<p>Equation (4b) in the paper gives:
<span class="math-container">$$
\Delta mc_t = \frac{E_t N_t W_t}{Y_t G(.)}\Delta w_t + \frac{E_t K_t R_t}{Y_t G(.)} \Delta r_t - \Delta e_t. \tag{I}
$$</span>
Next we also have equation (5) in the paper:
<span class="math-container">$$
(1-B)P_t = MC_T = \frac{G(.)}{E_t}. \tag{II}
$$</span>
Now substitute <span class="math-container">$(II)$</span> into <span class="math-container">$(I)$</span> to get:
<span class="math-container">$$
\Delta mc_t = \frac{N_t W_t}{P_t Y_t (1-B)}\Delta w_t + \frac{K_t R_t}{P_t Y_t (1-B)} \Delta r_t - \Delta e_t. \tag{III}
$$</span>
Now, we define the wage cost over the total revenue to be <span class="math-container">$\alpha_t$</span>.
<span class="math-container">$$
\alpha_t = \frac{N_t W_t}{P_t Y_t}. \tag{IV}
$$</span>
Also, by the CRS assumption, we have that <span class="math-container">$\frac{N_t W_t}{C(.)} + \frac{R_t K_t}{C(.)} = 1$</span>. Dividing this by <span class="math-container">$P_t Y_t/C(.)$</span> and using <span class="math-container">$C(.) = G(.)Y_t/E_t$</span> as equation (2) in the paper, we get:
<span class="math-container">$$
\frac{N_t W_t}{P_t Y_t} + \frac{R_t K_t}{P_t Y_t} = \frac{C(.)}{P_t Y_t} = \frac{G(.)Y_t}{E_t P_t Y_t} = \frac{G(.)}{E_t P_t} = (1-B) \tag{V}
$$</span>
The last equality comes from comes from <span class="math-container">$(II)$</span>
Using <span class="math-container">$(IV)$</span> in <span class="math-container">$(V)$</span> gives:
<span class="math-container">$$
\frac{R_t K_t}{P_t Y_t} = (1-B) - \alpha_t. \tag{VI}
$$</span>
Now substitute <span class="math-container">$(IV)$</span> and <span class="math-container">$(VI)$</span> into <span class="math-container">$(III)$</span> to get:
<span class="math-container">$$
\Delta mc_t = \frac{\alpha_t}{(1-B)} \Delta w_t + \frac{1 - \alpha_t - B}{(1-B)}\Delta r_t - \Delta e_t. \tag{VII}
$$</span>
Also log differentiating the first part of <span class="math-container">$(II)$</span> gives <span class="math-container">$\Delta p_t = \Delta mc_t$</span> so substituting this in <span class="math-container">$(VII)$</span>  (and multiplying both sides by <span class="math-container">$(1-B)$</span> finally produces:
<span class="math-container">$$
\Delta p_t (1-B) = \alpha_t \Delta w_t + (1 - \alpha_t - B) \Delta r_t - \Delta e_t. \tag{VIII}
$$</span>
The expression that you are looking for is a simple rearrangement of <span class="math-container">$(VIII)$</span>.</p>

CC BY-SA 4.0
2022-05-16T16:13:50.140Z
2022-05-16T16:13:50.140Z
tdm
<p>I am trying to get a good understanding of the steps involved in solving the dual of a maximization problem, namely cost minimization. At some point (last two steps), the author ends up with the following function for marginal cost growth:
<span class="math-container">$$\Delta mc=\left [\frac{WN}{C(\cdot )}  \right ]\Delta w+\left [1-\frac{WN}{C(\cdot )}  \right ]\Delta r -\Delta e$$</span>
Where the relation between price and marginal cost is given by:
<span class="math-container">$$(1-B)P=MC=\frac{G(W,R)}{E}$$</span>
From this latter equation, the difference between the change in price and a weighted average of changes in factor prices, the dual or price-based Solow residual is defined as:
<span class="math-container">$$\alpha \Delta w +(1-\alpha)\Delta r-\Delta p=-B(\Delta p-\Delta r)+(1-B)\Delta e$$</span>
Where the weights for the factor prices are the wage share in output for wages and its complement for capital costs (<span class="math-container">$\alpha$</span>) and (<span class="math-container">$1-\alpha$</span>).</p>
<p>I can't figure out how you get to this last step. I tried differentiating the marginal cost equation and substituting in the second one, but I'm missing some terms. The problem comes from the following paper: &quot;Can Imperfect Competition Explain the Difference between Primal and Dual Productivity Measures? Estimates for U.S. Manufacturing&quot; (1995).</p>
<p>*<span class="math-container">$\Delta x$</span> is the log difference of the variable X.</p>

