# Positive productivity shock with a fall in output

I am modeling a two-country world. Within each country there exists a representative infinitely-lived agent, a representative final good producer, a continuum of domestic monopolistic competitive firms (owned by consumers), and a continuum of local retailers that import goods. The representative agent supplies labor and capital to the production process. Domestic firms use labor and capital as inputs to produce intermediate goods; local retailers import differentiated goods. Domestic producers sell to the home and foreign final good producers. Each representative final good producer is a competitive firm that bundles home and foreign intermediate goods into a non-tradable final good, which is consumed and used for investment by the domestic agent. Preferences and technologies are symmetric across both countries.

$$\begin{gather} Z_{H,t} + Z^*_{H,t} = Y_t = A_t K_t^\alpha L^{1-\alpha}_t \end{gather}$$

so for the Foreign country, the production process is given by:

$$\begin{gather} Z^*_{F,t} + Z_{F,t} = Y^*_t = A^*_t K^{* \alpha}_t L^{* 1-\alpha}_t \end{gather}$$

where $$Z^*_{F,}t$$ is the intermediate good to produce final good $$Z^*_t$$ for Foreign country and $$Z_{F,t}$$ is the intermediate good that will be exported to the Home country.

When shocking the economy with a positive temporary productivity shock, I find that overall $$Y^*_t$$ decreases, while $$Z^*_{F,t}$$ decreases but $$Z_{F,t}$$ increases. However, $$L^*_t$$ increases, but $$K^*_t$$ decreases.

I understand that $$Z_{F,t}$$ increases as a response to the positive shock by the Home country, because there is an increase in the demand of goods. My question is, what would be the economic intuition in knowing that $$Y^*_t$$ decreases, but $$L^*_t$$ increases and $$K^*_t$$ decreases? I was thinking that a possible answer would be that, this occurs given that the Foreign country needs to produce more $$Z_{F,t}$$ to export to the Home country, and given that the share of labor needed in the production process is greater than the share of capital, in order to achieve the higher production of $$Z_{F,t}$$, only labor is needed as it is "relatively" more important than capital in the production process (I am assuming that capital share $$\alpha = 1/3$$).

Thank you,

Alejandro