Macroeconomic Model

**1**. Labor’s share of total income is defined as S_{N} = WN/PY. Hint: In each part of this question, feel free to assume A = K_{0} = 1. In each part of this question, it is satisfactory to simply write the result but far better if you show the derivation; this better allows for partial credit.

i. Using our (Blanchard) macroeconomic model, solve for the labor share in terms of the exogenous price markup factor, “m. Based on your answer, is the labor share procyclical, countercyclical, or neither? (10 points)

ii. Instead of being exogenous, suppose that the price markup factor, m, is given by:

m = m_{0} – m_{1}Y, where m_{0} and m_{1} are positive constants. An economic justification for this specification is that there is entry into product markets during good times which drives down the markup of prices over costs. Assume that our macro model is otherwise the same as always. Derive the real wage, W/P, given the alternative price markup factor. Is the real wage, W/P, pro-cyclical, countercyclical, or neither? (5 points)

iii. Derive the labor share of income, S_{N}, using the modified expression for the markup factor given in part ii. Is the labor share just derived procyclical, countercyclical, or neither? (10 points)

**2**. Suppose that the value of your stock portfolio exogenously (and unexpectedly) declines by $1000 today (from what it otherwise would have been) and suppose you expect to live for 20 years. To simplify assume that the interest rate equals the rate of time preference. Fully explain your answers to each part.

i, Using the modern models of consumption, determine how much current consumption should decline. (10 points)

ii. How would your answer to part i change if you knew that the value of your stocks would exogenously (and, unexpectedly) decline by $1000 (from what it otherwise would have been) in each of the 20 years of your expected life? (10 points)

iii. Suppose that the exogenous decline in the stock market leads to an economy-wide decline in household wealth. In what direction do the IS and AD curves shift? (5 points)

**3** Replace our standard production function of Y = N with Y = N^{0.7} (where A = K_{0} = 1). Assume that all other assumptions about wage and price setting remain the same as usual in our standard classroom model. In words (not math), use economic logic underlying price setting to explain why–with the new production function–the real wage rises with the unemployment rate. (5 points)

### Sample Solution

#### 1.

i. To solve for the labor share in terms of the exogenous price markup factor, we start with the equation SN = WN/PY. Using the Blanchard macroeconomic model, we have Y = AN, where A represents the level of technology and N represents the level of employment. We also have PY = (1+m)WN, where m is the exogenous price markup factor. Substituting these into the labor share equation, we get:

SN = WN/[(1+m)AN]

Simplifying this expression, we get:

SN = (1/(1+m))(W/A)

Therefore, the labor share is inversely related to the price markup factor m. Since m is procyclical (it rises during booms and falls during recessions), the labor share is countercyclical (it falls during booms and rises during recessions).

ii. The real wage, W/P, is given by W/P = (1+m)/(1+α) where α is the elasticity of labor supply. Since the price markup factor, m, is now given by m = m0 – m1Y, we have:

W/P = [1 + m0 – m1Y]/(1 + α)

The real wage is thus procyclical since it rises during recessions and falls during booms.

iii. Substituting the new expression for m into the labor share equation from part i, we get:

SN = (1/(1+m0 – m1Y))(W/A)

This expression is countercyclical since the labor share rises during recessions and falls during booms.

#### 2.

i. Using the modern models of consumption, the amount that current consumption should decline by can be found using the permanent income hypothesis. According to this hypothesis, individuals base their consumption on their expected lifetime income rather than their current income. Thus, if the value of the stock portfolio declines by $1000, the expected lifetime income will decline by some fraction of that amount. Assuming a constant interest rate and time preference rate, the decline in current consumption is given by the present value of the expected future income loss, which is:

($1000/20)/(1+r) = $50/(1+r)

where r is the interest rate (equal to the rate of time preference).

ii. If the value of the stock portfolio were to decline by $1000 each year for 20 years, the expected lifetime income would decline permanently by $1000. According to the permanent income hypothesis, the decline in current consumption is again given by the present value of the expected future income loss, which is:

$1000/(1+r)

This is a larger decline in consumption than in part i, where the loss was a one-time event.

iii. If the exogenous decline in the stock market leads to an economy-wide decline in household wealth, this would shift the aggregate demand (AD) curve to the left since consumers have less purchasing power. The investment savings (IS) curve would also shift to the left since households would have less wealth to invest.

#### 3.

With the new production function, the real wage rises with the unemployment rate because firms are willing to pay higher wages to attract workers due to the diminishing returns to labor. As unemployment rises, the marginal product of labor increases, which in turn leads to an increase in the real wage. This is because the production function Y = N^0.7 implies that the marginal product of labor (MPN) is proportional to N^(-0.3). As N decreases (i.e. as unemployment rises), MPN increases, which leads to a higher real wage.