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so for the Learning by doing model productivity is defined as $$A_t=BK_t$$.

Inserting this into the optimum conditions leads us to the following:$$r_t=\alpha BL^{1-\alpha}$$ and $$w_t=(1-\alpha)B^{1-\alpha} K_t/L$$.

As we can see, the interest rate here is constant and does not change with capital $K_t$. Intuitively, the reasoning is that more capital used in an individual firm reduces marginal product of capital. However, more capital of an individual firm level also leads to a higher aggregated capital stock (if all firms increase their stock) and hence also to higher productivity A, that can compensate for the decreasing marginal product of capital.

However, could someone explain to me why $w$ increases with $K_t$. What does this intuitively mean?

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You don’t explain what $w$ even is but I will assume it’s wage. The wage increases with the capital accumulation for several reasons. First, as the capital becomes more abundant the labor inputs become relatively more scarce and hence more valuable. Thus, also the price of labor relatively to capital should increase through higher wages.

Second, in most commonly used production function capital needs to be used with labor for example with standard Cobb-Douglass production function $AK^{\alpha}L^{1-\alpha}$ if you only use capital and no labor output is 0 no matter how much capital you employ. Hence using these kind of production function imply that when the capital stock increases, demand for labor increases as well putting more pressure on wages.

Another way how to think about it more intuitively is to think about labor productivity. It’s well established that wages vary positively with labor productivity and working with more capital makes people more productive hence increasing their wages.

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  • $\begingroup$ Hey, thank for your answer and the different explanations. I am sorry that I didn't mention what w was (I accidentally assumed that it's consistent through textbooks). Yes, it's wages. I do have some follow up questions. For the first reasoning, I don't understand why why abundance of capital should make labour more valuable (e.g. if labour demand stays constant). And in the second one, could you give me a function of labour dependent on capital? $\endgroup$ – randomname Apr 6 at 8:55
  • $\begingroup$ @randomname because price of all things is based on beside other things on relative scarcity. For example, assuming that both x and y are normal goods and there is no limit to demand if an economy produces 1 x and 2 y x must be twice as expensive as y because it’s more scarce, if economy starts producing 1x and 4 y x should be 4 times as expensive as y - the prices reflect beside preferences also relative scarcity of commodities. An example of such function is again Cobb Douglas function if K=0 output is also 0 so it also work vice versa more labor increases demand for capital - that’s why.. $\endgroup$ – 1muflon1 Apr 6 at 9:40
  • $\begingroup$ The second equation also shows r is increasing in L $\endgroup$ – 1muflon1 Apr 6 at 9:41

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