<macroeconomics><microeconomics><mathematical-economics><economic-growth>
Solow residual with cost minimization, calculus (Roeger, 1995)
Alessandro
# Solow residual with cost minimization, calculus (Roeger, 1995)

(Asked by: username_0 on 2022-05-07)

I am trying to get a good understanding of the steps involved in solving the dual of a maximization problem, namely cost minimization. At some point (last two steps), the author ends up with the following function for marginal cost growth:
$$\Delta mc=\left [\frac{WN}{C(\cdot )} \right ]\Delta w+\left [1-\frac{WN}{C(\cdot )} \right ]\Delta r -\Delta e$$
Where the relation between price and marginal cost is given by:
$$(1-B)P=MC=\frac{G(W,R)}{E}$$
From this latter equation, the difference between the change in price and a weighted average of changes in factor prices, the dual or price-based Solow residual is defined as:
$$\alpha \Delta w +(1-\alpha)\Delta r-\Delta p=-B(\Delta p-\Delta r)+(1-B)\Delta e$$
Where the weights for the factor prices are the wage share in output for wages and its complement for capital costs ($\alpha$) and ($1-\alpha$).

I can't figure out how you get to this last step. I tried differentiating the marginal cost equation and substituting in the second one, but I'm missing some terms. The problem comes from the following paper: "Can Imperfect Competition Explain the Difference between Primal and Dual Productivity Measures? Estimates for U.S. Manufacturing" (1995).

\*$\Delta x$ is the log difference of the variable X.

**Tags:** macroeconomics, microeconomics, mathematical-economics, economic-growth

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## 1 answers

### Answer by username_1 on 2022-05-16. Score: 2

Equation (4b) in the paper gives:
$$
\Delta mc\_t = \frac{E\_t N\_t W\_t}{Y\_t G(.)}\Delta w\_t + \frac{E\_t K\_t R\_t}{Y\_t G(.)} \Delta r\_t - \Delta e\_t. \tag{I}
$$
Next we also have equation (5) in the paper:
$$
(1-B)P\_t = MC\_T = \frac{G(.)}{E\_t}. \tag{II}
$$
Now substitute $(II)$ into $(I)$ to get:
$$
\Delta mc\_t = \frac{N\_t W\_t}{P\_t Y\_t (1-B)}\Delta w\_t + \frac{K\_t R\_t}{P\_t Y\_t (1-B)} \Delta r\_t - \Delta e\_t. \tag{III}
$$
Now, we define the wage cost over the total revenue to be $\alpha\_t$.
$$
\alpha\_t = \frac{N\_t W\_t}{P\_t Y\_t}. \tag{IV}
$$
Also, by the CRS assumption, we have that $\frac{N\_t W\_t}{C(.)} + \frac{R\_t K\_t}{C(.)} = 1$. Dividing this by $P\_t Y\_t/C(.)$ and using $C(.) = G(.)Y\_t/E\_t$ as equation (2) in the paper, we get:
$$
\frac{N\_t W\_t}{P\_t Y\_t} + \frac{R\_t K\_t}{P\_t Y\_t} = \frac{C(.)}{P\_t Y\_t} = \frac{G(.)Y\_t}{E\_t P\_t Y\_t} = \frac{G(.)}{E\_t P\_t} = (1-B) \tag{V}
$$
The last equality comes from comes from $(II)$
Using $(IV)$ in $(V)$ gives:
$$
\frac{R\_t K\_t}{P\_t Y\_t} = (1-B) - \alpha\_t. \tag{VI}
$$
Now substitute $(IV)$ and $(VI)$ into $(III)$ to get:
$$
\Delta mc\_t = \frac{\alpha\_t}{(1-B)} \Delta w\_t + \frac{1 - \alpha\_t - B}{(1-B)}\Delta r\_t - \Delta e\_t. \tag{VII}
$$
Also log differentiating the first part of $(II)$ gives $\Delta p\_t = \Delta mc\_t$ so substituting this in $(VII)$ (and multiplying both sides by $(1-B)$ finally produces:
$$
\Delta p\_t (1-B) = \alpha\_t \Delta w\_t + (1 - \alpha\_t - B) \Delta r\_t - \Delta e\_t. \tag{VIII}
$$
The expression that you are looking for is a simple rearrangement of $(VIII)$.


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