we would need both cost-minimizing inputs to increase by the same factor when going from <span class="math-container">$q$</span> to <span class="math-container">$q'$</span>. However, the proof then states that "for the above to true <strong>it is clear that the ratio must be independent of w</strong>, thus, setting <span class="math-container">$q' = 1$</span>"</p> <p><span class="math-container">$$\frac{H^1(w,q)}{H^1(w,1)}= \frac{H^2(w,q)}{H^2(w,1)} = \cdots =\frac{H^m(w,q)}{H^m(w,1)} = b(q)$$</span></p> <p>and so</p> <p><span class="math-container">$$H^i(w,q) = b(q)H^i(w, 1)$$</span></p> <p>The steps after this to get to <span class="math-container">$C(w, q) = a(w)b(q)$</span> is pretty understandable and straightforward for me.</p> <p>However, what I don't understand is how you can clearly see that the ratio must be independent of <strong>w</strong>. Surely the ratio just states that for a given <strong>w</strong> the inputs must increase by the same factor/ratio. But how does that imply that the ratio has to be exactly the same for all <strong>w</strong>? </p> <p><strong>Edit</strong>: for those who are asking, this comes from an exercise question in Frank Cowell's textbook: Microeconomics: Principles and Analysis.</p> https://economics.stackexchange.com/questions/26158/-/26187#26187 0 Answer by Bertrand for Derive the cost function for a Homothetic production function Bertrand https://economics.stackexchange.com/users/20576 2018-12-29T12:10:25Z 2018-12-30T15:40:17Z <p>There must be an error in your proof. Where does your proof come from? The part "for the above to true it is clear that the ratio must be independent of <span class="math-container">$w$</span>," cannot be True. Your ratio must be independent of <span class="math-container">$q$</span> but not of <span class="math-container">$w$</span>. Take for instance the Cobb-Douglas case (which corresponds to a homothetic production function) for a counter example. See Diewert or Chambers for a proof of your result:<br> Chambers, Robert G., 1988, Applied Production Analysis, Cambridge University Press.<br> Diewert, E., 1982, "Duality approaches to microeconomic theory", in Handbook of Mathematical Economics, Volume 2.<br> In the Cobb Douglas case, the production function <span class="math-container">$$q=x_1^\alpha x_2^\beta$$</span> yields the first order conditions <span class="math-container">$$\frac{w_1}{\lambda}=\alpha\frac{q}{x_1}$$</span> <span class="math-container">$$\frac{w_2}{\lambda}=\beta\frac{q}{x_2},$$</span> so that optimal input demand satisfy (in your notations) <span class="math-container">$$\frac{x_1}{x_2}=\frac{\alpha}{\beta}\frac{w_2}{w_1} \equiv \frac{c_1(w)}{c_2(w)},$$</span> which is independent of <span class="math-container">$q$</span> and implies (show it) that <span class="math-container">$$x_1=H^1(w,q)=c_1(w)b(q),$$</span> <span class="math-container">$$x_2=H^2(w,q)=c_2(w)b(q).$$</span> It follows that the cost function <span class="math-container">$$c(w,q)=w_1H^1(w,q)+w_2H^2(w,q)=a(w)b(q),$$</span> with <span class="math-container">$a(w)=...$</span>. A similar derivation holds for the more general homothetic case, because by definition of homotheticity, the production technology can then be written as <span class="math-container">$$y=f(x)=g(h(x))$$</span> where h is homogeneous of degree one...</p> https://economics.stackexchange.com/questions/26158/-/26239#26239 0 Answer by Robin Liao for Derive the cost function for a Homothetic production function Robin Liao https://economics.stackexchange.com/users/20326 2019-01-02T10:53:00Z 2019-01-02T10:53:00Z <p>The fact the ratio is independent of <strong>w</strong> comes from one of the properties of homothetic functions. A homothetic function by definition is a monotonic transformation of a homogenous function. Thus, for any homothetic function, a known result is that <span class="math-container">$Φ(z_1) = Φ(z_2)$</span> implies that <span class="math-container">$Φ(tz_1) = Φ(tz_2)$</span> for any input combination <span class="math-container">$z_1$</span> and <span class="math-container">$z_2$</span>. Thus, for any two input combinations/ratios along the same isoquant (i.e. <span class="math-container">$Φ(z_1) = Φ(z_2)$</span>) and <strong>therefore any price ratio</strong> (since for a homothetic function price ratio = MRTS is solely determined by input ratio) multiplying by the exact same factor <span class="math-container">$t$</span> gets you from the isoquant where <span class="math-container">$Φ(z_1) = Φ(z_2)$</span> to <span class="math-container">$Φ(tz_1) = Φ(tz_2)$</span>.</p> <p>As a result, the ratio above is independent of the price <strong>w</strong> (i.e. which point along the isoquant you are) because no matter where it is, you always have to multiply by the same t to get from q to q', which in my case is the ratio </p> <p><span class="math-container">$$t = \frac{H^i(w,q)}{H^i(w,q')} = \frac{H^j(w,q)}{H^j(w,q')}$$</span></p>